3520:
2842:
16484:
11819:
5160:
11325:
11308:
16731:
16751:
16741:
4273:
15128:
11814:{\displaystyle {\begin{aligned}f=\Phi _{Y,X}^{-1}(g)&=\varepsilon _{X}\circ F(g)&\in &\,\,\mathrm {hom} _{C}(F(Y),X)\\g=\Phi _{Y,X}(f)&=G(f)\circ \eta _{Y}&\in &\,\,\mathrm {hom} _{D}(Y,G(X))\\\Phi _{GX,X}^{-1}(1_{GX})&=\varepsilon _{X}&\in &\,\,\mathrm {hom} _{C}(FG(X),X)\\\Phi _{Y,FY}(1_{FY})&=\eta _{Y}&\in &\,\,\mathrm {hom} _{D}(Y,GF(Y))\\\end{aligned}}}
7356:, by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of
13292:
13074:
6454:. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor on the limit, in the functor category). Below are some specific examples.
14729:
2303:
The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be
1842:
Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the
12614:
A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)
4161:
This definition is a logical compromise in that it is more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counitβunit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.
3841:
These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.
14929:
11966:
5032:
13990:
12858:
4494:
4871:
6786:. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples.
13120:
12902:
1190:
15381:
5819:
is fully determined by its action on generators, another restatement of the universal property of free groups. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair
6300:
6029:
9374:
6757:
A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left
404:
13520:
8558:, quantifiers are identified with adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion of propositional logic using set theory but the general definition make for a richer range of logics.
2288:
The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving
15473:
However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of
14610:
4645:
4578:
597:
524:
10082:
4114:
10294:
15123:{\displaystyle {\begin{aligned}&1_{\mathcal {E}}{\xrightarrow {\eta '}}G'F'{\xrightarrow {G'\eta F'}}G'GFF'\\&FF'G'G{\xrightarrow {F\varepsilon 'G}}FG{\xrightarrow {\varepsilon }}1_{\mathcal {C}}.\end{aligned}}}
13713:
10178:
8390:
6744:, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group
3996:
9447:
11832:
4898:
2974:
773:
707:
8676:
3652:
14228:
14127:
8200:
210:
170:
1816:
13852:
1045:, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.
10959:
14934:
14615:
13857:
13793:
13600:
13125:
12907:
12733:
11837:
11330:
4903:
4788:
4413:
1335:
15691:
12728:
9568:
10868:
4408:
3519:
6197:
5927:
4783:
13394:
5876:
4735:
2387:). If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.
14858:
1667:
1610:
10914:
6581:
etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. More generally, any type of limit is right adjoint to a diagonal functor.
15746:
10987:
2659:
58:. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain
5771:
5660:
5209:
2829:
8061:
10808:
14910:
13287:{\displaystyle {\begin{aligned}\Phi \Psi g&=G(\varepsilon _{X})\circ GF(g)\circ \eta _{Y}\\&=G(\varepsilon _{X})\circ \eta _{GX}\circ g\\&=1_{GX}\circ g=g\end{aligned}}}
13069:{\displaystyle {\begin{aligned}\Psi \Phi f&=\varepsilon _{X}\circ FG(f)\circ F(\eta _{Y})\\&=f\circ \varepsilon _{FY}\circ F(\eta _{Y})\\&=f\circ 1_{FY}=f\end{aligned}}}
3344:
5710:
3514:
9095:
9046:
8211:
in predicate logics is in forming propositions and also in expressing sophisticated predicates by closing formulas with possibly more variables. For example, consider a predicate
6962:-bilinear product with (r,0)(0,1) = (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng.
6386:
5578:
5513:
15792:
9719:
9260:
6130:
2299:
The definition via counitβunit adjunction is convenient for proofs about functors which are known to be adjoint, because they provide formulas that can be directly manipulated.
1083:
5065:
15271:
10727:
10674:
10640:
8899:
8868:
5089:
3689:
3011:
1043:
1019:
991:
967:
817:
641:
301:
257:
123:
99:
10328:
8998:
8783:
11031:
11009:
10696:
10523:
9899:
9828:
9754:
9489:
9289:
8837:
8810:
6076:
9639:
7269:
be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a β A and morphisms preserve the distinguished elements). The forgetful functor G:
3466:
3425:
2781:
2740:
8925:
8087:
6342:
5318:
1710:
10771:
10567:
9674:
8508:
8481:
8236:
8007:
4761:
1358:
940:
14467:
10474:
10454:
8613:
3577:
3110:
2899:
2425:
1213:
12668:
11051:
7004:
14752:
10496:
The twin fact in probability can be understood as an adjunction: that expectation commutes with affine transform, and that the expectation is in some sense the best
9925:
9594:
8750:
8548:
6210:
5939:
5150:
1847:(which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.
14772:
10394:
9781:
9296:
8528:
5116:
3267:
2582:
313:
10587:
10434:
10414:
10368:
10348:
9954:
9848:
9801:
9215:
9195:
9175:
9155:
9135:
9115:
9018:
8965:
8945:
8724:
8704:
8581:
8454:
8434:
8414:
8296:
8276:
8256:
8127:
8107:
3836:
3816:
3796:
3776:
3753:
3729:
3709:
3545:
3384:
3364:
3287:
3238:
3218:
3198:
3178:
3154:
3134:
3075:
3051:
3031:
2867:
2699:
2679:
2602:
2553:
2533:
2513:
2493:
2469:
2449:
1760:
1740:
1630:
1573:
1553:
1533:
1510:
1490:
1470:
1446:
1426:
1402:
1382:
1280:
1259:
1239:
913:
887:
866:
840:
793:
617:
451:
431:
277:
233:
46:
may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as
11061:
There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.
6524:. The universal property of the product group shows that Ξ is right-adjoint to Ξ. The counit of this adjunction is the defining pair of projection maps from
2328:
which are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category
14724:{\displaystyle {\begin{aligned}\eta '&=(\tau \ast \sigma )\circ \eta \\\varepsilon '&=\varepsilon \circ (\sigma ^{-1}\ast \tau ^{-1}).\end{aligned}}}
13407:
5368:. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a
5254:, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as
1818:, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.
4584:
4517:
529:
456:
9961:
7657:
that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous
4008:
10188:
7010:
and 1 is an identity element. This construction gives a functor that is a left adjoint to the functor taking a monoid to the underlying semigroup.
15828:
Every monad arises from some adjunctionβin fact, typically from many adjunctionsβin the above fashion. Two constructions, called the category of
13608:
10091:
8303:
3903:
9385:
11961:{\displaystyle {\begin{aligned}1_{FY}&=\varepsilon _{FY}\circ F(\eta _{Y})\\1_{GX}&=G(\varepsilon _{X})\circ \eta _{GX}\end{aligned}}}
5222:
Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an
5027:{\displaystyle {\begin{aligned}1_{FY}&=\varepsilon _{FY}\circ F(\eta _{Y})\\1_{GX}&=G(\varepsilon _{X})\circ \eta _{GX}\end{aligned}}}
2841:
1945:
is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.
7197:
to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the
6857:, again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pair of inclusion maps from
712:
646:
8620:
10525:, with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". That is, for any affine function
15875:
1879:
way is to adjoin an element '1' to the rng, adjoin all (and only) the elements which are necessary for satisfying the ring axioms (e.g.
2904:
16128:
9783:
is given by the direct image. Here is a characterization of this result, which matches more the logical interpretation: The image of
16027:
12029:
In particular, the equations above allow one to define Ξ¦, Ξ΅, and Ξ· in terms of any one of the three. However, the adjoint functors
9048:, the category of sets and functions, the canonical subobjects are the subset (or rather their canonical injections). The pullback
1834:
5834:
One can also verify directly that Ξ΅ and Ξ· are natural. Then, a direct verification that they form a counitβunit adjunction
16413:
15506:
then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.
14165:
14064:
13985:{\displaystyle {\begin{aligned}\Phi _{Y,X}(f)=G(f)\circ \eta _{Y}\\\Phi _{Y,X}^{-1}(g)=\varepsilon _{X}\circ F(g)\end{aligned}}}
8132:
5167:
These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the
175:
135:
1765:
7678:
can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism from
3582:
1887:
in the ring), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is
12037:
alone are in general not sufficient to determine the adjunction. The equivalence of these situations is demonstrated below.
15183:
Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:
14245:
12853:{\displaystyle {\begin{aligned}\Phi _{Y,X}(f)=G(f)\circ \eta _{Y}\\\Psi _{Y,X}(g)=\varepsilon _{X}\circ F(g)\end{aligned}}}
10919:
13726:
13533:
4489:{\displaystyle {\begin{aligned}\varepsilon &:FG\to 1_{\mathcal {C}}\\\eta &:1_{\mathcal {D}}\to GF\end{aligned}}}
1285:
15656:
14283:
9494:
8009:
is a unary predicate expressing some property, then a sufficiently strong set theory may prove the existence of the set
4866:{\displaystyle {\begin{aligned}1_{F}&=\varepsilon F\circ F\eta \\1_{G}&=G\varepsilon \circ \eta G\end{aligned}}}
1843:
uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve
16787:
10813:
1894:
This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is
7549:
6042:. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow
67:
16040:
16004:
15971:
6157:
5887:
13355:
12057:; in the sense of initial morphisms, one may construct the induced hom-set adjunction by doing the following steps.
6144:
it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on
5837:
4696:
12234:
The commuting diagram of that factorization implies the commuting diagram of natural transformations, so Ξ· : 1
1635:
1578:
15156:
The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore
10880:
15710:
10964:
2607:
16838:
7553:
5731:
5620:
5182:
2786:
14820:
8012:
7728:
The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:
16121:
16070:
16032:
10776:
3292:
16325:
16280:
14434:
5673:
3471:
15836:
are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.
9051:
9031:
1185:{\displaystyle \varphi _{XY}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
16754:
16694:
15829:
6355:
5352:. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the
16403:
15376:{\displaystyle \Phi _{Y,X}:\mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
5538:
5473:
1910:. Universal properties come in two types: initial properties and terminal properties. Since these are
16853:
16744:
16530:
16394:
16302:
15757:
14869:
9679:
9220:
7303:
6439:
6105:
6427:
have generally the same description as in the detailed description of the free group situation above.
5041:
16894:
16703:
16347:
16285:
16208:
15503:
15188:
10701:
10648:
10592:
8873:
8842:
7950:
7890:
7740:
7658:
6771:
6435:
6404:
it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on
994:
20:
5070:
1024:
1000:
972:
948:
798:
622:
282:
238:
104:
80:
16734:
16690:
16295:
16114:
10303:
8970:
8755:
2292:
The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word
2202:
yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor
1917:
The idea of using an initial property is to set up the problem in terms of some auxiliary category
11014:
10992:
10679:
10506:
9853:
9806:
9724:
9459:
9268:
8815:
8788:
7411:
that associates to every topological space its underlying set (forgetting the topology, that is).
6045:
16290:
16272:
15639:
15177:
15165:
9599:
7736:
7732:
adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
7713:
As is the case for Galois groups, the real interest lies often in refining a correspondence to a
6783:
6451:
6443:
5235:
3430:
3389:
2745:
2704:
2050:
is a ring map (which preserves the identity). (Note that this is precisely the definition of the
1844:
10183:
The right adjoint to the inverse image functor is given (without doing the computation here) by
8904:
8066:
6317:
5297:
1686:
16780:
16497:
16263:
16243:
16166:
16096:
15230:
14531:
12251:
11147:
10732:
10528:
9644:
8486:
8459:
8214:
7985:
7857:
7840:
multiplication, but in situations where this is not possible, we often attempt to construct an
7833:
7383:
6295:{\displaystyle GX{\xrightarrow {\;\eta _{GX}\;}}GFGX{\xrightarrow {\;G(\varepsilon _{X})\,}}GX}
6024:{\displaystyle FY{\xrightarrow {\;F(\eta _{Y})\;}}FGFY{\xrightarrow {\;\varepsilon _{FY}\,}}FY}
4740:
4400:
3657:
2979:
2371:, and whenever possible such things will be referred to in order from left to right (a functor
1911:
1340:
919:
410:
15921:
14440:
10459:
10439:
9369:{\displaystyle {\operatorname {Hom} }(\exists _{f}S,T)\cong {\operatorname {Hom} }(S,f^{*}T),}
8586:
3550:
3083:
2872:
2398:
1198:
16868:
16379:
16218:
12653:
11036:
10500:
to the problem of finding a real-valued approximation to a distribution on the real numbers.
7714:
7675:
7624:
7461:
7345:
7100:
6989:
6909:
5337:
15596:
In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of
14737:
13296:
hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ = Ψ.
9904:
9573:
8729:
8533:
7364:. For the case of finitary algebraic structures, the existence by itself can be referred to
5515: be the set map given by "inclusion of generators". This is an initial morphism from
5125:
399:{\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)\cong \mathrm {hom} _{\mathcal {D}}(Y,GX)}
16818:
16191:
16186:
16100:
16064:
14757:
10373:
9759:
8513:
7923:
which assigns to a category its set of connected components is left-adjoint to the functor
7853:
7606:
7379:
7190:
6917:
6822:
5670:
they correspond to, which exists by the universal property of free groups. Then each
5348:; those two groups are not really identical but there is a way of identifying them that is
5094:
16050:
16014:
15168:
in the category theoretical sense); every functor that has a right adjoint (and therefore
12673:, we can construct a hom-set adjunction by finding the natural transformation Ξ¦ : hom
5589:
3523:
The existence of the unit, a universal morphism, can prove the existence of an adjunction.
3243:
2558:
1069:: one is taken directly from Latin, the other from Latin via French. In the classic text
8:
16535:
16483:
16409:
16213:
15410:
15209:
11120:
10874:
7969:
and uncurrying; in many special cases, they are also continuous and form a homeomorphism.
7861:
7593:
7221:
7210:
7202:
7198:
5369:
5251:
5250:
in 1958. Like many of the concepts in category theory, it was suggested by the needs of
4266:
3895:
2835:
2003:
453:. Naturality here means that there are natural isomorphisms between the pair of functors
15929:
16389:
16384:
16366:
16248:
16223:
15900:
15854:
15439:
15202:
14491:
14340:
13515:{\displaystyle \varepsilon _{X}=\Phi _{GX,X}^{-1}(1_{GX})\in \mathrm {hom} _{C}(FGX,X)}
12011:
11983:
10572:
10419:
10399:
10353:
10333:
9930:
9833:
9786:
9200:
9180:
9160:
9140:
9120:
9100:
9003:
8950:
8930:
8709:
8689:
8566:
8439:
8419:
8399:
8281:
8261:
8241:
8208:
8112:
8092:
7662:
7653:
describes an adjunction between the category of topological spaces and the category of
7529:
7446:
7353:
7307:
7253:
5404:
3821:
3801:
3781:
3761:
3738:
3714:
3694:
3530:
3369:
3349:
3272:
3223:
3203:
3183:
3163:
3157:
3139:
3119:
3060:
3036:
3016:
2852:
2684:
2664:
2587:
2538:
2518:
2498:
2478:
2472:
2454:
2434:
1899:
1872:
1745:
1725:
1615:
1558:
1538:
1518:
1495:
1475:
1455:
1431:
1411:
1387:
1367:
1265:
1244:
1224:
898:
872:
851:
825:
778:
602:
436:
416:
262:
218:
59:
15140:
Since there is also a natural way to define an identity adjunction between a category
7710:
and to inverse order-preserving bijections between the corresponding closed elements.
4272:
16848:
16773:
16698:
16635:
16623:
16525:
16450:
16445:
16399:
16181:
16176:
16036:
16022:
16000:
15967:
15959:
14267:
8551:
7695:
7582:
7492:
7491:. The suspension functor is therefore left adjoint to the loop space functor in the
7408:
7404:
7365:
7361:
7341:
7033:
6574:
6570:
6420:
5433:
5345:
4508:
2320:
at its foundation, and there are many components which live in one of two categories
1074:
126:
71:
10729:
as "affine functions evaluated at a distribution". That is, for any affine function
4280:
The vertical arrows in this diagram are those induced by composition. Formally, Hom(
1937:
solutionβmeans something rigorous and recognisable, rather like the attainment of a
16863:
16833:
16828:
16808:
16659:
16545:
16520:
16455:
16440:
16435:
16374:
16171:
16046:
16010:
15942:
15890:
15833:
15550:
15246:
7722:
7707:
7457:
7432:
6714:
6578:
6505:
6447:
5398:
5353:
4216:
2058:
over the inclusion of unitary rings into rng.) The existence of a morphism between
1720:
24:
15928:, 1969. The notation is different nowadays; an easier introduction by Peter Smith
13400:
which defines families of initial and terminal morphisms, in the following steps:
4640:{\displaystyle G{\xrightarrow {\;\eta G\;}}GFG{\xrightarrow {\;G\varepsilon \,}}G}
4573:{\displaystyle F{\xrightarrow {\;F\eta \;}}FGF{\xrightarrow {\;\varepsilon F\,}}F}
2260:
This gives the intuition behind the fact that adjoint functors occur in pairs: if
16858:
16823:
16571:
16137:
15917:
7747:
7513:
7496:
7357:
7281:
2101:
can have more adjoined elements and/or more relations not imposed by axioms than
592:{\displaystyle {\mathcal {D}}(-,GX):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
519:{\displaystyle {\mathcal {C}}(F-,X):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
35:
10077:{\displaystyle \exists _{f}S=\{y\in Y\mid \exists (x\in f^{-1}).\,x\in S\;\}=f.}
7913:
form an adjoint pair. The unit and counit are natural isomorphisms in this case.
7006:{1} and defining a binary operation on it such that it extends the operation on
6871:
into the direct sum, and the counit is the additive map from the direct sum of (
4109:{\displaystyle \Phi _{Y,X}:\mathrm {hom} _{C}(FY,X)\to \mathrm {hom} _{D}(Y,GX)}
16873:
16608:
16603:
16587:
16550:
16540:
16460:
15991:
15145:
14275:
10436:. Note how the predicate determining the set is the same as above, except that
7845:
7640:
7567:
7420:
7311:
7206:
7045:
6981:
5321:
5176:
2051:
1922:
15417:), then any pair of adjoint functors between them are automatically additive.
10289:{\displaystyle \forall _{f}S=\{y\in Y\mid \forall (x\in f^{-1}).\,x\in S\;\}.}
7280:
has a left adjoint β it assigns to every ring R the pair (R,x) where R is the
16888:
16843:
16598:
16430:
16307:
16233:
16074:
7735:
closure operators may indicate the presence of adjunctions, as corresponding
7703:
7694:). A pair of adjoint functors between two partially ordered sets is called a
7650:
7510:
7349:
7337:
6941:
5287:
1868:
1716:
5159:
1871:(which is like a ring that might not have a multiplicative identity) into a
16352:
16253:
15751:
is just the unit Ξ· of the adjunction and the multiplication transformation
13845:
The naturality of Ξ¦ implies the naturality of Ξ΅ and Ξ·, and the two formulas
13708:{\displaystyle \eta _{Y}=\Phi _{Y,FY}(1_{FY})\in \mathrm {hom} _{D}(Y,GFY)}
10173:{\displaystyle \{y\in Y\mid \exists x.\,\psi _{f}(x,y)\land \phi _{S}(x)\}}
8555:
8385:{\displaystyle \{y\in Y\mid \exists x.\,\psi _{f}(x,y)\land \phi _{S}(x)\}}
7702:
Galois connection). See that article for a number of examples: the case of
7369:
6921:
6913:
6424:
14430:
An analogous statement characterizes those functors with a right adjoint.
6423:
which assigns to an algebraic object its underlying set. These algebraic
5179:. A way to remember them is to first write down the nonsensical equation
3991:{\displaystyle \Phi :\mathrm {hom} _{C}(F-,-)\to \mathrm {hom} _{D}(-,G-)}
1061:
1055:
16613:
16593:
16465:
16335:
16084:
15853:
Baez, John C. (1996). "Higher-Dimensional
Algebra II: 2-Hilbert Spaces".
9442:{\displaystyle \exists _{f}S\subseteq T\leftrightarrow S\subseteq f^{-1}}
7654:
7224:
instead of the category of rings, to get the monoid and group rings over
6946:
This example was discussed in the motivation section above. Given a rng
6775:
6416:
4220:
1864:
31:
16645:
16583:
16196:
15904:
11307:
7849:
7480:
7475:
is naturally isomorphic to the space of homotopy classes of maps from
7386:. This example foreshadowed the general theory by about half a century.
7252:
from fields has a left adjointβit assigns to every integral domain its
7205:
to rings, left adjoint to the functor that assigns to a given ring its
5413:
5386:
5247:
2226:, and pose the following (vague) question: is there a problem to which
63:
16:
Relationship between two functors abstracting many common constructions
16080:
15859:
7762:
the power set of the set of all mathematical structures. For a theory
7721:
order isomorphism). A treatment of Galois theory along these lines by
16639:
16330:
15414:
15195:
9217:. It therefore turns out to be (in bijection with) the inverse image
8683:
7927:
which assigns to a set the discrete category on that set. Moreover,
6967:
6767:
4198:
768:{\displaystyle {\mathcal {D}}(Y,G-):{\mathcal {C}}\to \mathrm {Set} }
702:{\displaystyle {\mathcal {C}}(FY,-):{\mathcal {C}}\to \mathrm {Set} }
304:
15895:
15094:
15065:
14988:
14955:
14782:
Adjunctions can be composed in a natural fashion. Specifically, if γ
12862:
The transformations Ξ¦ and Ξ¨ are natural because Ξ· and Ξ΅ are natural.
12618:
8671:{\displaystyle f^{*}:{\text{Sub}}(Y)\longrightarrow {\text{Sub}}(X)}
7241:
of integral domains with injective morphisms. The forgetful functor
6261:
6225:
5996:
5954:
4622:
4596:
4555:
4529:
2120:, that is, that there is a morphism from it to any other element of
1891:
in the sense that it works in essentially the same way for any rng.
16708:
16340:
16238:
16088:
14270:, then the functors with left adjoints can be characterized by the
10483:
7966:
7333:
6779:
5238:
where it looks like the insertion of the identity 1 into a monoid.
5219:
in one of the two simple ways which make the compositions defined.
4224:
2171:
has an identity and considering it simply as a rng, so essentially
1938:
16106:
7725:
was influential in the recognition of the general structure here.
7372:; naturally there is also a proof adapted to category theory, too.
6038:
is the free group generated freely by the words of the free group
16796:
16678:
16668:
16317:
16228:
16092:
15818:
14286:
and a certain smallness condition is satisfied: for every object
11081:
6396:
which underlies the group homomorphism sending each generator of
4373:
3866:
3852:
1907:
1066:
43:
16765:
15517:, Ξ΅, Ξ·γ extends an equivalence of certain subcategories. Define
15450:. Conversely, if there exists a universal morphism to a functor
12040:
8785:. If this functor has a left- or right adjoint, they are called
7706:
of course is a leading one. Any Galois connection gives rise to
2969:{\displaystyle \epsilon _{X}\circ F(G(f))=f\circ \epsilon _{X'}}
16673:
15144:
and itself, one can then form a category whose objects are all
13087:
is a functor, that Ξ· is natural, and the counitβunit equation 1
12871:
is a functor, that Ξ΅ is natural, and the counitβunit equation 1
7935:
which assigns to each category its set of objects, and finally
7194:
6977:
5662: be the group homomorphism which sends the generators of
2367:, Ξ· will consistently denote things which live in the category
2351:, Ξ΅ will consistently denote things which live in the category
2336:, and also to write them down in this order whenever possible.
2284:
There are various equivalent definitions for adjoint functors:
2245:
is, in a certain rigorous sense, equivalent to the notion that
2151:
denote the above process of adjoining an identity to a rng, so
1632:
a right adjoint because it is applied to the right argument of
15201:
every right adjoint functor between two abelian categories is
13300:
7943:
which assigns to each set the indiscrete category on that set.
16555:
15208:
every left adjoint functor between two abelian categories is
14923:
with unit and counit given respectively by the compositions:
14754:
denotes vertical composition of natural transformations, and
9756:. We conclude that left adjoint to the inverse image functor
7758:
to be the set of all logical theories (axiomatizations), and
6344: is the "inclusion of generators" set map from the set
5118:
denotes the identity natural transformation from the functor
2143:
can be expressed simultaneously by saying that it defines an
2135:
The two facts that this method of turning rngs into rings is
1575:
a left adjoint because it is applied to the left argument of
1492:
may have itself a right adjoint that is quite different from
11318:
The transformations Ξ΅, Ξ·, and Ξ¦ are related by the equations
8483:-related, and which itself is characterized by the property
7336:, the point of departure is to observe that the category of
6411:
15990:
AdΓ‘mek, JiΕΓ; Herrlich, Horst; Strecker, George E. (1990).
9901:
is non-empty. This works because it neglects exactly those
9291:, let us figure out the left adjoint, which is defined via
6950:, a multiplicative identity element can be added by taking
3080:
Similarly, we may define right-adjoint functors. A functor
14469:
is a functor between locally presentable categories, then
14223:{\displaystyle 1_{GX}=G(\varepsilon _{X})\circ \eta _{GX}}
14160:
in the first formula gives the second counitβunit equation
14122:{\displaystyle 1_{FY}=\varepsilon _{FY}\circ F(\eta _{Y})}
14057:
in the second formula gives the first counitβunit equation
11825:
The transformations Ξ΅, Ξ· satisfy the counitβunit equations
9157:
is characterized as the largest set which knows all about
8195:{\displaystyle \phi _{T}(y)=\phi _{Y}(y)\land \varphi (y)}
205:{\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}}
165:{\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}}
15386:
are, in fact, isomorphisms of abelian groups. Dually, if
7965:
has a right adjoint β. This pair is often referred to as
1811:{\displaystyle \langle Ty,x\rangle =\langle y,Ux\rangle }
15932:, which also attribute the concept to the article cited.
7639:. Here a more subtle point is that the left adjoint for
4355:
3647:{\displaystyle G(F(g))\circ \eta _{Y}=\eta _{Y'}\circ g}
2383:
can be thought of as "living" where its outputs are, in
1921:, so that the problem at hand corresponds to finding an
15593:
and yield inverse equivalences of these subcategories.
15265:
is also an additive functor and the hom-set bijections
7778:) be the set of all structures that satisfy the axioms
15989:
14569:, Ξ΅, Ξ·γ is an adjunction (with counitβunit (Ξ΅,Ξ·)) and
6713:
be the functor which assigns to each homomorphism its
4693:, and may indicate this relationship by writing
1914:
notions, it is only necessary to discuss one of them.
1855:
In a sense, an adjoint functor is a way of giving the
15760:
15713:
15659:
15274:
14932:
14872:
14823:
14760:
14740:
14613:
14443:
14168:
14067:
13855:
13798:
The bijectivity and naturality of Ξ¦ imply that each (
13729:
13611:
13536:
13410:
13358:
13123:
12905:
12731:
12656:
11835:
11328:
11039:
11017:
10995:
10967:
10954:{\displaystyle \mathbb {E} :\mu \mapsto \mathbb {E} }
10922:
10883:
10816:
10779:
10735:
10704:
10682:
10651:
10595:
10575:
10531:
10509:
10462:
10442:
10422:
10402:
10376:
10356:
10336:
10306:
10191:
10094:
9964:
9933:
9907:
9856:
9836:
9809:
9789:
9762:
9727:
9682:
9647:
9602:
9576:
9497:
9462:
9388:
9299:
9271:
9223:
9203:
9183:
9163:
9143:
9123:
9103:
9054:
9034:
9006:
8973:
8953:
8933:
8907:
8876:
8845:
8818:
8791:
8758:
8732:
8712:
8692:
8623:
8589:
8569:
8536:
8516:
8489:
8462:
8442:
8422:
8402:
8306:
8284:
8264:
8244:
8217:
8135:
8115:
8095:
8069:
8015:
7988:
6992:
6358:
6320:
6213:
6160:
6108:
6048:
5942:
5890:
5840:
5734:
5676:
5623:
5541:
5476:
5300:
5185:
5128:
5097:
5073:
5044:
4901:
4786:
4743:
4699:
4587:
4520:
4411:
4223:. Explicitly, the naturality of Ξ¦ means that for all
4011:
3906:
3824:
3804:
3784:
3764:
3741:
3717:
3697:
3660:
3585:
3553:
3533:
3474:
3433:
3392:
3372:
3352:
3295:
3275:
3246:
3226:
3206:
3186:
3166:
3142:
3122:
3086:
3063:
3039:
3019:
2982:
2907:
2875:
2855:
2789:
2748:
2707:
2687:
2667:
2610:
2590:
2561:
2541:
2521:
2501:
2481:
2457:
2437:
2401:
2390:
1850:
1768:
1748:
1728:
1689:
1638:
1618:
1581:
1561:
1541:
1521:
1498:
1478:
1458:
1434:
1414:
1390:
1370:
1343:
1288:
1268:
1247:
1227:
1201:
1086:
1027:
1003:
975:
951:
922:
901:
875:
854:
828:
801:
781:
715:
649:
625:
605:
532:
459:
439:
419:
316:
285:
265:
241:
221:
178:
138:
107:
83:
15430:
As stated earlier, an adjunction between categories
11251:
such that the diagrams below commute, and for every
8530:
of two sets directly corresponds to the conjunction
8063:
of terms that fulfill the property. A proper subset
6837:
is the functor which assigns to every abelian group
3845:
2213:
1077:
makes a distinction between the two. Given a family
13788:{\displaystyle 1_{FY}\in \mathrm {hom} _{C}(FY,FY)}
13595:{\displaystyle 1_{GX}\in \mathrm {hom} _{D}(GX,GX)}
12265:
on morphisms preserves compositions and identities.
5720:, because any group homomorphism from a free group
5226:property whereas the counit morphisms will satisfy
1330:{\displaystyle \mathrm {hom} _{\mathcal {C}}(FY,X)}
15786:
15740:
15686:{\displaystyle T:{\mathcal {D}}\to {\mathcal {D}}}
15685:
15375:
15122:
14904:
14852:
14766:
14746:
14723:
14461:
14222:
14121:
13984:
13787:
13707:
13594:
13514:
13388:
13286:
13068:
12852:
12662:
11960:
11813:
11045:
11025:
11003:
10981:
10953:
10908:
10862:
10802:
10765:
10721:
10690:
10668:
10634:
10581:
10561:
10517:
10468:
10448:
10428:
10408:
10388:
10362:
10342:
10322:
10288:
10172:
10076:
9948:
9919:
9893:
9842:
9822:
9795:
9775:
9748:
9713:
9668:
9633:
9588:
9563:{\displaystyle S\subseteq f^{-1}]\subseteq f^{-1}}
9562:
9483:
9441:
9368:
9283:
9254:
9209:
9189:
9169:
9149:
9129:
9109:
9089:
9040:
9012:
8992:
8959:
8939:
8919:
8893:
8862:
8831:
8804:
8777:
8744:
8718:
8698:
8670:
8607:
8575:
8542:
8522:
8502:
8475:
8448:
8428:
8408:
8384:
8290:
8270:
8250:
8230:
8194:
8121:
8101:
8081:
8055:
8001:
7661:of sober spaces and spatial locales, exploited in
7614:from the category of sheaves of abelian groups on
6998:
6380:
6336:
6294:
6191:
6124:
6070:
6023:
5921:
5870:
5765:
5704:
5654:
5572:
5507:
5312:
5203:
5144:
5110:
5083:
5059:
5026:
4865:
4755:
4729:
4639:
4572:
4488:
4108:
3990:
3830:
3810:
3790:
3770:
3747:
3723:
3703:
3683:
3646:
3571:
3539:
3508:
3460:
3419:
3378:
3358:
3338:
3281:
3261:
3232:
3212:
3192:
3172:
3148:
3128:
3104:
3069:
3045:
3025:
3005:
2968:
2893:
2861:
2834:The latter equation is expressed by the following
2823:
2775:
2734:
2693:
2673:
2653:
2596:
2576:
2547:
2527:
2507:
2487:
2463:
2443:
2419:
2198:Note however that we haven't actually constructed
2167:denote the process of βforgettingβ³ whether a ring
1827:The slogan is "Adjoint functors arise everywhere".
1810:
1754:
1734:
1704:
1661:
1624:
1604:
1567:
1547:
1527:
1504:
1484:
1464:
1440:
1420:
1396:
1376:
1352:
1329:
1274:
1253:
1233:
1207:
1184:
1037:
1013:
985:
961:
934:
907:
881:
860:
834:
811:
787:
767:
701:
635:
611:
591:
518:
445:
425:
398:
295:
271:
251:
227:
204:
164:
117:
93:
15993:Abstract and Concrete Categories. The joy of cats
15883:Transactions of the American Mathematical Society
12619:counitβunit adjunction induces hom-set adjunction
10863:{\displaystyle (\mu ,f):\mu \to \mu \circ f^{-1}}
8510:. Set theoretic operations like the intersection
6762:
16886:
8202:expressing a strictly more restrictive property.
7618:to the category of sheaves of abelian groups on
6782:are all examples of the categorical notion of a
6450:are all examples of the categorical notion of a
6034:should be the identity. The intermediate group
77:By definition, an adjunction between categories
16035:. Vol. 5 (2nd ed.). Springer-Verlag.
14915:More precisely, there is an adjunction between
13349:-), one can construct a counitβunit adjunction
10370:'s with the property that the inverse image of
6430:
6192:{\displaystyle 1_{G}=G\varepsilon \circ \eta G}
5922:{\displaystyle 1_{F}=\varepsilon F\circ F\eta }
5246:The idea of adjoint functors was introduced by
5175:because of the appearance of the corresponding
3200:. Spelled out, this means that for each object
2515:. Spelled out, this means that for each object
1859:solution to some problem via a method which is
15198:of objects yields the coproduct of the images;
13389:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
10916:, and the expectation defines another functor
6976:, we can add an identity element and obtain a
6306:should be the identity. The intermediate set
5871:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
4730:{\displaystyle (\varepsilon ,\eta ):F\dashv G}
1821:
16781:
16122:
16087:. Manipulation and visualization of objects,
15485:
15413:(i.e. preadditive categories with all finite
12647:, and a counitβunit adjunction (Ξ΅, Ξ·) :
12041:Universal morphisms induce hom-set adjunction
10698:with finite expectation. Define morphisms on
7596:(of sets, or abelian groups, or rings...) on
5462:is just the underlying set of the free group
5067:denotes the identify functor on the category
3758:It is true, as the terminology implies, that
1972:a ring having a multiplicative identity. The
1662:{\displaystyle \mathrm {hom} _{\mathcal {D}}}
1605:{\displaystyle \mathrm {hom} _{\mathcal {C}}}
1555:has a right adjoint" are equivalent. We call
15191:of objects yields the product of the images;
12270:Construct a natural isomorphism Ξ¦ : hom
10909:{\displaystyle \delta :x\mapsto \delta _{x}}
10383:
10377:
10280:
10257:
10251:
10208:
10167:
10095:
10053:
10030:
10024:
9981:
9879:
9873:
9000:and returns the thereby specified subset of
8379:
8307:
8050:
8022:
7643:will differ from that for sheaves (of sets).
7600:to the corresponding category of sheaves on
7528:be the inclusion functor to the category of
6924:of groups and by the disjoint union of sets.
6552:which define the limit, and the unit is the
5580: via a unique group homomorphism from
5152:denotes the identity morphism of the object
1805:
1790:
1784:
1769:
15741:{\displaystyle \eta :1_{\mathcal {D}}\to T}
15614:) need not be a right (or left) adjoint of
13301:Hom-set adjunction induces all of the above
11314:From this assertion, one can recover that:
8583:in a category with pullbacks. Any morphism
4766:In equation form, the above conditions on (
2304:repeated separately in every subject area.
1048:
16788:
16774:
16750:
16740:
16496:
16129:
16115:
15425:
12608:, and then Ξ¦ is natural in both arguments.
12257:Uniqueness of that factorization and that
12173:) is an initial morphism, then factorize Ξ·
10982:{\displaystyle \mathbb {E} \dashv \delta }
10279:
10052:
7397:A functor with a left and a right adjoint.
6262:
6240:
6226:
5997:
5975:
5955:
5803:correspond precisely to maps from the set
4623:
4604:
4597:
4556:
4537:
4530:
2654:{\displaystyle \epsilon _{X}:F(G(X))\to X}
2108:. Therefore, the assertion that an object
19:For the construction in field theory, see
16073:β seven short lectures on adjunctions by
15894:
15858:
15783:
12132:on objects and the family of morphisms Ξ·.
11761:
11760:
11648:
11647:
11530:
11529:
11412:
11411:
11019:
10997:
10969:
10938:
10924:
10793:
10712:
10684:
10676:, the set of probability distribution on
10659:
10511:
10269:
10119:
10042:
8331:
7826:is right adjoint to the "syntax functor"
7552:. The unit of this adjoint pair yields a
7344:has a commutative monoid structure under
7076:is left adjoint to the forgetful functor
6412:Free constructions and forgetful functors
6282:
6094:to the corresponding word of length one (
6011:
5766:{\displaystyle \varepsilon _{X}:FGX\to X}
5655:{\displaystyle \varepsilon _{X}:FGX\to X}
5204:{\displaystyle 1=\varepsilon \circ \eta }
4630:
4563:
2824:{\displaystyle \epsilon _{X}\circ F(g)=f}
2339:In this article for example, the letters
1863:. For example, an elementary problem in
16028:Categories for the Working Mathematician
16021:
14853:{\displaystyle F\circ F':E\rightarrow C}
14282:has a left adjoint if and only if it is
13078:hence ΨΦ is the identity transformation.
8682:on the category that is the preorder of
8056:{\displaystyle Y=\{y\mid \phi _{Y}(y)\}}
6469:the functor which assigns to each pair (
6419:are all examples of a left adjoint to a
5795:Group homomorphisms from the free group
5590:universal property of the free group on
5158:
2845:Here the counit is a universal morphism.
1948:Back to our example: take the given rng
1835:Categories for the Working Mathematician
1071:Categories for the Working Mathematician
993:is somewhat akin to a "weak form" of an
23:. For the construction in topology, see
13329:, and a hom-set adjunction Ξ¦ : hom
10803:{\displaystyle \mu \in M(\mathbb {R} )}
7782:; for a set of mathematical structures
6593:of homomorphisms of abelian groups. If
2087:is at least as efficient a solution as
409:such that this family of bijections is
66:in algebra, or the construction of the
16887:
15187:applying a right adjoint functor to a
14534:. The same is true for left adjoints.
11056:
10088:Put this in analogy to our motivation
8726:(technically: monomorphism classes of
7931:is left-adjoint to the object functor
7893:, then we have an inverse equivalence
7360:; but there is the other option of an
7117:-module, then the tensor product with
6140:sending each generator to the word of
6132: is the group homomorphism from
6078: is the group homomorphism from
5609:is the free group generated freely by
5389:is a common and illuminating example.
4002:This specifies a family of bijections
3547:can be uniquely turned into a functor
3339:{\displaystyle \eta _{Y}:Y\to G(F(Y))}
1933:βthe sense that the process finds the
16769:
16495:
16148:
16110:
15194:applying a left adjoint functor to a
15151:
15148:and whose morphisms are adjunctions.
14433:An important special case is that of
12261:is a functor implies that the map of
11301:such that the diagrams below commute:
8927:to quantify a relation expressed via
7564:Direct and inverse images of sheaves.
7560:into its StoneβΔech compactification.
5705:{\displaystyle (GX,\varepsilon _{X})}
5408:be the functor assigning to each set
4356:Definition via counitβunit adjunction
3509:{\displaystyle G(f)\circ \eta _{Y}=g}
2849:In this situation, one can show that
2312:The theory of adjoints has the terms
2279:
307:between the respective morphism sets
125:is a pair of functors (assumed to be
15852:
15638:, Ξ΅, Ξ·γ gives rise to an associated
14806:β², Ξ΅β², Ξ·β²γ is an adjunction between
14246:Formal criteria for adjoint functors
10350:is characterized as the full set of
9090:{\displaystyle f^{*}T=X\times _{Y}T}
9041:{\displaystyle \operatorname {Set} }
8947:over, the functor/quantifier closes
7973:
7848:is adjoint to the multiplication by
6908:Analogous examples are given by the
6748:with the kernel of the homomorphism
6728:be the functor which maps the group
5234:here is borrowed from the theory of
4507:of the adjunction (terminology from
4197:as functors. In fact, they are both
16136:
15873:
14476:has a right adjoint if and only if
14294:there exists a family of morphisms
12384:is a functor, then for any objects
11166:A natural transformation Ξ· : 1
8839:, respectively. They both map from
7794:) be the minimal axiomatization of
7201:construction yields a functor from
6928:
6891:) of the direct sum to the element
6381:{\displaystyle G(\varepsilon _{X})}
5712: is a terminal morphism from
4219:). For details, see the article on
13:
15726:
15678:
15668:
15349:
15343:
15340:
15337:
15308:
15302:
15299:
15296:
15276:
15133:This new adjunction is called the
15107:
14944:
14604:β², Ξ΅β², Ξ·β²γ is an adjunction where
14486:has a left adjoint if and only if
13918:
13861:
13754:
13751:
13748:
13674:
13671:
13668:
13626:
13561:
13558:
13555:
13481:
13478:
13475:
13425:
13131:
13128:
12913:
12910:
12794:
12737:
11770:
11767:
11764:
11699:
11657:
11654:
11651:
11578:
11539:
11536:
11533:
11466:
11421:
11418:
11415:
11340:
11306:
10463:
10443:
10308:
10223:
10193:
10110:
9996:
9966:
9811:
9390:
9309:
8820:
8793:
8322:
7868:
7193:construction gives a functor from
5573:{\displaystyle \eta _{Y}:Y\to GFY}
5508:{\displaystyle \eta _{Y}:Y\to GFY}
5230:properties, and dually. The term
5076:
5051:
4650:are the identity transformations 1
4467:
4441:
4271:
4078:
4075:
4072:
4039:
4036:
4033:
4013:
3960:
3957:
3954:
3921:
3918:
3915:
3907:
3518:
2840:
2391:Definition via universal morphisms
1929:. This has an advantage that the
1851:Solutions to optimization problems
1653:
1647:
1644:
1641:
1596:
1590:
1587:
1584:
1303:
1297:
1294:
1291:
1158:
1152:
1149:
1146:
1117:
1111:
1108:
1105:
1030:
1006:
978:
954:
804:
761:
758:
755:
746:
718:
695:
692:
689:
680:
652:
628:
579:
575:
572:
563:
535:
506:
502:
499:
490:
462:
372:
366:
363:
360:
331:
325:
322:
319:
288:
244:
197:
187:
157:
147:
110:
86:
14:
16906:
16795:
16083:is a category theory package for
16058:
15787:{\displaystyle \mu :T^{2}\to T\,}
14905:{\displaystyle G'\circ G:C\to E.}
14790:, Ξ΅, Ξ·γ is an adjunction between
14490:preserves small limits and is an
11183:An equivalent formulation, where
11011:is the left adjoint, even though
9714:{\displaystyle S\subseteq f^{-1}}
9255:{\displaystyle f^{-1}\subseteq X}
7556:map from every topological space
7290:. Consider the inclusion functor
7187:From monoids and groups to rings.
6125:{\displaystyle \varepsilon _{FY}}
5773: via a unique set map from
3846:Definition via Hom-set adjunction
2214:Symmetry of optimization problems
945:An adjunction between categories
62:), such as the construction of a
16749:
16739:
16730:
16729:
16482:
16149:
15420:
15390:is additive with a left adjoint
14774:denotes horizontal composition.
13795: is the identity morphism.
13602: is the identity morphism.
12323:) is an initial morphism, then Ξ¦
12073:and a natural transformation Ξ·.
8238:with two open variables of sort
8129:is characterized by a predicate
8089:and the associated injection of
7698:(or, if it is contravariant, an
6199: says that for each group
5060:{\displaystyle 1_{\mathcal {C}}}
2742:there exists a unique morphism
2230:is the most efficient solution?
643:, and also the pair of functors
15478:(equivalently, every object of
14596:are natural isomorphisms then γ
14018:(which completely determine Ξ¦).
12380:Ξ· is a natural transformation,
10722:{\displaystyle M(\mathbb {R} )}
10669:{\displaystyle M(\mathbb {R} )}
10635:{\displaystyle (r,f):r\to f(r)}
9927:which are in the complement of
8901:. Very roughly, given a domain
8894:{\displaystyle {\text{Sub}}(Y)}
8863:{\displaystyle {\text{Sub}}(X)}
7836:is (in general) the attempt to
7380:representation theory of groups
7310:. It has a left adjoint called
6152:The second counitβunit equation
3427:there exists a unique morphism
1715:The terminology comes from the
1195:of hom-set bijections, we call
15953:
15935:
15911:
15867:
15846:
15777:
15732:
15673:
15542:is an isomorphism, and define
15370:
15355:
15329:
15314:
15137:of the two given adjunctions.
14893:
14844:
14777:
14711:
14679:
14645:
14633:
14453:
14435:locally presentable categories
14201:
14188:
14116:
14103:
13975:
13969:
13947:
13941:
13897:
13891:
13882:
13876:
13830:) is an initial morphism from
13808:) is a terminal morphism from
13782:
13764:
13702:
13684:
13660:
13644:
13589:
13571:
13509:
13491:
13467:
13451:
13371:
13359:
13220:
13207:
13178:
13172:
13160:
13147:
13024:
13011:
12973:
12960:
12951:
12945:
12843:
12837:
12815:
12809:
12773:
12767:
12758:
12752:
12084:, choose an initial morphism (
12045:Given a right adjoint functor
11935:
11922:
11892:
11879:
11804:
11801:
11795:
11780:
11733:
11717:
11691:
11682:
11676:
11667:
11620:
11604:
11570:
11567:
11561:
11549:
11506:
11500:
11487:
11481:
11452:
11443:
11437:
11431:
11401:
11395:
11369:
11363:
10948:
10942:
10934:
10893:
10838:
10829:
10817:
10797:
10789:
10745:
10739:
10716:
10708:
10663:
10655:
10629:
10623:
10617:
10608:
10596:
10541:
10535:
10491:
10263:
10260:
10248:
10226:
10164:
10158:
10142:
10130:
10068:
10062:
10036:
10033:
10021:
9999:
9943:
9937:
9882:
9870:
9737:
9731:
9708:
9702:
9657:
9651:
9628:
9622:
9557:
9551:
9532:
9529:
9523:
9517:
9472:
9466:
9436:
9430:
9408:
9360:
9338:
9327:
9305:
9243:
9237:
8888:
8882:
8857:
8851:
8736:
8665:
8659:
8651:
8648:
8642:
8599:
8376:
8370:
8354:
8342:
8278:. Using a quantifier to close
8189:
8183:
8174:
8168:
8152:
8146:
8047:
8041:
6763:Colimits and diagonal functors
6375:
6362:
6310:is just the underlying set of
6279:
6266:
6065:
6052:
5972:
5959:
5929: says that for each set
5882:The first counitβunit equation
5853:
5841:
5757:
5699:
5677:
5646:
5558:
5493:
5436:, which assigns to each group
5380:
5336:,β) (this is now known as the
5163:String diagram for adjunction.
5084:{\displaystyle {\mathcal {C}}}
5001:
4988:
4958:
4945:
4712:
4700:
4673:In this situation we say that
4511:), such that the compositions
4473:
4432:
4103:
4088:
4067:
4064:
4049:
3985:
3970:
3949:
3946:
3931:
3670:
3604:
3601:
3595:
3589:
3563:
3484:
3478:
3452:
3449:
3443:
3414:
3408:
3402:
3333:
3330:
3324:
3318:
3312:
3256:
3250:
3096:
2997:
2939:
2936:
2930:
2924:
2885:
2812:
2806:
2770:
2764:
2758:
2726:
2723:
2717:
2645:
2642:
2639:
2633:
2627:
2571:
2565:
2411:
2307:
1324:
1309:
1179:
1164:
1138:
1123:
1038:{\displaystyle {\mathcal {D}}}
1014:{\displaystyle {\mathcal {C}}}
986:{\displaystyle {\mathcal {D}}}
962:{\displaystyle {\mathcal {C}}}
812:{\displaystyle {\mathcal {D}}}
751:
738:
723:
685:
672:
657:
636:{\displaystyle {\mathcal {C}}}
568:
555:
540:
495:
482:
467:
393:
378:
352:
337:
296:{\displaystyle {\mathcal {D}}}
252:{\displaystyle {\mathcal {C}}}
192:
152:
118:{\displaystyle {\mathcal {D}}}
94:{\displaystyle {\mathcal {C}}}
1:
16033:Graduate Texts in Mathematics
15983:
15964:Sheaves in Geometry and Logic
15216:
14497:
14234:
10989:. (Somewhat disconcertingly,
10323:{\displaystyle \forall _{f}S}
8993:{\displaystyle X\times _{Y}T}
8778:{\displaystyle X\times _{Y}T}
7314:which assigns to every group
7216:and consider the category of
6972:Similarly, given a semigroup
6579:product of topological spaces
6508:which assigns to every group
5416:generated by the elements of
4165:In order to interpret Ξ¦ as a
2869:can be turned into a functor
1512:; see below for an example.)
15608:is naturally isomorphic to 1
15557:consisting of those objects
15528:consisting of those objects
14239:
11026:{\displaystyle \mathbb {E} }
11004:{\displaystyle \mathbb {E} }
10691:{\displaystyle \mathbb {R} }
10518:{\displaystyle \mathbb {R} }
9894:{\displaystyle f^{-1}\cap S}
9823:{\displaystyle \exists _{f}}
9749:{\displaystyle f\subseteq T}
9484:{\displaystyle f\subseteq T}
9284:{\displaystyle S\subseteq X}
9097:of an injection of a subset
8832:{\displaystyle \forall _{f}}
8805:{\displaystyle \exists _{f}}
7610:. It also induces a functor
7503:StoneβΔech compactification.
7443:Suspensions and loop spaces.
7209:. One can also start with a
6577:, the product of rings, the
6431:Diagonal functors and limits
6071:{\displaystyle F(\eta _{Y})}
2661:such that for every object
2332:or the "righthand" category
7:
16424:Constructions on categories
15524:as the full subcategory of
15446:and one for each object in
14549:is naturally isomorphic to
14344:, such that every morphism
12693:-) in the following steps:
10645:Define a category based on
10503:Define a category based on
9634:{\displaystyle x\in f^{-1}}
7550:StoneβΔech compactification
7390:
6966:Adjoining an identity to a
6940:Adjoining an identity to a
6388: is the set map from
5523:, because any set map from
5375:
3461:{\displaystyle f:F(Y)\to X}
3420:{\displaystyle g:Y\to G(X)}
3346:such that for every object
2901:in a unique way such that
2776:{\displaystyle g:Y\to G(X)}
2735:{\displaystyle f:F(Y)\to X}
1822:Introduction and Motivation
68:StoneβΔech compactification
42:is a relationship that two
10:
16911:
16531:Higher-dimensional algebra
15922:Adjointness in foundations
15486:Equivalences of categories
15438:gives rise to a family of
14262:admits a left adjoint. If
14243:
10416:is fully contained within
8920:{\displaystyle S\subset X}
8082:{\displaystyle T\subset Y}
7822:: the "semantics functor"
7746:a very general comment of
7304:category of abelian groups
7175:) for every abelian group
6933:
6821:) of abelian groups their
6520:) in the product category
6337:{\displaystyle \eta _{GX}}
5728:will factor through
5535:will factor through
5440:its underlying set. Then
5313:{\displaystyle -\otimes A}
5241:
3240:, there exists an object
2395:By definition, a functor
1705:{\displaystyle F\dashv G.}
1384:(p. 81). The functor
18:
16804:
16725:
16658:
16622:
16570:
16563:
16514:
16504:
16491:
16480:
16423:
16365:
16316:
16271:
16262:
16159:
16155:
16144:
15999:. John Wiley & Sons.
15625:
15504:equivalence of categories
15442:, one for each object in
10766:{\displaystyle f(x)=ax+b}
10562:{\displaystyle f(x)=ax+b}
9669:{\displaystyle f(x)\in T}
8503:{\displaystyle \phi _{S}}
8476:{\displaystyle \psi _{f}}
8231:{\displaystyle \psi _{f}}
8002:{\displaystyle \phi _{Y}}
7951:cartesian closed category
7891:equivalence of categories
7741:Kuratowski closure axioms
7669:
7284:with coefficients from R.
6671:) of morphisms such that
5811:: each homomorphism from
5588:. This is precisely the
4876:which mean that for each
4756:{\displaystyle F\dashv G}
3684:{\displaystyle g:Y\to Y'}
3006:{\displaystyle f:X'\to X}
2132:solution to our problem.
1515:In general, the phrases "
1353:{\displaystyle \varphi f}
935:{\displaystyle F\dashv G}
21:Adjunction (field theory)
15962:; Moerdijk, Ieke (1992)
15839:
15830:EilenbergβMoore algebras
15704:. The unit of the monad
15646:, Ξ·, ΞΌγ in the category
15571:is an isomorphism. Then
14557:is also left adjoint to
14480:preserves small colimits
14462:{\displaystyle F:C\to D}
10961:, and they are adjoint:
10469:{\displaystyle \forall }
10449:{\displaystyle \exists }
9570:. Conversely, If for an
8608:{\displaystyle f:X\to Y}
7917:A series of adjunctions.
7352:out of this monoid, the
7179:, is a right adjoint to
7040:can be seen as a (left)
7024:are rings, and Ο :
5211:and then fill in either
4737: , or simply
4499:respectively called the
3572:{\displaystyle F:D\to C}
3105:{\displaystyle G:C\to D}
2894:{\displaystyle G:C\to D}
2555:there exists an object
2420:{\displaystyle F:D\to C}
2241:to the problem posed by
2147:. More explicitly: Let
1535:is a left adjoint" and "
1208:{\displaystyle \varphi }
1049:Terminology and notation
16341:Cokernels and quotients
16264:Universal constructions
16097:natural transformations
15874:Kan, Daniel M. (1958).
15426:Universal constructions
14561:β². More generally, if γ
14514:has two right adjoints
14272:adjoint functor theorem
12663:{\displaystyle \dashv }
12333:is a bijection, where Ξ¦
12128:)). We have the map of
11046:{\displaystyle \delta }
7905:, and the two functors
7798:. We can then say that
7495:, an important fact in
6999:{\displaystyle \sqcup }
6807:assigns to every pair (
6639:, then a morphism from
6086:sending each generator
5832:counitβunit adjunction.
4401:natural transformations
4364:between two categories
3857:between two categories
2239:most efficient solution
2218:It is also possible to
1065:are both used, and are
16839:Essentially surjective
16498:Higher category theory
16244:Natural transformation
15930:in these lecture notes
15805:. Dually, the triple γ
15788:
15742:
15687:
15618:. Adjoints generalize
15377:
15231:preadditive categories
15124:
14906:
14854:
14768:
14748:
14747:{\displaystyle \circ }
14725:
14463:
14224:
14123:
13986:
13789:
13709:
13596:
13516:
13390:
13288:
13070:
12867:Using, in order, that
12854:
12664:
12252:natural transformation
12227:). This is the map of
11962:
11815:
11311:
11195:denotes any object of
11187:denotes any object of
11148:natural transformation
11047:
11027:
11005:
10983:
10955:
10910:
10864:
10804:
10767:
10723:
10692:
10670:
10636:
10583:
10563:
10519:
10470:
10450:
10430:
10410:
10390:
10364:
10344:
10324:
10290:
10174:
10078:
9950:
9921:
9920:{\displaystyle y\in Y}
9895:
9844:
9824:
9797:
9777:
9750:
9715:
9670:
9635:
9590:
9589:{\displaystyle x\in S}
9564:
9485:
9443:
9380:which here just means
9370:
9285:
9256:
9211:
9191:
9171:
9151:
9131:
9111:
9091:
9042:
9014:
8994:
8961:
8941:
8921:
8895:
8864:
8833:
8806:
8779:
8746:
8745:{\displaystyle T\to Y}
8720:
8700:
8672:
8609:
8577:
8563:So consider an object
8544:
8543:{\displaystyle \land }
8524:
8504:
8477:
8450:
8436:for which there is an
8430:
8410:
8386:
8298:, we can form the set
8292:
8272:
8252:
8232:
8196:
8123:
8103:
8083:
8057:
8003:
7427:, and a right adjoint
7384:induced representation
7330:The Grothendieck group
7234:Consider the category
7000:
6589:Consider the category
6382:
6338:
6296:
6193:
6126:
6072:
6025:
5923:
5878: is as follows:
5872:
5789:) is an adjoint pair.
5767:
5706:
5656:
5574:
5527:to the underlying set
5509:
5314:
5205:
5164:
5146:
5145:{\displaystyle 1_{FY}}
5112:
5085:
5061:
5028:
4867:
4757:
4731:
4641:
4574:
4490:
4362:counitβunit adjunction
4277:
4265:the following diagram
4110:
3992:
3832:
3812:
3792:
3772:
3749:
3725:
3705:
3685:
3648:
3573:
3541:
3524:
3510:
3462:
3421:
3380:
3360:
3340:
3283:
3263:
3234:
3214:
3194:
3174:
3150:
3130:
3106:
3071:
3047:
3027:
3007:
2970:
2895:
2863:
2846:
2825:
2777:
2736:
2695:
2675:
2655:
2598:
2578:
2549:
2529:
2509:
2489:
2465:
2445:
2421:
2251:most difficult problem
2124:, means that the ring
1960:are rng homomorphisms
1952:, and make a category
1840:
1812:
1756:
1736:
1706:
1663:
1626:
1606:
1569:
1549:
1529:
1506:
1486:
1466:
1442:
1422:
1398:
1378:
1354:
1331:
1276:
1255:
1235:
1209:
1186:
1039:
1015:
987:
963:
936:
909:
883:
862:
836:
813:
789:
769:
703:
637:
613:
593:
520:
447:
427:
400:
297:
273:
253:
229:
206:
166:
119:
95:
15943:"Indiscrete category"
15789:
15743:
15688:
15579:can be restricted to
15462:from every object of
15378:
15249:with a right adjoint
15125:
14907:
14855:
14769:
14767:{\displaystyle \ast }
14749:
14726:
14464:
14225:
14124:
13987:
13790:
13710:
13597:
13517:
13391:
13289:
13071:
12855:
12665:
11963:
11816:
11310:
11048:
11028:
11006:
10984:
10956:
10911:
10865:
10805:
10768:
10724:
10693:
10671:
10637:
10584:
10564:
10520:
10471:
10451:
10431:
10411:
10391:
10389:{\displaystyle \{y\}}
10365:
10345:
10325:
10291:
10175:
10079:
9951:
9922:
9896:
9845:
9825:
9798:
9778:
9776:{\displaystyle f^{*}}
9751:
9716:
9671:
9636:
9591:
9565:
9486:
9444:
9371:
9286:
9257:
9212:
9192:
9177:and the injection of
9172:
9152:
9132:
9112:
9092:
9043:
9015:
8995:
8962:
8942:
8922:
8896:
8865:
8834:
8807:
8780:
8747:
8721:
8701:
8686:. It maps subobjects
8673:
8610:
8578:
8545:
8525:
8523:{\displaystyle \cap }
8505:
8478:
8451:
8431:
8411:
8387:
8293:
8273:
8253:
8233:
8197:
8124:
8104:
8084:
8058:
8004:
7676:partially ordered set
7625:inverse image functor
7592:from the category of
7376:Frobenius reciprocity
7001:
6883:(sending an element (
6564:(mapping x to (x,x)).
6383:
6339:
6297:
6194:
6127:
6073:
6026:
5924:
5873:
5768:
5707:
5657:
5575:
5510:
5338:tensor-hom adjunction
5315:
5206:
5162:
5147:
5113:
5111:{\displaystyle 1_{F}}
5086:
5062:
5029:
4868:
4776:counitβunit equations
4758:
4732:
4642:
4575:
4491:
4275:
4169:, one must recognize
4111:
3993:
3833:
3813:
3793:
3773:
3750:
3726:
3706:
3686:
3649:
3574:
3542:
3522:
3511:
3463:
3422:
3381:
3361:
3341:
3284:
3264:
3235:
3215:
3195:
3175:
3151:
3131:
3114:right adjoint functor
3107:
3072:
3048:
3028:
3008:
2971:
2896:
2864:
2844:
2826:
2778:
2737:
2696:
2676:
2656:
2599:
2579:
2550:
2530:
2510:
2490:
2466:
2446:
2422:
2004:commutative triangles
1825:
1813:
1757:
1737:
1707:
1664:
1627:
1607:
1570:
1550:
1530:
1507:
1487:
1467:
1443:
1423:
1399:
1379:
1355:
1332:
1277:
1256:
1236:
1210:
1187:
1040:
1016:
988:
964:
937:
910:
891:right adjoint functor
884:
863:
837:
814:
790:
770:
704:
638:
614:
594:
521:
448:
428:
401:
298:
274:
254:
230:
215:and, for all objects
207:
167:
120:
96:
16367:Algebraic categories
16101:universal properties
15758:
15711:
15657:
15470:has a left adjoint.
15272:
15176:(i.e. commutes with
15164:(i.e. commutes with
15160:a right adjoint) is
14930:
14870:
14821:
14758:
14738:
14611:
14532:naturally isomorphic
14441:
14166:
14065:
13853:
13727:
13609:
13534:
13408:
13356:
13121:
12903:
12729:
12654:
12061:Construct a functor
11833:
11326:
11267:, there is a unique
11221:, there is a unique
11037:
11015:
10993:
10965:
10920:
10881:
10814:
10810:, define a morphism
10777:
10733:
10702:
10680:
10649:
10593:
10589:, define a morphism
10573:
10569:and any real number
10529:
10507:
10460:
10440:
10420:
10400:
10374:
10354:
10334:
10304:
10189:
10092:
9962:
9931:
9905:
9854:
9834:
9807:
9787:
9760:
9725:
9680:
9645:
9600:
9574:
9495:
9460:
9386:
9297:
9269:
9221:
9201:
9181:
9161:
9141:
9121:
9101:
9052:
9032:
9004:
8971:
8951:
8931:
8905:
8874:
8843:
8816:
8789:
8756:
8730:
8710:
8690:
8621:
8587:
8567:
8534:
8514:
8487:
8460:
8440:
8420:
8400:
8304:
8282:
8262:
8242:
8215:
8133:
8113:
8093:
8067:
8013:
7986:
7818:) logically implies
7752:syntax and semantics
7607:direct image functor
7403:be the functor from
7191:integral monoid ring
6990:
6740:is right adjoint to
6732:to the homomorphism
6483:) the product group
6356:
6318:
6211:
6158:
6106:
6098:) as a generator of
6046:
5940:
5888:
5838:
5781:. This means that (
5732:
5674:
5621:
5539:
5474:
5385:The construction of
5332:was the functor hom(
5298:
5183:
5126:
5095:
5071:
5042:
4899:
4784:
4741:
4697:
4688:is right adjoint to
4585:
4518:
4409:
4153:is right adjoint to
4009:
3904:
3822:
3818:is right adjoint to
3802:
3782:
3762:
3739:
3715:
3695:
3658:
3583:
3551:
3531:
3472:
3431:
3390:
3370:
3350:
3293:
3273:
3262:{\displaystyle F(Y)}
3244:
3224:
3204:
3184:
3164:
3140:
3120:
3084:
3061:
3037:
3017:
2980:
2905:
2873:
2853:
2787:
2746:
2705:
2685:
2665:
2608:
2588:
2577:{\displaystyle G(X)}
2559:
2539:
2519:
2499:
2479:
2455:
2435:
2429:left adjoint functor
2399:
2272:is right adjoint to
2189:left adjoint functor
1766:
1746:
1726:
1687:
1636:
1616:
1579:
1559:
1539:
1519:
1496:
1476:
1456:
1432:
1412:
1388:
1368:
1341:
1286:
1266:
1245:
1225:
1199:
1084:
1025:
1001:
973:
949:
920:
899:
873:
852:
844:left adjoint functor
826:
799:
779:
713:
647:
623:
603:
530:
457:
437:
417:
314:
283:
263:
239:
219:
176:
136:
105:
81:
16536:Homotopy hypothesis
16214:Commutative diagram
15966:, Springer-Verlag.
15918:Lawvere, F. William
15440:universal morphisms
15411:additive categories
15172:a left adjoint) is
15098:
15080:
15008:
14964:
14863:is left adjoint to
14541:is left adjoint to
13940:
13450:
13083:Dually, using that
11603:
11362:
11121:natural isomorphism
11068:between categories
11057:Adjunctions in full
11033:is "forgetful" and
10877:defines a functor:
10875:Dirac delta measure
9830:is the full set of
7939:is left-adjoint to
7862:logical conjunction
7858:propositional logic
7632:is left adjoint to
7536:has a left adjoint
7509:be the category of
7415:has a left adjoint
7318:the quotient group
7232:Field of fractions.
7199:integral group ring
6853:is left adjoint to
6635:are two objects of
6497:, and let Ξ :
6352:. The arrow
6314:. The arrow
6283:
6241:
6102:. The arrow
6012:
5976:
5793:Hom-set adjunction.
5666:to the elements of
5599:Terminal morphisms.
5444:is left adjoint to
5370:natural isomorphism
5286:in the category of
5252:homological algebra
5171:, or sometimes the
5169:triangle identities
4678:is left adjoint to
4631:
4605:
4564:
4538:
4167:natural isomorphism
4143:is left adjoint to
4138:In this situation,
3896:natural isomorphism
3778:is left adjoint to
3386:and every morphism
3116:if for each object
2836:commutative diagram
2701:and every morphism
2431:if for each object
2264:is left adjoint to
1832:Saunders Mac Lane,
1676:is left adjoint to
1221:adjunction between
64:free group on a set
16249:Universal property
16023:Mac Lane, Saunders
15960:Mac Lane, Saunders
15876:"Adjoint Functors"
15784:
15738:
15683:
15630:Every adjunction γ
15509:Every adjunction γ
15502:is one half of an
15401:Moreover, if both
15398:is also additive.
15373:
15152:Limit preservation
15120:
15118:
14902:
14850:
14764:
14744:
14721:
14719:
14492:accessible functor
14459:
14402:and some morphism
14367:can be written as
14327:where the indices
14250:Not every functor
14220:
14119:
13982:
13980:
13917:
13785:
13705:
13592:
13512:
13424:
13386:
13284:
13282:
13066:
13064:
12850:
12848:
12660:
11958:
11956:
11811:
11809:
11577:
11339:
11312:
11043:
11023:
11001:
10979:
10951:
10906:
10860:
10800:
10763:
10719:
10688:
10666:
10632:
10579:
10559:
10515:
10466:
10446:
10426:
10406:
10386:
10360:
10340:
10320:
10286:
10170:
10074:
9946:
9917:
9891:
9840:
9820:
9793:
9773:
9746:
9711:
9666:
9631:
9586:
9560:
9481:
9439:
9366:
9281:
9252:
9207:
9187:
9167:
9147:
9127:
9107:
9087:
9038:
9010:
8990:
8957:
8937:
8917:
8891:
8860:
8829:
8802:
8775:
8752:) to the pullback
8742:
8716:
8696:
8668:
8615:induces a functor
8605:
8573:
8550:of predicates. In
8540:
8520:
8500:
8473:
8446:
8426:
8406:
8382:
8288:
8268:
8248:
8228:
8192:
8119:
8099:
8079:
8053:
7999:
7947:Exponential object
7754:are adjoint: take
7663:pointless topology
7585:induces a functor
7583:topological spaces
7530:topological spaces
7447:topological spaces
7405:topological spaces
7354:Grothendieck group
7348:. One may make an
7308:category of groups
7254:field of fractions
6996:
6556:of a group X into
6554:diagonal inclusion
6400:to the element of
6378:
6334:
6292:
6189:
6122:
6068:
6021:
5919:
5868:
5763:
5702:
5652:
5613:, the elements of
5570:
5505:
5452:Initial morphisms.
5340:). The use of the
5310:
5201:
5165:
5142:
5108:
5081:
5057:
5024:
5022:
4863:
4861:
4753:
4727:
4637:
4570:
4486:
4484:
4278:
4244:and all morphisms
4106:
3988:
3828:
3808:
3788:
3768:
3745:
3721:
3701:
3681:
3644:
3569:
3537:
3525:
3506:
3458:
3417:
3376:
3356:
3336:
3279:
3259:
3230:
3210:
3190:
3170:
3158:universal morphism
3156:, there exists a
3146:
3126:
3102:
3067:
3043:
3023:
3003:
2976:for all morphisms
2966:
2891:
2859:
2847:
2821:
2773:
2732:
2691:
2671:
2651:
2594:
2574:
2545:
2525:
2505:
2485:
2473:universal morphism
2461:
2441:
2417:
2280:Formal definitions
1900:universal property
1898:if it satisfies a
1808:
1752:
1732:
1702:
1659:
1622:
1602:
1565:
1545:
1525:
1502:
1482:
1462:
1438:
1418:
1394:
1374:
1350:
1327:
1272:
1251:
1231:
1205:
1182:
1035:
1011:
983:
959:
932:
905:
879:
858:
832:
809:
785:
765:
699:
633:
609:
589:
516:
443:
423:
396:
293:
269:
249:
225:
202:
172: and
162:
115:
91:
60:universal property
54:and the other the
16882:
16881:
16854:Full and faithful
16763:
16762:
16721:
16720:
16717:
16716:
16699:monoidal category
16654:
16653:
16526:Enriched category
16478:
16477:
16474:
16473:
16451:Quotient category
16446:Opposite category
16361:
16360:
15099:
15081:
15009:
14965:
14814:then the functor
14268:complete category
14132:and substituting
13106:
12885:
12604:
12580:
12565:
12553:
12542:
12521:
12506:
12495:
12484:
12478:
12370:
12246:
12184:
11984:terminal morphism
11199:, is as follows:
10582:{\displaystyle r}
10429:{\displaystyle S}
10409:{\displaystyle f}
10363:{\displaystyle y}
10343:{\displaystyle Y}
9949:{\displaystyle f}
9843:{\displaystyle y}
9796:{\displaystyle S}
9210:{\displaystyle Y}
9190:{\displaystyle T}
9170:{\displaystyle f}
9150:{\displaystyle f}
9130:{\displaystyle Y}
9110:{\displaystyle T}
9013:{\displaystyle Y}
8960:{\displaystyle X}
8940:{\displaystyle f}
8880:
8849:
8719:{\displaystyle Y}
8699:{\displaystyle T}
8657:
8640:
8576:{\displaystyle Y}
8552:categorical logic
8449:{\displaystyle x}
8429:{\displaystyle Y}
8409:{\displaystyle y}
8291:{\displaystyle X}
8271:{\displaystyle Y}
8251:{\displaystyle X}
8122:{\displaystyle Y}
8102:{\displaystyle T}
7974:Categorical logic
7810:) if and only if
7708:closure operators
7696:Galois connection
7493:homotopy category
7460:of maps from the
7366:universal algebra
7362:existence theorem
7342:topological space
7121:yields a functor
7052:yields a functor
7044:-module, and the
7034:ring homomorphism
6772:fibred coproducts
6571:cartesian product
6421:forgetful functor
6284:
6242:
6013:
5977:
5434:forgetful functor
5354:bilinear mappings
5346:abuse of notation
5173:zig-zag equations
4632:
4606:
4565:
4539:
4509:universal algebra
4325:
4321:
3831:{\displaystyle F}
3811:{\displaystyle G}
3791:{\displaystyle G}
3771:{\displaystyle F}
3748:{\displaystyle F}
3731:is then called a
3724:{\displaystyle G}
3704:{\displaystyle D}
3540:{\displaystyle F}
3379:{\displaystyle C}
3359:{\displaystyle X}
3282:{\displaystyle C}
3233:{\displaystyle D}
3213:{\displaystyle Y}
3193:{\displaystyle G}
3173:{\displaystyle Y}
3149:{\displaystyle D}
3129:{\displaystyle Y}
3070:{\displaystyle G}
3053:is then called a
3046:{\displaystyle F}
3026:{\displaystyle C}
2862:{\displaystyle G}
2694:{\displaystyle D}
2674:{\displaystyle Y}
2597:{\displaystyle D}
2548:{\displaystyle C}
2528:{\displaystyle X}
2508:{\displaystyle X}
2488:{\displaystyle F}
2464:{\displaystyle C}
2444:{\displaystyle X}
2222:with the functor
2210:actually exists.
1867:is how to turn a
1755:{\displaystyle U}
1735:{\displaystyle T}
1721:adjoint operators
1625:{\displaystyle G}
1568:{\displaystyle F}
1548:{\displaystyle F}
1528:{\displaystyle F}
1505:{\displaystyle F}
1485:{\displaystyle G}
1465:{\displaystyle F}
1441:{\displaystyle G}
1421:{\displaystyle G}
1397:{\displaystyle F}
1377:{\displaystyle f}
1275:{\displaystyle f}
1254:{\displaystyle G}
1234:{\displaystyle F}
908:{\displaystyle F}
895:right adjoint to
882:{\displaystyle G}
861:{\displaystyle G}
835:{\displaystyle F}
788:{\displaystyle Y}
612:{\displaystyle X}
585:
512:
446:{\displaystyle Y}
426:{\displaystyle X}
272:{\displaystyle Y}
228:{\displaystyle X}
72:topological space
16902:
16895:Adjoint functors
16790:
16783:
16776:
16767:
16766:
16753:
16752:
16743:
16742:
16733:
16732:
16568:
16567:
16546:Simplex category
16521:Categorification
16512:
16511:
16493:
16492:
16486:
16456:Product category
16441:Kleisli category
16436:Functor category
16281:Terminal objects
16269:
16268:
16204:Adjoint functors
16157:
16156:
16146:
16145:
16131:
16124:
16117:
16108:
16107:
16067:
16054:
16018:
15998:
15977:
15957:
15951:
15950:
15939:
15933:
15915:
15909:
15908:
15898:
15880:
15871:
15865:
15864:
15862:
15850:
15834:Kleisli category
15797:is given by ΞΌ =
15793:
15791:
15790:
15785:
15776:
15775:
15747:
15745:
15744:
15739:
15731:
15730:
15729:
15692:
15690:
15689:
15684:
15682:
15681:
15672:
15671:
15600:(i.e. a functor
15551:full subcategory
15382:
15380:
15379:
15374:
15354:
15353:
15352:
15346:
15313:
15312:
15311:
15305:
15290:
15289:
15247:additive functor
15146:small categories
15129:
15127:
15126:
15121:
15119:
15112:
15111:
15110:
15100:
15090:
15082:
15076:
15061:
15056:
15048:
15036:
15032:
15018:
15010:
15007:
14996:
14984:
14982:
14974:
14966:
14963:
14951:
14949:
14948:
14947:
14936:
14911:
14909:
14908:
14903:
14880:
14859:
14857:
14856:
14851:
14837:
14773:
14771:
14770:
14765:
14753:
14751:
14750:
14745:
14730:
14728:
14727:
14722:
14720:
14710:
14709:
14694:
14693:
14665:
14625:
14468:
14466:
14465:
14460:
14401:
14337:
14229:
14227:
14226:
14221:
14219:
14218:
14200:
14199:
14181:
14180:
14128:
14126:
14125:
14120:
14115:
14114:
14096:
14095:
14080:
14079:
13991:
13989:
13988:
13983:
13981:
13962:
13961:
13939:
13931:
13912:
13911:
13875:
13874:
13794:
13792:
13791:
13786:
13763:
13762:
13757:
13742:
13741:
13715: for each
13714:
13712:
13711:
13706:
13683:
13682:
13677:
13659:
13658:
13643:
13642:
13621:
13620:
13601:
13599:
13598:
13593:
13570:
13569:
13564:
13549:
13548:
13522: for each
13521:
13519:
13518:
13513:
13490:
13489:
13484:
13466:
13465:
13449:
13441:
13420:
13419:
13395:
13393:
13392:
13387:
13293:
13291:
13290:
13285:
13283:
13267:
13266:
13248:
13238:
13237:
13219:
13218:
13197:
13193:
13192:
13159:
13158:
13104:
13075:
13073:
13072:
13067:
13065:
13055:
13054:
13030:
13023:
13022:
13004:
13003:
12979:
12972:
12971:
12935:
12934:
12883:
12859:
12857:
12856:
12851:
12849:
12830:
12829:
12808:
12807:
12788:
12787:
12751:
12750:
12669:
12667:
12666:
12661:
12602:
12578:
12563:
12551:
12540:
12519:
12504:
12493:
12482:
12476:
12368:
12293:For each object
12244:
12182:
12076:For each object
12012:initial morphism
11967:
11965:
11964:
11959:
11957:
11953:
11952:
11934:
11933:
11911:
11910:
11891:
11890:
11872:
11871:
11852:
11851:
11820:
11818:
11817:
11812:
11810:
11779:
11778:
11773:
11752:
11751:
11732:
11731:
11716:
11715:
11666:
11665:
11660:
11639:
11638:
11619:
11618:
11602:
11594:
11548:
11547:
11542:
11521:
11520:
11480:
11479:
11430:
11429:
11424:
11388:
11387:
11361:
11353:
11052:
11050:
11049:
11044:
11032:
11030:
11029:
11024:
11022:
11010:
11008:
11007:
11002:
11000:
10988:
10986:
10985:
10980:
10972:
10960:
10958:
10957:
10952:
10941:
10927:
10915:
10913:
10912:
10907:
10905:
10904:
10869:
10867:
10866:
10861:
10859:
10858:
10809:
10807:
10806:
10801:
10796:
10772:
10770:
10769:
10764:
10728:
10726:
10725:
10720:
10715:
10697:
10695:
10694:
10689:
10687:
10675:
10673:
10672:
10667:
10662:
10641:
10639:
10638:
10633:
10588:
10586:
10585:
10580:
10568:
10566:
10565:
10560:
10524:
10522:
10521:
10516:
10514:
10475:
10473:
10472:
10467:
10455:
10453:
10452:
10447:
10435:
10433:
10432:
10427:
10415:
10413:
10412:
10407:
10396:with respect to
10395:
10393:
10392:
10387:
10369:
10367:
10366:
10361:
10349:
10347:
10346:
10341:
10329:
10327:
10326:
10321:
10316:
10315:
10295:
10293:
10292:
10287:
10247:
10246:
10201:
10200:
10179:
10177:
10176:
10171:
10157:
10156:
10129:
10128:
10083:
10081:
10080:
10075:
10020:
10019:
9974:
9973:
9955:
9953:
9952:
9947:
9926:
9924:
9923:
9918:
9900:
9898:
9897:
9892:
9869:
9868:
9849:
9847:
9846:
9841:
9829:
9827:
9826:
9821:
9819:
9818:
9802:
9800:
9799:
9794:
9782:
9780:
9779:
9774:
9772:
9771:
9755:
9753:
9752:
9747:
9720:
9718:
9717:
9712:
9701:
9700:
9675:
9673:
9672:
9667:
9640:
9638:
9637:
9632:
9621:
9620:
9595:
9593:
9592:
9587:
9569:
9567:
9566:
9561:
9550:
9549:
9516:
9515:
9490:
9488:
9487:
9482:
9448:
9446:
9445:
9440:
9429:
9428:
9398:
9397:
9375:
9373:
9372:
9367:
9356:
9355:
9337:
9317:
9316:
9304:
9290:
9288:
9287:
9282:
9261:
9259:
9258:
9253:
9236:
9235:
9216:
9214:
9213:
9208:
9196:
9194:
9193:
9188:
9176:
9174:
9173:
9168:
9156:
9154:
9153:
9148:
9136:
9134:
9133:
9128:
9116:
9114:
9113:
9108:
9096:
9094:
9093:
9088:
9083:
9082:
9064:
9063:
9047:
9045:
9044:
9039:
9019:
9017:
9016:
9011:
8999:
8997:
8996:
8991:
8986:
8985:
8966:
8964:
8963:
8958:
8946:
8944:
8943:
8938:
8926:
8924:
8923:
8918:
8900:
8898:
8897:
8892:
8881:
8878:
8869:
8867:
8866:
8861:
8850:
8847:
8838:
8836:
8835:
8830:
8828:
8827:
8811:
8809:
8808:
8803:
8801:
8800:
8784:
8782:
8781:
8776:
8771:
8770:
8751:
8749:
8748:
8743:
8725:
8723:
8722:
8717:
8705:
8703:
8702:
8697:
8677:
8675:
8674:
8669:
8658:
8655:
8641:
8638:
8633:
8632:
8614:
8612:
8611:
8606:
8582:
8580:
8579:
8574:
8554:, a subfield of
8549:
8547:
8546:
8541:
8529:
8527:
8526:
8521:
8509:
8507:
8506:
8501:
8499:
8498:
8482:
8480:
8479:
8474:
8472:
8471:
8455:
8453:
8452:
8447:
8435:
8433:
8432:
8427:
8415:
8413:
8412:
8407:
8396:of all elements
8391:
8389:
8388:
8383:
8369:
8368:
8341:
8340:
8297:
8295:
8294:
8289:
8277:
8275:
8274:
8269:
8257:
8255:
8254:
8249:
8237:
8235:
8234:
8229:
8227:
8226:
8201:
8199:
8198:
8193:
8167:
8166:
8145:
8144:
8128:
8126:
8125:
8120:
8108:
8106:
8105:
8100:
8088:
8086:
8085:
8080:
8062:
8060:
8059:
8054:
8040:
8039:
8008:
8006:
8005:
8000:
7998:
7997:
7953:the endofunctor
7641:coherent sheaves
7514:Hausdorff spaces
7458:homotopy classes
7456:, the space of
7433:trivial topology
7358:negative numbers
7260:Polynomial rings
7014:Ring extensions.
7005:
7003:
7002:
6997:
6929:Further examples
6506:diagonal functor
6387:
6385:
6384:
6379:
6374:
6373:
6343:
6341:
6340:
6335:
6333:
6332:
6301:
6299:
6298:
6293:
6285:
6278:
6277:
6257:
6243:
6239:
6238:
6221:
6203:the composition
6198:
6196:
6195:
6190:
6170:
6169:
6131:
6129:
6128:
6123:
6121:
6120:
6077:
6075:
6074:
6069:
6064:
6063:
6030:
6028:
6027:
6022:
6014:
6010:
6009:
5992:
5978:
5971:
5970:
5950:
5933:the composition
5928:
5926:
5925:
5920:
5900:
5899:
5877:
5875:
5874:
5869:
5772:
5770:
5769:
5764:
5744:
5743:
5711:
5709:
5708:
5703:
5698:
5697:
5661:
5659:
5658:
5653:
5633:
5632:
5579:
5577:
5576:
5571:
5551:
5550:
5514:
5512:
5511:
5506:
5486:
5485:
5319:
5317:
5316:
5311:
5294:was the functor
5210:
5208:
5207:
5202:
5151:
5149:
5148:
5143:
5141:
5140:
5117:
5115:
5114:
5109:
5107:
5106:
5090:
5088:
5087:
5082:
5080:
5079:
5066:
5064:
5063:
5058:
5056:
5055:
5054:
5033:
5031:
5030:
5025:
5023:
5019:
5018:
5000:
4999:
4977:
4976:
4957:
4956:
4938:
4937:
4918:
4917:
4872:
4870:
4869:
4864:
4862:
4836:
4835:
4800:
4799:
4762:
4760:
4759:
4754:
4736:
4734:
4733:
4728:
4646:
4644:
4643:
4638:
4633:
4618:
4607:
4592:
4579:
4577:
4576:
4571:
4566:
4551:
4540:
4525:
4495:
4493:
4492:
4487:
4485:
4472:
4471:
4470:
4446:
4445:
4444:
4372:consists of two
4323:
4319:
4260:
4239:
4217:category of sets
4210:
4196:
4182:
4119:for all objects
4115:
4113:
4112:
4107:
4087:
4086:
4081:
4048:
4047:
4042:
4027:
4026:
3997:
3995:
3994:
3989:
3969:
3968:
3963:
3930:
3929:
3924:
3893:
3865:consists of two
3837:
3835:
3834:
3829:
3817:
3815:
3814:
3809:
3797:
3795:
3794:
3789:
3777:
3775:
3774:
3769:
3754:
3752:
3751:
3746:
3730:
3728:
3727:
3722:
3710:
3708:
3707:
3702:
3690:
3688:
3687:
3682:
3680:
3653:
3651:
3650:
3645:
3637:
3636:
3635:
3619:
3618:
3578:
3576:
3575:
3570:
3546:
3544:
3543:
3538:
3515:
3513:
3512:
3507:
3499:
3498:
3467:
3465:
3464:
3459:
3426:
3424:
3423:
3418:
3385:
3383:
3382:
3377:
3365:
3363:
3362:
3357:
3345:
3343:
3342:
3337:
3305:
3304:
3288:
3286:
3285:
3280:
3268:
3266:
3265:
3260:
3239:
3237:
3236:
3231:
3219:
3217:
3216:
3211:
3199:
3197:
3196:
3191:
3179:
3177:
3176:
3171:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3111:
3109:
3108:
3103:
3076:
3074:
3073:
3068:
3052:
3050:
3049:
3044:
3032:
3030:
3029:
3024:
3012:
3010:
3009:
3004:
2996:
2975:
2973:
2972:
2967:
2965:
2964:
2963:
2917:
2916:
2900:
2898:
2897:
2892:
2868:
2866:
2865:
2860:
2830:
2828:
2827:
2822:
2799:
2798:
2782:
2780:
2779:
2774:
2741:
2739:
2738:
2733:
2700:
2698:
2697:
2692:
2680:
2678:
2677:
2672:
2660:
2658:
2657:
2652:
2620:
2619:
2603:
2601:
2600:
2595:
2583:
2581:
2580:
2575:
2554:
2552:
2551:
2546:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2470:
2468:
2467:
2462:
2450:
2448:
2447:
2442:
2426:
2424:
2423:
2418:
2233:The notion that
2094:to our problem:
1941:. The category
1906:if it defines a
1838:
1817:
1815:
1814:
1809:
1761:
1759:
1758:
1753:
1741:
1739:
1738:
1733:
1711:
1709:
1708:
1703:
1680:, we also write
1668:
1666:
1665:
1660:
1658:
1657:
1656:
1650:
1631:
1629:
1628:
1623:
1611:
1609:
1608:
1603:
1601:
1600:
1599:
1593:
1574:
1572:
1571:
1566:
1554:
1552:
1551:
1546:
1534:
1532:
1531:
1526:
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1471:
1469:
1468:
1463:
1447:
1445:
1444:
1439:
1427:
1425:
1424:
1419:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1359:
1357:
1356:
1351:
1336:
1334:
1333:
1328:
1308:
1307:
1306:
1300:
1281:
1279:
1278:
1273:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1214:
1212:
1211:
1206:
1191:
1189:
1188:
1183:
1163:
1162:
1161:
1155:
1122:
1121:
1120:
1114:
1099:
1098:
1044:
1042:
1041:
1036:
1034:
1033:
1020:
1018:
1017:
1012:
1010:
1009:
992:
990:
989:
984:
982:
981:
968:
966:
965:
960:
958:
957:
941:
939:
938:
933:
914:
912:
911:
906:
888:
886:
885:
880:
867:
865:
864:
859:
848:left adjoint to
841:
839:
838:
833:
818:
816:
815:
810:
808:
807:
794:
792:
791:
786:
774:
772:
771:
766:
764:
750:
749:
722:
721:
708:
706:
705:
700:
698:
684:
683:
656:
655:
642:
640:
639:
634:
632:
631:
618:
616:
615:
610:
598:
596:
595:
590:
588:
587:
586:
583:
567:
566:
539:
538:
525:
523:
522:
517:
515:
514:
513:
510:
494:
493:
466:
465:
452:
450:
449:
444:
432:
430:
429:
424:
405:
403:
402:
397:
377:
376:
375:
369:
336:
335:
334:
328:
302:
300:
299:
294:
292:
291:
278:
276:
275:
270:
258:
256:
255:
250:
248:
247:
234:
232:
231:
226:
211:
209:
208:
203:
201:
200:
191:
190:
171:
169:
168:
163:
161:
160:
151:
150:
124:
122:
121:
116:
114:
113:
100:
98:
97:
92:
90:
89:
50:, one being the
48:adjoint functors
25:Adjunction space
16910:
16909:
16905:
16904:
16903:
16901:
16900:
16899:
16885:
16884:
16883:
16878:
16800:
16794:
16764:
16759:
16713:
16683:
16650:
16627:
16618:
16575:
16559:
16510:
16500:
16487:
16470:
16419:
16357:
16326:Initial objects
16312:
16258:
16151:
16140:
16138:Category theory
16135:
16077:of The Catsters
16065:
16061:
16043:
16007:
15996:
15986:
15981:
15980:
15958:
15954:
15941:
15940:
15936:
15916:
15912:
15896:10.2307/1993102
15878:
15872:
15868:
15851:
15847:
15842:
15771:
15767:
15759:
15756:
15755:
15725:
15724:
15720:
15712:
15709:
15708:
15677:
15676:
15667:
15666:
15658:
15655:
15654:
15628:
15613:
15592:
15585:
15570:
15548:
15541:
15523:
15488:
15428:
15423:
15348:
15347:
15336:
15335:
15307:
15306:
15295:
15294:
15279:
15275:
15273:
15270:
15269:
15219:
15154:
15117:
15116:
15106:
15105:
15101:
15089:
15069:
15060:
15049:
15041:
15034:
15033:
15025:
15011:
15000:
14989:
14983:
14975:
14967:
14956:
14950:
14943:
14942:
14938:
14933:
14931:
14928:
14927:
14873:
14871:
14868:
14867:
14830:
14822:
14819:
14818:
14780:
14759:
14756:
14755:
14739:
14736:
14735:
14718:
14717:
14702:
14698:
14686:
14682:
14666:
14658:
14655:
14654:
14626:
14618:
14614:
14612:
14609:
14608:
14537:Conversely, if
14502:If the functor
14500:
14442:
14439:
14438:
14417:
14399:
14390:
14335:
14322:
14305:
14248:
14242:
14237:
14211:
14207:
14195:
14191:
14173:
14169:
14167:
14164:
14163:
14155:
14149:
14143:
14110:
14106:
14088:
14084:
14072:
14068:
14066:
14063:
14062:
14052:
14046:
14036:
13979:
13978:
13957:
13953:
13932:
13921:
13914:
13913:
13907:
13903:
13864:
13860:
13856:
13854:
13851:
13850:
13829:
13807:
13758:
13747:
13746:
13734:
13730:
13728:
13725:
13724:
13678:
13667:
13666:
13651:
13647:
13629:
13625:
13616:
13612:
13610:
13607:
13606:
13565:
13554:
13553:
13541:
13537:
13535:
13532:
13531:
13485:
13474:
13473:
13458:
13454:
13442:
13428:
13415:
13411:
13409:
13406:
13405:
13357:
13354:
13353:
13344:
13334:
13305:Given functors
13303:
13281:
13280:
13259:
13255:
13246:
13245:
13230:
13226:
13214:
13210:
13195:
13194:
13188:
13184:
13154:
13150:
13137:
13124:
13122:
13119:
13118:
13112:
13102:
13092:
13063:
13062:
13047:
13043:
13028:
13027:
13018:
13014:
12996:
12992:
12977:
12976:
12967:
12963:
12930:
12926:
12919:
12906:
12904:
12901:
12900:
12894:
12882:
12876:
12847:
12846:
12825:
12821:
12797:
12793:
12790:
12789:
12783:
12779:
12740:
12736:
12732:
12730:
12727:
12726:
12688:
12678:
12655:
12652:
12651:
12623:Given functors
12621:
12596:
12595:
12588:
12562:
12561:
12530:
12529:
12471:
12470:
12463:
12455:
12448:
12437:
12430:
12415:
12408:
12397:
12390:
12376:
12342:
12332:
12322:
12285:
12275:
12239:
12226:
12215:
12196:
12195:
12181:
12180:
12172:
12171:
12163:
12152:
12145:
12111:
12097:
12043:
12009:
11981:
11955:
11954:
11945:
11941:
11929:
11925:
11912:
11903:
11899:
11896:
11895:
11886:
11882:
11864:
11860:
11853:
11844:
11840:
11836:
11834:
11831:
11830:
11808:
11807:
11774:
11763:
11762:
11758:
11753:
11747:
11743:
11736:
11724:
11720:
11702:
11698:
11695:
11694:
11661:
11650:
11649:
11645:
11640:
11634:
11630:
11623:
11611:
11607:
11595:
11581:
11574:
11573:
11543:
11532:
11531:
11527:
11522:
11516:
11512:
11490:
11469:
11465:
11456:
11455:
11425:
11414:
11413:
11409:
11404:
11383:
11379:
11372:
11354:
11343:
11329:
11327:
11324:
11323:
11280:
11234:
11171:
11159:
11138:
11128:
11059:
11038:
11035:
11034:
11018:
11016:
11013:
11012:
10996:
10994:
10991:
10990:
10968:
10966:
10963:
10962:
10937:
10923:
10921:
10918:
10917:
10900:
10896:
10882:
10879:
10878:
10851:
10847:
10815:
10812:
10811:
10792:
10778:
10775:
10774:
10734:
10731:
10730:
10711:
10703:
10700:
10699:
10683:
10681:
10678:
10677:
10658:
10650:
10647:
10646:
10594:
10591:
10590:
10574:
10571:
10570:
10530:
10527:
10526:
10510:
10508:
10505:
10504:
10494:
10461:
10458:
10457:
10456:is replaced by
10441:
10438:
10437:
10421:
10418:
10417:
10401:
10398:
10397:
10375:
10372:
10371:
10355:
10352:
10351:
10335:
10332:
10331:
10311:
10307:
10305:
10302:
10301:
10239:
10235:
10196:
10192:
10190:
10187:
10186:
10152:
10148:
10124:
10120:
10093:
10090:
10089:
10012:
10008:
9969:
9965:
9963:
9960:
9959:
9932:
9929:
9928:
9906:
9903:
9902:
9861:
9857:
9855:
9852:
9851:
9835:
9832:
9831:
9814:
9810:
9808:
9805:
9804:
9788:
9785:
9784:
9767:
9763:
9761:
9758:
9757:
9726:
9723:
9722:
9693:
9689:
9681:
9678:
9677:
9646:
9643:
9642:
9641:, then clearly
9613:
9609:
9601:
9598:
9597:
9575:
9572:
9571:
9542:
9538:
9508:
9504:
9496:
9493:
9492:
9461:
9458:
9457:
9421:
9417:
9393:
9389:
9387:
9384:
9383:
9351:
9347:
9333:
9312:
9308:
9300:
9298:
9295:
9294:
9270:
9267:
9266:
9228:
9224:
9222:
9219:
9218:
9202:
9199:
9198:
9182:
9179:
9178:
9162:
9159:
9158:
9142:
9139:
9138:
9122:
9119:
9118:
9102:
9099:
9098:
9078:
9074:
9059:
9055:
9053:
9050:
9049:
9033:
9030:
9029:
9005:
9002:
9001:
8981:
8977:
8972:
8969:
8968:
8952:
8949:
8948:
8932:
8929:
8928:
8906:
8903:
8902:
8877:
8875:
8872:
8871:
8846:
8844:
8841:
8840:
8823:
8819:
8817:
8814:
8813:
8796:
8792:
8790:
8787:
8786:
8766:
8762:
8757:
8754:
8753:
8731:
8728:
8727:
8711:
8708:
8707:
8691:
8688:
8687:
8654:
8637:
8628:
8624:
8622:
8619:
8618:
8588:
8585:
8584:
8568:
8565:
8564:
8535:
8532:
8531:
8515:
8512:
8511:
8494:
8490:
8488:
8485:
8484:
8467:
8463:
8461:
8458:
8457:
8456:to which it is
8441:
8438:
8437:
8421:
8418:
8417:
8401:
8398:
8397:
8364:
8360:
8336:
8332:
8305:
8302:
8301:
8283:
8280:
8279:
8263:
8260:
8259:
8243:
8240:
8239:
8222:
8218:
8216:
8213:
8212:
8162:
8158:
8140:
8136:
8134:
8131:
8130:
8114:
8111:
8110:
8094:
8091:
8090:
8068:
8065:
8064:
8035:
8031:
8014:
8011:
8010:
7993:
7989:
7987:
7984:
7983:
7980:Quantification.
7976:
7922:
7871:
7869:Category theory
7802:is a subset of
7748:William Lawvere
7686:if and only if
7672:
7649:The article on
7647:Soberification.
7638:
7591:
7497:homotopy theory
7419:, creating the
7393:
7282:polynomial ring
7275:
7268:
7251:
7240:
7166:
7101:Tensor products
6991:
6988:
6987:
6958:and defining a
6936:
6931:
6870:
6863:
6820:
6813:
6765:
6700:
6692:
6685:
6679:
6670:
6661:
6652:
6645:
6634:
6627:
6620:
6613:
6606:
6599:
6551:
6544:
6537:
6530:
6496:
6489:
6481:
6475:
6440:fibred products
6433:
6414:
6369:
6365:
6357:
6354:
6353:
6325:
6321:
6319:
6316:
6315:
6273:
6269:
6256:
6231:
6227:
6220:
6212:
6209:
6208:
6165:
6161:
6159:
6156:
6155:
6113:
6109:
6107:
6104:
6103:
6059:
6055:
6047:
6044:
6043:
6002:
5998:
5991:
5966:
5962:
5949:
5941:
5938:
5937:
5895:
5891:
5889:
5886:
5885:
5839:
5836:
5835:
5739:
5735:
5733:
5730:
5729:
5693:
5689:
5675:
5672:
5671:
5628:
5624:
5622:
5619:
5618:
5601:For each group
5546:
5542:
5540:
5537:
5536:
5481:
5477:
5475:
5472:
5471:
5383:
5378:
5320:(i.e. take the
5299:
5296:
5295:
5244:
5184:
5181:
5180:
5177:string diagrams
5133:
5129:
5127:
5124:
5123:
5122:to itself, and
5102:
5098:
5096:
5093:
5092:
5075:
5074:
5072:
5069:
5068:
5050:
5049:
5045:
5043:
5040:
5039:
5021:
5020:
5011:
5007:
4995:
4991:
4978:
4969:
4965:
4962:
4961:
4952:
4948:
4930:
4926:
4919:
4910:
4906:
4902:
4900:
4897:
4896:
4860:
4859:
4837:
4831:
4827:
4824:
4823:
4801:
4795:
4791:
4787:
4785:
4782:
4781:
4742:
4739:
4738:
4698:
4695:
4694:
4661:
4655:
4617:
4591:
4586:
4583:
4582:
4550:
4524:
4519:
4516:
4515:
4483:
4482:
4466:
4465:
4461:
4454:
4448:
4447:
4440:
4439:
4435:
4419:
4412:
4410:
4407:
4406:
4358:
4335:
4303:
4291:
4276:Naturality of Ξ¦
4245:
4227:
4202:
4190:
4184:
4176:
4170:
4082:
4071:
4070:
4043:
4032:
4031:
4016:
4012:
4010:
4007:
4006:
3964:
3953:
3952:
3925:
3914:
3913:
3905:
3902:
3901:
3881:
3848:
3823:
3820:
3819:
3803:
3800:
3799:
3798:if and only if
3783:
3780:
3779:
3763:
3760:
3759:
3740:
3737:
3736:
3716:
3713:
3712:
3696:
3693:
3692:
3673:
3659:
3656:
3655:
3628:
3627:
3623:
3614:
3610:
3584:
3581:
3580:
3552:
3549:
3548:
3532:
3529:
3528:
3494:
3490:
3473:
3470:
3469:
3432:
3429:
3428:
3391:
3388:
3387:
3371:
3368:
3367:
3351:
3348:
3347:
3300:
3296:
3294:
3291:
3290:
3289:and a morphism
3274:
3271:
3270:
3245:
3242:
3241:
3225:
3222:
3221:
3205:
3202:
3201:
3185:
3182:
3181:
3165:
3162:
3161:
3141:
3138:
3137:
3121:
3118:
3117:
3085:
3082:
3081:
3062:
3059:
3058:
3038:
3035:
3034:
3018:
3015:
3014:
2989:
2981:
2978:
2977:
2956:
2955:
2951:
2912:
2908:
2906:
2903:
2902:
2874:
2871:
2870:
2854:
2851:
2850:
2794:
2790:
2788:
2785:
2784:
2747:
2744:
2743:
2706:
2703:
2702:
2686:
2683:
2682:
2666:
2663:
2662:
2615:
2611:
2609:
2606:
2605:
2604:and a morphism
2589:
2586:
2585:
2560:
2557:
2556:
2540:
2537:
2536:
2520:
2517:
2516:
2500:
2497:
2496:
2480:
2477:
2476:
2471:there exists a
2456:
2453:
2452:
2436:
2433:
2432:
2400:
2397:
2396:
2393:
2310:
2282:
2216:
2145:adjoint functor
2107:
2100:
2093:
2086:
2079:
2068:
2049:
2045:
2041:
2034:
2027:
2016:
2001:
1990:
1853:
1845:colimits/limits
1839:
1831:
1824:
1767:
1764:
1763:
1747:
1744:
1743:
1727:
1724:
1723:
1688:
1685:
1684:
1652:
1651:
1640:
1639:
1637:
1634:
1633:
1617:
1614:
1613:
1595:
1594:
1583:
1582:
1580:
1577:
1576:
1560:
1557:
1556:
1540:
1537:
1536:
1520:
1517:
1516:
1497:
1494:
1493:
1477:
1474:
1473:
1457:
1454:
1453:
1433:
1430:
1429:
1413:
1410:
1409:
1389:
1386:
1385:
1369:
1366:
1365:
1342:
1339:
1338:
1302:
1301:
1290:
1289:
1287:
1284:
1283:
1282:is an arrow in
1267:
1264:
1263:
1246:
1243:
1242:
1226:
1223:
1222:
1200:
1197:
1196:
1157:
1156:
1145:
1144:
1116:
1115:
1104:
1103:
1091:
1087:
1085:
1082:
1081:
1051:
1029:
1028:
1026:
1023:
1022:
1005:
1004:
1002:
999:
998:
977:
976:
974:
971:
970:
953:
952:
950:
947:
946:
921:
918:
917:
900:
897:
896:
874:
871:
870:
853:
850:
849:
827:
824:
823:
803:
802:
800:
797:
796:
780:
777:
776:
754:
745:
744:
717:
716:
714:
711:
710:
688:
679:
678:
651:
650:
648:
645:
644:
627:
626:
624:
621:
620:
604:
601:
600:
582:
578:
571:
562:
561:
534:
533:
531:
528:
527:
509:
505:
498:
489:
488:
461:
460:
458:
455:
454:
438:
435:
434:
418:
415:
414:
371:
370:
359:
358:
330:
329:
318:
317:
315:
312:
311:
287:
286:
284:
281:
280:
264:
261:
260:
243:
242:
240:
237:
236:
220:
217:
216:
196:
195:
186:
185:
177:
174:
173:
156:
155:
146:
145:
137:
134:
133:
109:
108:
106:
103:
102:
85:
84:
82:
79:
78:
36:category theory
34:, specifically
28:
17:
12:
11:
5:
16908:
16898:
16897:
16880:
16879:
16877:
16876:
16871:
16866:
16861:
16856:
16851:
16846:
16841:
16836:
16831:
16826:
16821:
16816:
16811:
16805:
16802:
16801:
16793:
16792:
16785:
16778:
16770:
16761:
16760:
16758:
16757:
16747:
16737:
16726:
16723:
16722:
16719:
16718:
16715:
16714:
16712:
16711:
16706:
16701:
16687:
16681:
16676:
16671:
16665:
16663:
16656:
16655:
16652:
16651:
16649:
16648:
16643:
16632:
16630:
16625:
16620:
16619:
16617:
16616:
16611:
16606:
16601:
16596:
16591:
16580:
16578:
16573:
16565:
16561:
16560:
16558:
16553:
16551:String diagram
16548:
16543:
16541:Model category
16538:
16533:
16528:
16523:
16518:
16516:
16509:
16508:
16505:
16502:
16501:
16489:
16488:
16481:
16479:
16476:
16475:
16472:
16471:
16469:
16468:
16463:
16461:Comma category
16458:
16453:
16448:
16443:
16438:
16433:
16427:
16425:
16421:
16420:
16418:
16417:
16407:
16397:
16395:Abelian groups
16392:
16387:
16382:
16377:
16371:
16369:
16363:
16362:
16359:
16358:
16356:
16355:
16350:
16345:
16344:
16343:
16333:
16328:
16322:
16320:
16314:
16313:
16311:
16310:
16305:
16300:
16299:
16298:
16288:
16283:
16277:
16275:
16266:
16260:
16259:
16257:
16256:
16251:
16246:
16241:
16236:
16231:
16226:
16221:
16216:
16211:
16206:
16201:
16200:
16199:
16194:
16189:
16184:
16179:
16174:
16163:
16161:
16153:
16152:
16142:
16141:
16134:
16133:
16126:
16119:
16111:
16105:
16104:
16091:, categories,
16078:
16060:
16059:External links
16057:
16056:
16055:
16041:
16019:
16005:
15985:
15982:
15979:
15978:
15952:
15934:
15910:
15889:(2): 294β329.
15866:
15844:
15843:
15841:
15838:
15795:
15794:
15782:
15779:
15774:
15770:
15766:
15763:
15749:
15748:
15737:
15734:
15728:
15723:
15719:
15716:
15694:
15693:
15680:
15675:
15670:
15665:
15662:
15650:. The functor
15627:
15624:
15609:
15590:
15583:
15566:
15546:
15537:
15521:
15487:
15484:
15427:
15424:
15422:
15419:
15384:
15383:
15372:
15369:
15366:
15363:
15360:
15357:
15351:
15345:
15342:
15339:
15334:
15331:
15328:
15325:
15322:
15319:
15316:
15310:
15304:
15301:
15298:
15293:
15288:
15285:
15282:
15278:
15218:
15215:
15214:
15213:
15206:
15199:
15192:
15153:
15150:
15131:
15130:
15115:
15109:
15104:
15097:
15093:
15088:
15085:
15079:
15075:
15072:
15068:
15064:
15059:
15055:
15052:
15047:
15044:
15040:
15037:
15035:
15031:
15028:
15024:
15021:
15017:
15014:
15006:
15003:
14999:
14995:
14992:
14987:
14981:
14978:
14973:
14970:
14962:
14959:
14954:
14946:
14941:
14937:
14935:
14913:
14912:
14901:
14898:
14895:
14892:
14889:
14886:
14883:
14879:
14876:
14861:
14860:
14849:
14846:
14843:
14840:
14836:
14833:
14829:
14826:
14779:
14776:
14763:
14743:
14732:
14731:
14716:
14713:
14708:
14705:
14701:
14697:
14692:
14689:
14685:
14681:
14678:
14675:
14672:
14669:
14667:
14664:
14661:
14657:
14656:
14653:
14650:
14647:
14644:
14641:
14638:
14635:
14632:
14629:
14627:
14624:
14621:
14617:
14616:
14594:
14593:
14582:
14499:
14496:
14495:
14494:
14481:
14458:
14455:
14452:
14449:
14446:
14428:
14427:
14413:
14392:
14391:
14386:
14365:
14364:
14325:
14324:
14318:
14301:
14276:Peter J. Freyd
14241:
14238:
14236:
14233:
14232:
14231:
14217:
14214:
14210:
14206:
14203:
14198:
14194:
14190:
14187:
14184:
14179:
14176:
14172:
14161:
14151:
14145:
14141:
14130:
14118:
14113:
14109:
14105:
14102:
14099:
14094:
14091:
14087:
14083:
14078:
14075:
14071:
14059:
14058:
14048:
14038:
14032:
14020:
14019:
13992:
13977:
13974:
13971:
13968:
13965:
13960:
13956:
13952:
13949:
13946:
13943:
13938:
13935:
13930:
13927:
13924:
13920:
13916:
13915:
13910:
13906:
13902:
13899:
13896:
13893:
13890:
13887:
13884:
13881:
13878:
13873:
13870:
13867:
13863:
13859:
13858:
13847:
13846:
13843:
13825:
13803:
13796:
13784:
13781:
13778:
13775:
13772:
13769:
13766:
13761:
13756:
13753:
13750:
13745:
13740:
13737:
13733:
13723:, where
13704:
13701:
13698:
13695:
13692:
13689:
13686:
13681:
13676:
13673:
13670:
13665:
13662:
13657:
13654:
13650:
13646:
13641:
13638:
13635:
13632:
13628:
13624:
13619:
13615:
13603:
13591:
13588:
13585:
13582:
13579:
13576:
13573:
13568:
13563:
13560:
13557:
13552:
13547:
13544:
13540:
13530:, where
13511:
13508:
13505:
13502:
13499:
13496:
13493:
13488:
13483:
13480:
13477:
13472:
13469:
13464:
13461:
13457:
13453:
13448:
13445:
13440:
13437:
13434:
13431:
13427:
13423:
13418:
13414:
13398:
13397:
13385:
13382:
13379:
13376:
13373:
13370:
13367:
13364:
13361:
13340:
13330:
13302:
13299:
13298:
13297:
13294:
13279:
13276:
13273:
13270:
13265:
13262:
13258:
13254:
13251:
13249:
13247:
13244:
13241:
13236:
13233:
13229:
13225:
13222:
13217:
13213:
13209:
13206:
13203:
13200:
13198:
13196:
13191:
13187:
13183:
13180:
13177:
13174:
13171:
13168:
13165:
13162:
13157:
13153:
13149:
13146:
13143:
13140:
13138:
13136:
13133:
13130:
13127:
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13088:
13080:
13079:
13076:
13061:
13058:
13053:
13050:
13046:
13042:
13039:
13036:
13033:
13031:
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13026:
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13017:
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13010:
13007:
13002:
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12988:
12985:
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12909:
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12890:
12878:
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12864:
12863:
12860:
12845:
12842:
12839:
12836:
12833:
12828:
12824:
12820:
12817:
12814:
12811:
12806:
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12800:
12796:
12792:
12791:
12786:
12782:
12778:
12775:
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12769:
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12763:
12760:
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12609:
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12586:
12582:
12559:
12555:
12527:
12523:
12468:
12461:
12457:
12453:
12446:
12435:
12428:
12413:
12406:
12402:, any objects
12395:
12388:
12378:
12372:
12334:
12324:
12318:
12301:, each object
12281:
12271:
12268:
12267:
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12193:
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12178:
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11998:
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10978:
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10930:
10926:
10903:
10899:
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10850:
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10840:
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9983:
9980:
9977:
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9968:
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9913:
9910:
9890:
9887:
9884:
9881:
9878:
9875:
9872:
9867:
9864:
9860:
9850:'s, such that
9839:
9817:
9813:
9792:
9770:
9766:
9745:
9742:
9739:
9736:
9733:
9730:
9710:
9707:
9704:
9699:
9696:
9692:
9688:
9685:
9665:
9662:
9659:
9656:
9653:
9650:
9630:
9627:
9624:
9619:
9616:
9612:
9608:
9605:
9585:
9582:
9579:
9559:
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9548:
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9541:
9537:
9534:
9531:
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9525:
9522:
9519:
9514:
9511:
9507:
9503:
9500:
9480:
9477:
9474:
9471:
9468:
9465:
9453:
9452:
9451:
9450:
9438:
9435:
9432:
9427:
9424:
9420:
9416:
9413:
9410:
9407:
9404:
9401:
9396:
9392:
9378:
9377:
9376:
9365:
9362:
9359:
9354:
9350:
9346:
9343:
9340:
9336:
9332:
9329:
9326:
9323:
9320:
9315:
9311:
9307:
9303:
9280:
9277:
9274:
9263:
9251:
9248:
9245:
9242:
9239:
9234:
9231:
9227:
9206:
9186:
9166:
9146:
9126:
9106:
9086:
9081:
9077:
9073:
9070:
9067:
9062:
9058:
9037:
9022:
9021:
9009:
8989:
8984:
8980:
8976:
8956:
8936:
8916:
8913:
8910:
8890:
8887:
8884:
8859:
8856:
8853:
8826:
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8774:
8769:
8765:
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8715:
8695:
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8679:
8678:
8667:
8664:
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8650:
8647:
8644:
8636:
8631:
8627:
8604:
8601:
8598:
8595:
8592:
8572:
8560:
8559:
8539:
8519:
8497:
8493:
8470:
8466:
8445:
8425:
8405:
8394:
8393:
8392:
8381:
8378:
8375:
8372:
8367:
8363:
8359:
8356:
8353:
8350:
8347:
8344:
8339:
8335:
8330:
8327:
8324:
8321:
8318:
8315:
8312:
8309:
8287:
8267:
8247:
8225:
8221:
8204:
8203:
8191:
8188:
8185:
8182:
8179:
8176:
8173:
8170:
8165:
8161:
8157:
8154:
8151:
8148:
8143:
8139:
8118:
8098:
8078:
8075:
8072:
8052:
8049:
8046:
8043:
8038:
8034:
8030:
8027:
8024:
8021:
8018:
7996:
7992:
7975:
7972:
7971:
7970:
7944:
7920:
7914:
7870:
7867:
7866:
7865:
7860:is adjoint to
7846:ideal quotient
7831:
7744:
7733:
7671:
7668:
7667:
7666:
7644:
7636:
7589:
7568:continuous map
7561:
7500:
7440:
7421:discrete space
7392:
7389:
7388:
7387:
7373:
7338:vector bundles
7327:
7312:abelianization
7288:Abelianization
7285:
7273:
7266:
7257:
7249:
7238:
7229:
7207:group of units
7184:
7162:
7137:. The functor
7109:is a ring and
7097:
7046:tensor product
7011:
6995:
6982:disjoint union
6980:by taking the
6963:
6935:
6932:
6930:
6927:
6926:
6925:
6905:
6904:
6868:
6861:
6818:
6811:
6764:
6761:
6760:
6759:
6754:
6753:
6696:
6690:
6683:
6675:
6666:
6657:
6650:
6643:
6632:
6625:
6618:
6611:
6604:
6597:
6583:
6582:
6566:
6565:
6549:
6542:
6535:
6528:
6494:
6487:
6479:
6473:
6432:
6429:
6413:
6410:
6377:
6372:
6368:
6364:
6361:
6331:
6328:
6324:
6304:
6303:
6291:
6288:
6281:
6276:
6272:
6268:
6265:
6260:
6255:
6252:
6249:
6246:
6237:
6234:
6230:
6224:
6219:
6216:
6188:
6185:
6182:
6179:
6176:
6173:
6168:
6164:
6119:
6116:
6112:
6067:
6062:
6058:
6054:
6051:
6032:
6031:
6020:
6017:
6008:
6005:
6001:
5995:
5990:
5987:
5984:
5981:
5974:
5969:
5965:
5961:
5958:
5953:
5948:
5945:
5918:
5915:
5912:
5909:
5906:
5903:
5898:
5894:
5867:
5864:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5762:
5759:
5756:
5753:
5750:
5747:
5742:
5738:
5701:
5696:
5692:
5688:
5685:
5682:
5679:
5651:
5648:
5645:
5642:
5639:
5636:
5631:
5627:
5569:
5566:
5563:
5560:
5557:
5554:
5549:
5545:
5531:of some group
5504:
5501:
5498:
5495:
5492:
5489:
5484:
5480:
5382:
5379:
5377:
5374:
5322:tensor product
5309:
5306:
5303:
5288:abelian groups
5284:
5283:
5243:
5240:
5200:
5197:
5194:
5191:
5188:
5139:
5136:
5132:
5105:
5101:
5078:
5053:
5048:
5036:
5035:
5017:
5014:
5010:
5006:
5003:
4998:
4994:
4990:
4987:
4984:
4981:
4979:
4975:
4972:
4968:
4964:
4963:
4960:
4955:
4951:
4947:
4944:
4941:
4936:
4933:
4929:
4925:
4922:
4920:
4916:
4913:
4909:
4905:
4904:
4874:
4873:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4838:
4834:
4830:
4826:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4802:
4798:
4794:
4790:
4789:
4752:
4749:
4746:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4670:respectively.
4657:
4651:
4648:
4647:
4636:
4629:
4626:
4621:
4616:
4613:
4610:
4603:
4600:
4595:
4590:
4580:
4569:
4562:
4559:
4554:
4549:
4546:
4543:
4536:
4533:
4528:
4523:
4497:
4496:
4481:
4478:
4475:
4469:
4464:
4460:
4457:
4455:
4453:
4450:
4449:
4443:
4438:
4434:
4431:
4428:
4425:
4422:
4420:
4418:
4415:
4414:
4357:
4354:
4352:) is similar.
4333:
4312:) is given by
4301:
4289:
4186:
4172:
4117:
4116:
4105:
4102:
4099:
4096:
4093:
4090:
4085:
4080:
4077:
4074:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4046:
4041:
4038:
4035:
4030:
4025:
4022:
4019:
4015:
4000:
3999:
3987:
3984:
3981:
3978:
3975:
3972:
3967:
3962:
3959:
3956:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3928:
3923:
3920:
3917:
3912:
3909:
3847:
3844:
3827:
3807:
3787:
3767:
3744:
3720:
3700:
3691:a morphism in
3679:
3676:
3672:
3669:
3666:
3663:
3643:
3640:
3634:
3631:
3626:
3622:
3617:
3613:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3568:
3565:
3562:
3559:
3556:
3536:
3505:
3502:
3497:
3493:
3489:
3486:
3483:
3480:
3477:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3375:
3355:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3303:
3299:
3278:
3258:
3255:
3252:
3249:
3229:
3209:
3189:
3169:
3145:
3125:
3101:
3098:
3095:
3092:
3089:
3066:
3042:
3022:
3002:
2999:
2995:
2992:
2988:
2985:
2962:
2959:
2954:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2915:
2911:
2890:
2887:
2884:
2881:
2878:
2858:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2797:
2793:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2690:
2670:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2618:
2614:
2593:
2573:
2570:
2567:
2564:
2544:
2524:
2504:
2484:
2460:
2440:
2416:
2413:
2410:
2407:
2404:
2392:
2389:
2355:, the letters
2309:
2306:
2301:
2300:
2297:
2290:
2289:optimizations.
2281:
2278:
2215:
2212:
2137:most efficient
2130:most efficient
2116:is initial in
2105:
2098:
2091:
2084:
2077:
2066:
2052:comma category
2047:
2043:
2039:
2032:
2025:
2014:
1999:
1988:
1935:most efficient
1923:initial object
1896:most efficient
1877:most efficient
1857:most efficient
1852:
1849:
1829:
1823:
1820:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1751:
1731:
1713:
1712:
1701:
1698:
1695:
1692:
1655:
1649:
1646:
1643:
1621:
1598:
1592:
1589:
1586:
1564:
1544:
1524:
1501:
1481:
1461:
1437:
1417:
1393:
1373:
1349:
1346:
1326:
1323:
1320:
1317:
1314:
1311:
1305:
1299:
1296:
1293:
1271:
1250:
1230:
1204:
1193:
1192:
1181:
1178:
1175:
1172:
1169:
1166:
1160:
1154:
1151:
1148:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1119:
1113:
1110:
1107:
1102:
1097:
1094:
1090:
1050:
1047:
1032:
1008:
980:
956:
931:
928:
925:
904:
878:
857:
831:
806:
784:
763:
760:
757:
753:
748:
743:
740:
737:
734:
731:
728:
725:
720:
697:
694:
691:
687:
682:
677:
674:
671:
668:
665:
662:
659:
654:
630:
608:
581:
577:
574:
570:
565:
560:
557:
554:
551:
548:
545:
542:
537:
508:
504:
501:
497:
492:
487:
484:
481:
478:
475:
472:
469:
464:
442:
422:
407:
406:
395:
392:
389:
386:
383:
380:
374:
368:
365:
362:
357:
354:
351:
348:
345:
342:
339:
333:
327:
324:
321:
290:
268:
246:
224:
213:
212:
199:
194:
189:
184:
181:
159:
154:
149:
144:
141:
112:
88:
15:
9:
6:
4:
3:
2:
16907:
16896:
16893:
16892:
16890:
16875:
16872:
16870:
16869:Representable
16867:
16865:
16862:
16860:
16857:
16855:
16852:
16850:
16847:
16845:
16842:
16840:
16837:
16835:
16832:
16830:
16827:
16825:
16822:
16820:
16817:
16815:
16812:
16810:
16807:
16806:
16803:
16798:
16791:
16786:
16784:
16779:
16777:
16772:
16771:
16768:
16756:
16748:
16746:
16738:
16736:
16728:
16727:
16724:
16710:
16707:
16705:
16702:
16700:
16696:
16692:
16688:
16686:
16684:
16677:
16675:
16672:
16670:
16667:
16666:
16664:
16661:
16657:
16647:
16644:
16641:
16637:
16634:
16633:
16631:
16629:
16621:
16615:
16612:
16610:
16607:
16605:
16602:
16600:
16599:Tetracategory
16597:
16595:
16592:
16589:
16588:pseudofunctor
16585:
16582:
16581:
16579:
16577:
16569:
16566:
16562:
16557:
16554:
16552:
16549:
16547:
16544:
16542:
16539:
16537:
16534:
16532:
16529:
16527:
16524:
16522:
16519:
16517:
16513:
16507:
16506:
16503:
16499:
16494:
16490:
16485:
16467:
16464:
16462:
16459:
16457:
16454:
16452:
16449:
16447:
16444:
16442:
16439:
16437:
16434:
16432:
16431:Free category
16429:
16428:
16426:
16422:
16415:
16414:Vector spaces
16411:
16408:
16405:
16401:
16398:
16396:
16393:
16391:
16388:
16386:
16383:
16381:
16378:
16376:
16373:
16372:
16370:
16368:
16364:
16354:
16351:
16349:
16346:
16342:
16339:
16338:
16337:
16334:
16332:
16329:
16327:
16324:
16323:
16321:
16319:
16315:
16309:
16308:Inverse limit
16306:
16304:
16301:
16297:
16294:
16293:
16292:
16289:
16287:
16284:
16282:
16279:
16278:
16276:
16274:
16270:
16267:
16265:
16261:
16255:
16252:
16250:
16247:
16245:
16242:
16240:
16237:
16235:
16234:Kan extension
16232:
16230:
16227:
16225:
16222:
16220:
16217:
16215:
16212:
16210:
16207:
16205:
16202:
16198:
16195:
16193:
16190:
16188:
16185:
16183:
16180:
16178:
16175:
16173:
16170:
16169:
16168:
16165:
16164:
16162:
16158:
16154:
16147:
16143:
16139:
16132:
16127:
16125:
16120:
16118:
16113:
16112:
16109:
16102:
16098:
16094:
16090:
16086:
16082:
16079:
16076:
16075:Eugenia Cheng
16072:
16068:
16063:
16062:
16052:
16048:
16044:
16042:0-387-98403-8
16038:
16034:
16030:
16029:
16024:
16020:
16016:
16012:
16008:
16006:0-471-60922-6
16002:
15995:
15994:
15988:
15987:
15976:
15973:
15972:0-387-97710-4
15969:
15965:
15961:
15956:
15948:
15944:
15938:
15931:
15927:
15923:
15919:
15914:
15906:
15902:
15897:
15892:
15888:
15884:
15877:
15870:
15861:
15860:q-alg/9609018
15856:
15849:
15845:
15837:
15835:
15831:
15826:
15824:
15820:
15816:
15812:
15808:
15804:
15800:
15780:
15772:
15768:
15764:
15761:
15754:
15753:
15752:
15735:
15721:
15717:
15714:
15707:
15706:
15705:
15703:
15699:
15663:
15660:
15653:
15652:
15651:
15649:
15645:
15641:
15637:
15633:
15623:
15621:
15617:
15612:
15607:
15603:
15599:
15594:
15589:
15582:
15578:
15574:
15569:
15564:
15560:
15556:
15552:
15545:
15540:
15535:
15531:
15527:
15520:
15516:
15512:
15507:
15505:
15501:
15497:
15493:
15490:If a functor
15483:
15481:
15477:
15471:
15469:
15465:
15461:
15457:
15453:
15449:
15445:
15441:
15437:
15433:
15421:Relationships
15418:
15416:
15412:
15408:
15404:
15399:
15397:
15393:
15389:
15367:
15364:
15361:
15358:
15332:
15326:
15323:
15320:
15317:
15291:
15286:
15283:
15280:
15268:
15267:
15266:
15264:
15260:
15256:
15252:
15248:
15244:
15240:
15236:
15232:
15228:
15224:
15211:
15207:
15204:
15200:
15197:
15193:
15190:
15186:
15185:
15184:
15181:
15179:
15175:
15171:
15167:
15163:
15159:
15149:
15147:
15143:
15138:
15136:
15113:
15102:
15095:
15091:
15086:
15083:
15077:
15073:
15070:
15066:
15062:
15057:
15053:
15050:
15045:
15042:
15038:
15029:
15026:
15022:
15019:
15015:
15012:
15004:
15001:
14997:
14993:
14990:
14985:
14979:
14976:
14971:
14968:
14960:
14957:
14952:
14939:
14926:
14925:
14924:
14922:
14918:
14899:
14896:
14890:
14887:
14884:
14881:
14877:
14874:
14866:
14865:
14864:
14847:
14841:
14838:
14834:
14831:
14827:
14824:
14817:
14816:
14815:
14813:
14809:
14805:
14801:
14797:
14793:
14789:
14785:
14775:
14761:
14741:
14714:
14706:
14703:
14699:
14695:
14690:
14687:
14683:
14676:
14673:
14670:
14668:
14662:
14659:
14651:
14648:
14642:
14639:
14636:
14630:
14628:
14622:
14619:
14607:
14606:
14605:
14603:
14599:
14591:
14587:
14583:
14580:
14576:
14572:
14571:
14570:
14568:
14564:
14560:
14556:
14552:
14548:
14544:
14540:
14535:
14533:
14529:
14525:
14521:
14517:
14513:
14509:
14505:
14493:
14489:
14485:
14482:
14479:
14475:
14472:
14471:
14470:
14456:
14450:
14447:
14444:
14436:
14431:
14425:
14421:
14416:
14412:
14408:
14405:
14404:
14403:
14397:
14389:
14385:
14381:
14377:
14373:
14370:
14369:
14368:
14362:
14358:
14354:
14350:
14347:
14346:
14345:
14343:
14342:
14334:
14330:
14321:
14317:
14313:
14309:
14304:
14300:
14297:
14296:
14295:
14293:
14289:
14285:
14281:
14277:
14273:
14269:
14265:
14261:
14257:
14253:
14247:
14215:
14212:
14208:
14204:
14196:
14192:
14185:
14182:
14177:
14174:
14170:
14162:
14159:
14154:
14148:
14139:
14135:
14131:
14111:
14107:
14100:
14097:
14092:
14089:
14085:
14081:
14076:
14073:
14069:
14061:
14060:
14056:
14051:
14045:
14041:
14035:
14030:
14026:
14023:Substituting
14022:
14021:
14017:
14013:
14009:
14005:
14001:
13997:
13993:
13972:
13966:
13963:
13958:
13954:
13950:
13944:
13936:
13933:
13928:
13925:
13922:
13908:
13904:
13900:
13894:
13888:
13885:
13879:
13871:
13868:
13865:
13849:
13848:
13844:
13841:
13837:
13833:
13828:
13823:
13819:
13815:
13811:
13806:
13801:
13797:
13779:
13776:
13773:
13770:
13767:
13759:
13743:
13738:
13735:
13731:
13722:
13718:
13699:
13696:
13693:
13690:
13687:
13679:
13663:
13655:
13652:
13648:
13639:
13636:
13633:
13630:
13622:
13617:
13613:
13604:
13586:
13583:
13580:
13577:
13574:
13566:
13550:
13545:
13542:
13538:
13529:
13525:
13506:
13503:
13500:
13497:
13494:
13486:
13470:
13462:
13459:
13455:
13446:
13443:
13438:
13435:
13432:
13429:
13421:
13416:
13412:
13403:
13402:
13401:
13383:
13380:
13377:
13374:
13368:
13365:
13362:
13352:
13351:
13350:
13348:
13343:
13338:
13333:
13328:
13324:
13320:
13316:
13312:
13308:
13295:
13277:
13274:
13271:
13268:
13263:
13260:
13256:
13252:
13250:
13242:
13239:
13234:
13231:
13227:
13223:
13215:
13211:
13204:
13201:
13199:
13189:
13185:
13181:
13175:
13169:
13166:
13163:
13155:
13151:
13144:
13141:
13139:
13134:
13117:
13116:
13111:
13101:
13096:
13091:
13086:
13082:
13081:
13077:
13059:
13056:
13051:
13048:
13044:
13040:
13037:
13034:
13032:
13019:
13015:
13008:
13005:
13000:
12997:
12993:
12989:
12986:
12983:
12981:
12968:
12964:
12957:
12954:
12948:
12942:
12939:
12936:
12931:
12927:
12923:
12921:
12916:
12899:
12898:
12893:
12888:
12881:
12875:
12870:
12866:
12865:
12861:
12840:
12834:
12831:
12826:
12822:
12818:
12812:
12804:
12801:
12798:
12784:
12780:
12776:
12770:
12764:
12761:
12755:
12747:
12744:
12741:
12725:
12724:
12720:
12716:
12712:
12708:
12704:
12700:
12696:
12695:
12694:
12692:
12687:
12682:
12677:
12672:
12657:
12650:
12646:
12642:
12638:
12634:
12630:
12626:
12616:
12607:
12600:
12592:
12585:
12576:
12572:
12568:
12558:
12549:
12545:
12538:
12534:
12526:
12517:
12513:
12509:
12502:
12498:
12491:
12487:
12481:
12475:
12467:
12460:
12452:
12445:
12441:
12434:
12427:
12423:
12419:
12412:
12405:
12401:
12394:
12387:
12383:
12379:
12375:
12366:
12362:
12358:
12354:
12350:
12346:
12341:
12337:
12331:
12327:
12321:
12316:
12312:
12308:
12304:
12300:
12296:
12292:
12291:
12289:
12284:
12279:
12274:
12269:
12264:
12260:
12256:
12253:
12249:
12243:
12238:
12233:
12231:on morphisms.
12230:
12223:
12219:
12212:
12208:
12204:
12200:
12192:
12187:
12177:
12168:
12160:
12156:
12149:
12142:
12138:
12134:
12131:
12127:
12123:
12119:
12115:
12110:
12105:
12101:
12096:
12091:
12087:
12083:
12079:
12075:
12074:
12072:
12068:
12064:
12060:
12059:
12058:
12056:
12052:
12048:
12038:
12036:
12032:
12025:
12021:
12017:
12013:
12008:
12003:
11999:
11997:
11993:
11989:
11985:
11980:
11975:
11971:
11970:
11949:
11946:
11942:
11938:
11930:
11926:
11919:
11916:
11914:
11907:
11904:
11900:
11887:
11883:
11876:
11873:
11868:
11865:
11861:
11857:
11855:
11848:
11845:
11841:
11829:
11828:
11824:
11823:
11798:
11792:
11789:
11786:
11783:
11775:
11755:
11748:
11744:
11740:
11738:
11728:
11725:
11721:
11712:
11709:
11706:
11703:
11688:
11685:
11679:
11673:
11670:
11662:
11642:
11635:
11631:
11627:
11625:
11615:
11612:
11608:
11599:
11596:
11591:
11588:
11585:
11582:
11564:
11558:
11555:
11552:
11544:
11524:
11517:
11513:
11509:
11503:
11497:
11494:
11492:
11484:
11476:
11473:
11470:
11462:
11459:
11449:
11446:
11440:
11434:
11426:
11406:
11398:
11392:
11389:
11384:
11380:
11376:
11374:
11366:
11358:
11355:
11350:
11347:
11344:
11336:
11333:
11322:
11321:
11317:
11316:
11315:
11309:
11300:
11296:
11292:
11288:
11284:
11279:
11275:
11270:
11266:
11262:
11258:
11254:
11250:
11246:
11242:
11238:
11233:
11229:
11224:
11220:
11216:
11212:
11208:
11204:
11203:
11202:
11201:
11200:
11198:
11194:
11190:
11186:
11179:
11175:
11170:
11165:
11163:
11158:
11153:
11149:
11145:
11142:
11137:
11132:
11127:
11122:
11118:
11116:
11115:right adjoint
11112:
11108:
11104:
11100:
11098:
11094:
11090:
11086:
11083:
11079:
11078:
11077:
11075:
11071:
11067:
11062:
11054:
11040:
10976:
10973:
10945:
10931:
10928:
10901:
10897:
10890:
10887:
10884:
10876:
10871:
10855:
10852:
10848:
10844:
10841:
10835:
10832:
10826:
10823:
10820:
10786:
10783:
10780:
10760:
10757:
10754:
10751:
10748:
10742:
10736:
10705:
10652:
10643:
10626:
10620:
10614:
10611:
10605:
10602:
10599:
10576:
10556:
10553:
10550:
10547:
10544:
10538:
10532:
10501:
10499:
10487:
10485:
10480:
10479:
10423:
10403:
10380:
10357:
10337:
10317:
10312:
10299:
10283:
10276:
10273:
10270:
10266:
10254:
10243:
10240:
10236:
10232:
10229:
10220:
10217:
10214:
10211:
10205:
10202:
10197:
10185:
10184:
10182:
10161:
10153:
10149:
10145:
10139:
10136:
10133:
10125:
10121:
10116:
10113:
10107:
10104:
10101:
10098:
10087:
10071:
10065:
10059:
10056:
10049:
10046:
10043:
10039:
10027:
10016:
10013:
10009:
10005:
10002:
9993:
9990:
9987:
9984:
9978:
9975:
9970:
9958:
9957:
9940:
9934:
9914:
9911:
9908:
9888:
9885:
9876:
9865:
9862:
9858:
9837:
9815:
9790:
9768:
9764:
9743:
9740:
9734:
9728:
9705:
9697:
9694:
9690:
9686:
9683:
9663:
9660:
9654:
9648:
9625:
9617:
9614:
9610:
9606:
9603:
9596:we also have
9583:
9580:
9577:
9554:
9546:
9543:
9539:
9535:
9526:
9520:
9512:
9509:
9505:
9501:
9498:
9478:
9475:
9469:
9463:
9455:
9454:
9433:
9425:
9422:
9418:
9414:
9411:
9405:
9402:
9399:
9394:
9382:
9381:
9379:
9363:
9357:
9352:
9348:
9344:
9341:
9334:
9330:
9324:
9321:
9318:
9313:
9301:
9293:
9292:
9278:
9275:
9272:
9264:
9249:
9246:
9240:
9232:
9229:
9225:
9204:
9184:
9164:
9144:
9124:
9104:
9084:
9079:
9075:
9071:
9068:
9065:
9060:
9056:
9035:
9027:
9024:
9023:
9007:
8987:
8982:
8978:
8974:
8954:
8934:
8914:
8911:
8908:
8885:
8854:
8824:
8797:
8772:
8767:
8763:
8759:
8739:
8733:
8713:
8693:
8685:
8681:
8662:
8645:
8634:
8629:
8625:
8617:
8616:
8602:
8596:
8593:
8590:
8570:
8562:
8561:
8557:
8553:
8537:
8517:
8495:
8491:
8468:
8464:
8443:
8423:
8403:
8395:
8373:
8365:
8361:
8357:
8351:
8348:
8345:
8337:
8333:
8328:
8325:
8319:
8316:
8313:
8310:
8300:
8299:
8285:
8265:
8245:
8223:
8219:
8210:
8206:
8205:
8186:
8180:
8177:
8171:
8163:
8159:
8155:
8149:
8141:
8137:
8116:
8096:
8076:
8073:
8070:
8044:
8036:
8032:
8028:
8025:
8019:
8016:
7994:
7990:
7981:
7978:
7977:
7968:
7964:
7960:
7956:
7952:
7948:
7945:
7942:
7938:
7934:
7930:
7926:
7919:The functor Ο
7918:
7915:
7912:
7908:
7904:
7900:
7896:
7892:
7888:
7884:
7880:
7876:
7875:Equivalences.
7873:
7872:
7863:
7859:
7855:
7851:
7847:
7844:instead: the
7843:
7839:
7835:
7832:
7829:
7825:
7821:
7817:
7813:
7809:
7805:
7801:
7797:
7793:
7789:
7785:
7781:
7777:
7773:
7769:
7765:
7761:
7757:
7753:
7749:
7745:
7742:
7738:
7734:
7731:
7730:
7729:
7726:
7724:
7720:
7716:
7711:
7709:
7705:
7704:Galois theory
7701:
7697:
7693:
7689:
7685:
7681:
7677:
7664:
7660:
7656:
7652:
7651:Stone duality
7648:
7645:
7642:
7635:
7631:
7627:
7626:
7621:
7617:
7613:
7609:
7608:
7603:
7599:
7595:
7588:
7584:
7580:
7576:
7572:
7569:
7565:
7562:
7559:
7555:
7551:
7547:
7543:
7539:
7535:
7531:
7527:
7523:
7519:
7515:
7512:
7508:
7504:
7501:
7498:
7494:
7490:
7486:
7482:
7478:
7474:
7470:
7466:
7463:
7459:
7455:
7451:
7448:
7444:
7441:
7438:
7434:
7431:creating the
7430:
7426:
7422:
7418:
7414:
7410:
7406:
7402:
7398:
7395:
7394:
7385:
7381:
7377:
7374:
7371:
7367:
7363:
7359:
7355:
7351:
7350:abelian group
7347:
7343:
7339:
7335:
7331:
7328:
7325:
7321:
7317:
7313:
7309:
7305:
7301:
7297:
7293:
7289:
7286:
7283:
7279:
7272:
7265:
7261:
7258:
7255:
7248:
7244:
7237:
7233:
7230:
7227:
7223:
7219:
7215:
7212:
7208:
7204:
7200:
7196:
7192:
7188:
7185:
7182:
7178:
7174:
7170:
7165:
7160:
7156:
7153:, defined by
7152:
7148:
7144:
7140:
7136:
7132:
7128:
7124:
7120:
7116:
7112:
7108:
7104:
7102:
7098:
7095:
7091:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7059:
7055:
7051:
7047:
7043:
7039:
7035:
7031:
7027:
7023:
7019:
7015:
7012:
7009:
6993:
6986:
6983:
6979:
6975:
6971:
6969:
6964:
6961:
6957:
6953:
6949:
6945:
6943:
6938:
6937:
6923:
6919:
6915:
6914:vector spaces
6911:
6907:
6906:
6902:
6898:
6894:
6890:
6886:
6882:
6879:) to back to
6878:
6874:
6867:
6860:
6856:
6852:
6848:
6844:
6840:
6836:
6832:
6828:
6824:
6817:
6810:
6806:
6803:
6800:
6796:
6792:
6789:
6788:
6787:
6785:
6781:
6777:
6773:
6769:
6756:
6755:
6751:
6747:
6743:
6739:
6735:
6731:
6727:
6724:
6720:
6716:
6712:
6708:
6704:
6699:
6695:
6689:
6682:
6678:
6674:
6669:
6665:
6660:
6656:
6649:
6642:
6638:
6631:
6624:
6617:
6610:
6603:
6596:
6592:
6588:
6585:
6584:
6580:
6576:
6572:
6568:
6567:
6563:
6559:
6555:
6548:
6541:
6534:
6527:
6523:
6519:
6515:
6511:
6507:
6503:
6500:
6493:
6486:
6482:
6472:
6468:
6464:
6461:Let Ξ :
6460:
6457:
6456:
6455:
6453:
6449:
6445:
6441:
6437:
6428:
6426:
6425:free functors
6422:
6418:
6409:
6407:
6403:
6399:
6395:
6391:
6370:
6366:
6359:
6351:
6347:
6329:
6326:
6322:
6313:
6309:
6289:
6286:
6274:
6270:
6263:
6258:
6253:
6250:
6247:
6244:
6235:
6232:
6228:
6222:
6217:
6214:
6206:
6205:
6204:
6202:
6186:
6183:
6180:
6177:
6174:
6171:
6166:
6162:
6153:
6149:
6147:
6143:
6139:
6135:
6117:
6114:
6110:
6101:
6097:
6093:
6089:
6085:
6081:
6060:
6056:
6049:
6041:
6037:
6018:
6015:
6006:
6003:
5999:
5993:
5988:
5985:
5982:
5979:
5967:
5963:
5956:
5951:
5946:
5943:
5936:
5935:
5934:
5932:
5916:
5913:
5910:
5907:
5904:
5901:
5896:
5892:
5883:
5879:
5865:
5862:
5859:
5856:
5850:
5847:
5844:
5833:
5829:
5827:
5823:
5818:
5814:
5810:
5806:
5802:
5798:
5794:
5790:
5788:
5784:
5780:
5776:
5760:
5754:
5751:
5748:
5745:
5740:
5736:
5727:
5723:
5719:
5715:
5694:
5690:
5686:
5683:
5680:
5669:
5665:
5649:
5643:
5640:
5637:
5634:
5629:
5625:
5616:
5612:
5608:
5604:
5600:
5596:
5594:
5593:
5587:
5583:
5567:
5564:
5561:
5555:
5552:
5547:
5543:
5534:
5530:
5526:
5522:
5518:
5502:
5499:
5496:
5490:
5487:
5482:
5478:
5470:. Let
5469:
5466:generated by
5465:
5461:
5457:
5454:For each set
5453:
5449:
5447:
5443:
5439:
5435:
5431:
5427:
5423:
5419:
5415:
5411:
5407:
5406:
5401:
5400:
5395:
5390:
5388:
5373:
5371:
5367:
5363:
5359:
5355:
5351:
5347:
5343:
5339:
5335:
5331:
5327:
5323:
5307:
5304:
5301:
5293:
5289:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5256:
5255:
5253:
5249:
5239:
5237:
5233:
5229:
5225:
5220:
5218:
5214:
5198:
5195:
5192:
5189:
5186:
5178:
5174:
5170:
5161:
5157:
5155:
5137:
5134:
5130:
5121:
5103:
5099:
5046:
5015:
5012:
5008:
5004:
4996:
4992:
4985:
4982:
4980:
4973:
4970:
4966:
4953:
4949:
4942:
4939:
4934:
4931:
4927:
4923:
4921:
4914:
4911:
4907:
4895:
4894:
4893:
4891:
4887:
4883:
4879:
4856:
4853:
4850:
4847:
4844:
4841:
4839:
4832:
4828:
4820:
4817:
4814:
4811:
4808:
4805:
4803:
4796:
4792:
4780:
4779:
4778:
4777:
4773:
4769:
4764:
4750:
4747:
4744:
4724:
4721:
4718:
4715:
4709:
4706:
4703:
4692:
4691:
4687:
4682:
4681:
4677:
4671:
4669:
4665:
4660:
4654:
4634:
4627:
4624:
4619:
4614:
4611:
4608:
4601:
4598:
4593:
4588:
4581:
4567:
4560:
4557:
4552:
4547:
4544:
4541:
4534:
4531:
4526:
4521:
4514:
4513:
4512:
4510:
4506:
4502:
4479:
4476:
4462:
4458:
4456:
4451:
4436:
4429:
4426:
4423:
4421:
4416:
4405:
4404:
4403:
4402:
4398:
4394:
4390:
4386:
4382:
4378:
4375:
4371:
4367:
4363:
4353:
4351:
4347:
4343:
4339:
4331:
4327:
4315:
4311:
4307:
4299:
4295:
4287:
4283:
4274:
4270:
4268:
4264:
4259:
4255:
4252:
4248:
4243:
4238:
4234:
4230:
4226:
4222:
4218:
4214:
4209:
4205:
4200:
4194:
4189:
4180:
4175:
4168:
4163:
4159:
4157:
4156:
4152:
4147:
4146:
4142:
4136:
4134:
4130:
4126:
4122:
4100:
4097:
4094:
4091:
4083:
4061:
4058:
4055:
4052:
4044:
4028:
4023:
4020:
4017:
4005:
4004:
4003:
3982:
3979:
3976:
3973:
3965:
3943:
3940:
3937:
3934:
3926:
3910:
3900:
3899:
3898:
3897:
3892:
3888:
3884:
3879:
3875:
3871:
3868:
3864:
3860:
3856:
3854:
3843:
3839:
3825:
3805:
3785:
3765:
3756:
3742:
3734:
3733:right adjoint
3718:
3698:
3677:
3674:
3667:
3664:
3661:
3641:
3638:
3632:
3629:
3624:
3620:
3615:
3611:
3607:
3598:
3592:
3586:
3566:
3560:
3557:
3554:
3534:
3521:
3517:
3503:
3500:
3495:
3491:
3487:
3481:
3475:
3455:
3446:
3440:
3437:
3434:
3411:
3405:
3399:
3396:
3393:
3373:
3353:
3327:
3321:
3315:
3309:
3306:
3301:
3297:
3276:
3253:
3247:
3227:
3207:
3187:
3167:
3159:
3143:
3123:
3115:
3099:
3093:
3090:
3087:
3078:
3064:
3056:
3040:
3020:
3000:
2993:
2990:
2986:
2983:
2960:
2957:
2952:
2948:
2945:
2942:
2933:
2927:
2921:
2918:
2913:
2909:
2888:
2882:
2879:
2876:
2856:
2843:
2839:
2837:
2832:
2818:
2815:
2809:
2803:
2800:
2795:
2791:
2767:
2761:
2755:
2752:
2749:
2729:
2720:
2714:
2711:
2708:
2688:
2668:
2648:
2636:
2630:
2624:
2621:
2616:
2612:
2591:
2568:
2562:
2542:
2522:
2502:
2482:
2474:
2458:
2438:
2430:
2414:
2408:
2405:
2402:
2388:
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2337:
2335:
2331:
2327:
2323:
2319:
2315:
2305:
2298:
2295:
2291:
2287:
2286:
2285:
2277:
2275:
2271:
2267:
2263:
2258:
2256:
2252:
2248:
2244:
2240:
2236:
2231:
2229:
2225:
2221:
2211:
2209:
2205:
2201:
2196:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2133:
2131:
2127:
2123:
2119:
2115:
2111:
2104:
2097:
2090:
2083:
2080:implies that
2076:
2072:
2065:
2061:
2057:
2053:
2038:
2031:
2024:
2020:
2013:
2009:
2006:of the form (
2005:
1998:
1994:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1946:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1915:
1913:
1909:
1905:
1901:
1897:
1892:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1848:
1846:
1837:
1836:
1828:
1819:
1802:
1799:
1796:
1793:
1787:
1781:
1778:
1775:
1772:
1749:
1729:
1722:
1718:
1717:Hilbert space
1699:
1696:
1693:
1690:
1683:
1682:
1681:
1679:
1675:
1670:
1619:
1562:
1542:
1522:
1513:
1499:
1479:
1472:. (Note that
1459:
1451:
1450:right adjoint
1435:
1415:
1407:
1391:
1371:
1363:
1360:is the right
1347:
1344:
1321:
1318:
1315:
1312:
1269:
1261:
1248:
1228:
1218:
1202:
1176:
1173:
1170:
1167:
1141:
1135:
1132:
1129:
1126:
1100:
1095:
1092:
1088:
1080:
1079:
1078:
1076:
1072:
1068:
1064:
1063:
1058:
1057:
1046:
996:
943:
929:
926:
923:
915:
902:
892:
876:
868:
855:
845:
829:
820:
782:
741:
735:
732:
729:
726:
675:
669:
666:
663:
660:
606:
558:
552:
549:
546:
543:
485:
479:
476:
473:
470:
440:
420:
412:
390:
387:
384:
381:
355:
349:
346:
343:
340:
310:
309:
308:
306:
266:
222:
182:
179:
142:
139:
132:
131:
130:
128:
75:
74:in topology.
73:
69:
65:
61:
57:
56:right adjoint
53:
49:
45:
41:
37:
33:
26:
22:
16819:Conservative
16813:
16679:
16660:Categorified
16564:n-categories
16515:Key concepts
16353:Direct limit
16336:Coequalizers
16254:Yoneda lemma
16203:
16160:Key concepts
16150:Key concepts
16069:playlist on
16026:
15992:
15974:
15963:
15955:
15946:
15937:
15925:
15913:
15886:
15882:
15869:
15848:
15827:
15822:
15817:γ defines a
15814:
15810:
15806:
15802:
15798:
15796:
15750:
15701:
15697:
15696:is given by
15695:
15647:
15643:
15635:
15631:
15629:
15619:
15615:
15610:
15605:
15601:
15597:
15595:
15587:
15580:
15576:
15572:
15567:
15562:
15558:
15554:
15543:
15538:
15533:
15529:
15525:
15518:
15514:
15510:
15508:
15499:
15495:
15491:
15489:
15479:
15475:
15472:
15467:
15463:
15459:
15455:
15451:
15447:
15443:
15435:
15431:
15429:
15406:
15402:
15400:
15395:
15391:
15387:
15385:
15262:
15258:
15254:
15250:
15242:
15238:
15234:
15226:
15222:
15220:
15182:
15174:cocontinuous
15173:
15169:
15161:
15157:
15155:
15141:
15139:
15134:
15132:
14920:
14916:
14914:
14862:
14811:
14807:
14803:
14799:
14795:
14791:
14787:
14783:
14781:
14733:
14601:
14597:
14595:
14589:
14585:
14578:
14574:
14566:
14562:
14558:
14554:
14550:
14546:
14542:
14538:
14536:
14527:
14523:
14519:
14515:
14511:
14507:
14503:
14501:
14487:
14483:
14477:
14473:
14432:
14429:
14423:
14419:
14414:
14410:
14406:
14395:
14393:
14387:
14383:
14379:
14375:
14371:
14366:
14360:
14356:
14352:
14348:
14341:proper class
14339:
14332:
14331:come from a
14328:
14326:
14319:
14315:
14311:
14307:
14302:
14298:
14291:
14287:
14279:
14271:
14263:
14259:
14255:
14251:
14249:
14157:
14152:
14146:
14137:
14133:
14054:
14049:
14043:
14039:
14033:
14028:
14024:
14015:
14011:
14007:
14003:
13999:
13995:
13839:
13835:
13831:
13826:
13821:
13820:, and each (
13817:
13813:
13809:
13804:
13799:
13720:
13716:
13527:
13523:
13399:
13346:
13341:
13336:
13331:
13326:
13322:
13318:
13314:
13310:
13306:
13304:
13109:
13099:
13094:
13089:
13084:
12895:), we obtain
12891:
12886:
12879:
12873:
12868:
12718:
12714:
12710:
12706:
12702:
12698:
12690:
12685:
12680:
12675:
12670:
12648:
12644:
12640:
12636:
12632:
12628:
12624:
12622:
12613:
12605:
12598:
12590:
12583:
12574:
12570:
12566:
12556:
12547:
12543:
12536:
12532:
12524:
12515:
12511:
12507:
12500:
12496:
12489:
12485:
12479:
12473:
12465:
12458:
12450:
12443:
12439:
12432:
12425:
12421:
12417:
12410:
12403:
12399:
12392:
12385:
12381:
12373:
12364:
12360:
12356:
12352:
12348:
12344:
12339:
12335:
12329:
12325:
12319:
12314:
12310:
12306:
12302:
12298:
12294:
12287:
12282:
12277:
12272:
12262:
12258:
12247:
12241:
12236:
12228:
12221:
12217:
12210:
12206:
12202:
12198:
12190:
12185:
12175:
12166:
12158:
12154:
12147:
12140:
12136:
12129:
12125:
12121:
12117:
12113:
12108:
12103:
12099:
12094:
12089:
12085:
12081:
12077:
12070:
12066:
12062:
12054:
12050:
12046:
12044:
12034:
12030:
12028:
12023:
12019:
12015:
12006:
12001:
11995:
11991:
11987:
11978:
11973:
11313:
11298:
11294:
11290:
11286:
11282:
11277:
11273:
11268:
11264:
11260:
11256:
11252:
11248:
11244:
11240:
11236:
11231:
11227:
11222:
11218:
11214:
11210:
11206:
11196:
11192:
11188:
11184:
11182:
11177:
11173:
11168:
11161:
11156:
11151:
11140:
11135:
11130:
11125:
11123:Ξ¦ : hom
11114:
11110:
11106:
11102:
11097:left adjoint
11096:
11092:
11088:
11084:
11076:consists of
11073:
11069:
11065:
11063:
11060:
11053:is "free".)
10872:
10644:
10502:
10497:
10495:
10481:
9025:
8556:topos theory
8207:The role of
7979:
7962:
7958:
7954:
7946:
7940:
7936:
7932:
7928:
7924:
7916:
7910:
7906:
7902:
7898:
7894:
7886:
7882:
7878:
7874:
7841:
7837:
7827:
7823:
7819:
7815:
7811:
7807:
7803:
7799:
7795:
7791:
7787:
7783:
7779:
7775:
7771:
7767:
7763:
7759:
7755:
7751:
7727:
7718:
7712:
7699:
7691:
7687:
7683:
7679:
7673:
7655:sober spaces
7646:
7633:
7629:
7623:
7619:
7615:
7611:
7605:
7601:
7597:
7586:
7578:
7574:
7570:
7563:
7557:
7545:
7541:
7537:
7533:
7525:
7521:
7517:
7506:
7502:
7488:
7484:
7476:
7472:
7468:
7464:
7453:
7449:
7442:
7436:
7428:
7424:
7416:
7412:
7400:
7396:
7375:
7370:model theory
7329:
7323:
7319:
7315:
7299:
7295:
7291:
7287:
7277:
7270:
7263:
7259:
7246:
7242:
7235:
7231:
7225:
7217:
7213:
7186:
7180:
7176:
7172:
7168:
7163:
7158:
7154:
7150:
7146:
7142:
7138:
7134:
7130:
7126:
7122:
7118:
7114:
7110:
7106:
7099:
7093:
7089:
7085:
7081:
7077:
7073:
7069:
7065:
7061:
7057:
7053:
7049:
7041:
7037:
7029:
7025:
7021:
7017:
7013:
7007:
6984:
6973:
6965:
6959:
6955:
6951:
6947:
6939:
6922:free product
6900:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6865:
6858:
6854:
6850:
6846:
6842:
6838:
6834:
6830:
6826:
6815:
6808:
6804:
6801:
6798:
6794:
6790:
6776:coequalizers
6766:
6749:
6745:
6741:
6737:
6733:
6729:
6725:
6722:
6718:
6710:
6706:
6702:
6697:
6693:
6687:
6680:
6676:
6672:
6667:
6663:
6658:
6654:
6647:
6640:
6636:
6629:
6622:
6615:
6608:
6601:
6594:
6590:
6586:
6561:
6557:
6553:
6546:
6539:
6532:
6525:
6521:
6517:
6513:
6509:
6501:
6498:
6491:
6484:
6477:
6470:
6466:
6462:
6458:
6434:
6417:Free objects
6415:
6405:
6401:
6397:
6393:
6389:
6349:
6345:
6311:
6307:
6305:
6200:
6151:
6150:
6145:
6141:
6137:
6133:
6099:
6095:
6091:
6087:
6083:
6079:
6039:
6035:
6033:
5930:
5881:
5880:
5831:
5830:
5825:
5821:
5816:
5812:
5808:
5804:
5800:
5796:
5792:
5791:
5786:
5782:
5778:
5774:
5725:
5721:
5717:
5713:
5667:
5663:
5617:. Let
5614:
5610:
5606:
5605:, the group
5602:
5598:
5597:
5591:
5585:
5581:
5532:
5528:
5524:
5520:
5516:
5467:
5463:
5459:
5455:
5451:
5450:
5445:
5441:
5437:
5429:
5425:
5421:
5417:
5409:
5403:
5397:
5393:
5391:
5384:
5365:
5361:
5357:
5349:
5341:
5333:
5329:
5325:
5291:
5285:
5279:
5275:
5271:
5267:
5263:
5259:
5245:
5231:
5227:
5223:
5221:
5216:
5212:
5172:
5168:
5166:
5153:
5119:
5037:
4889:
4885:
4881:
4877:
4875:
4775:
4771:
4767:
4765:
4689:
4685:
4684:
4679:
4675:
4674:
4672:
4667:
4663:
4658:
4652:
4649:
4504:
4500:
4498:
4396:
4392:
4388:
4384:
4380:
4376:
4369:
4365:
4361:
4359:
4349:
4345:
4341:
4337:
4329:
4317:
4313:
4309:
4305:
4297:
4293:
4288:) : Hom
4285:
4281:
4279:
4262:
4257:
4254:
4250:
4246:
4241:
4236:
4232:
4228:
4221:hom functors
4212:
4207:
4203:
4192:
4187:
4178:
4173:
4166:
4164:
4160:
4154:
4150:
4149:
4144:
4140:
4139:
4137:
4132:
4128:
4124:
4120:
4118:
4001:
3890:
3886:
3882:
3877:
3873:
3869:
3862:
3858:
3851:
3849:
3840:
3757:
3732:
3527:Again, this
3526:
3113:
3079:
3055:left adjoint
3054:
2848:
2833:
2428:
2394:
2384:
2380:
2376:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2338:
2333:
2329:
2325:
2321:
2317:
2313:
2311:
2302:
2293:
2283:
2273:
2269:
2265:
2261:
2259:
2254:
2250:
2246:
2242:
2238:
2234:
2232:
2227:
2223:
2219:
2217:
2207:
2203:
2199:
2197:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2134:
2129:
2125:
2121:
2117:
2113:
2109:
2102:
2095:
2088:
2081:
2074:
2070:
2063:
2059:
2055:
2036:
2029:
2022:
2018:
2011:
2007:
1996:
1992:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1947:
1942:
1934:
1931:optimization
1930:
1926:
1918:
1916:
1903:
1895:
1893:
1888:
1884:
1883:+1 for each
1880:
1876:
1860:
1856:
1854:
1841:
1833:
1826:
1714:
1677:
1673:
1671:
1514:
1449:
1406:left adjoint
1405:
1361:
1220:
1216:
1194:
1070:
1060:
1054:
1052:
944:
894:
890:
889:is called a
847:
843:
842:is called a
822:The functor
821:
775:for a fixed
599:for a fixed
408:
214:
76:
55:
52:left adjoint
51:
47:
39:
29:
16628:-categories
16604:Kan complex
16594:Tricategory
16576:-categories
16466:Subcategory
16224:Exponential
16192:Preadditive
16187:Pre-abelian
16085:Mathematica
16066:Adjunctions
15975:See page 58
15565:for which Ξ·
15536:for which Ξ΅
15210:right exact
15135:composition
14778:Composition
13113:, we obtain
12456:, we have Ξ¦
12106:, so that Ξ·
12000:Each pair (
11972:Each pair (
11271:-morphism Ξ¦
11225:-morphism Ξ¦
11176:called the
11160:called the
11113:called the
11095:called the
10492:Probability
10300:The subset
8209:quantifiers
7961:given by βΓ
7854:implication
7850:ring ideals
7113:is a right
6791:Coproducts.
6653:is a pair (
6348:to the set
5807:to the set
5799:to a group
5387:free groups
5381:Free groups
5344:sign is an
2308:Conventions
1865:ring theory
995:equivalence
916:. We write
32:mathematics
16646:3-category
16636:2-category
16609:β-groupoid
16584:Bicategory
16331:Coproducts
16291:Equalizers
16197:Bicategory
16051:0906.18001
16015:0695.18001
15984:References
15926:Dialectica
15622:inverses.
15604:such that
15415:biproducts
15217:Additivity
15203:left exact
15162:continuous
14498:Uniqueness
14284:continuous
14244:See also:
14235:Properties
13605:Let
13404:Let
13339:-,-) β hom
12683:-,-) β hom
12492:)) = G(x)
12280:-,-) β hom
11255:-morphism
11209:-morphism
11205:For every
11133:β,β) β hom
11101:A functor
11066:adjunction
10873:Then, the
8684:subobjects
7852:, and the
7554:continuous
7481:loop space
7462:suspension
7346:direct sum
6910:direct sum
6841:the pair (
6823:direct sum
6768:Coproducts
6736:β 0. Then
6512:the pair (
6444:equalizers
5458:, the set
5420:, and let
5414:free group
5248:Daniel Kan
5038:Note that
4774:) are the
4199:bifunctors
3855:adjunction
3579:such that
2249:poses the
1217:adjunction
1053:The terms
40:adjunction
16849:Forgetful
16695:Symmetric
16640:2-functor
16380:Relations
16303:Pullbacks
16089:morphisms
15778:→
15762:μ
15733:→
15715:η
15674:→
15620:two-sided
15333:≅
15277:Φ
15196:coproduct
15096:ε
15071:ε
14998:η
14958:η
14894:→
14882:∘
14845:→
14828:∘
14762:∗
14742:∘
14704:−
14700:τ
14696:∗
14688:−
14684:σ
14677:∘
14674:ε
14660:ε
14652:η
14649:∘
14643:σ
14640:∗
14637:τ
14620:η
14584:Ο :
14573:Ο :
14454:→
14394:for some
14240:Existence
14209:η
14205:∘
14193:ε
14108:η
14098:∘
14086:ε
13994:for each
13964:∘
13955:ε
13934:−
13919:Φ
13905:η
13901:∘
13862:Φ
13744:∈
13664:∈
13627:Φ
13614:η
13551:∈
13471:∈
13444:−
13426:Φ
13413:ε
13381:⊣
13369:η
13363:ε
13269:∘
13240:∘
13228:η
13224:∘
13212:ε
13186:η
13182:∘
13164:∘
13152:ε
13132:Ψ
13129:Φ
13041:∘
13016:η
13006:∘
12994:ε
12990:∘
12965:η
12955:∘
12937:∘
12928:ε
12914:Φ
12911:Ψ
12832:∘
12823:ε
12795:Ψ
12781:η
12777:∘
12738:Φ
12709:and each
12697:For each
12658:⊣
12205:) :
12135:For each
11943:η
11939:∘
11927:ε
11884:η
11874:∘
11862:ε
11756:∈
11745:η
11700:Φ
11643:∈
11632:ε
11597:−
11579:Φ
11525:∈
11514:η
11510:∘
11467:Φ
11407:∈
11390:∘
11381:ε
11356:−
11341:Φ
11150:Ξ΅ :
11041:δ
10977:δ
10974:⊣
10946:μ
10935:↦
10932:μ
10898:δ
10894:↦
10885:δ
10853:−
10845:∘
10842:μ
10839:→
10836:μ
10821:μ
10784:∈
10781:μ
10618:→
10482:See also
10464:∀
10444:∃
10309:∀
10274:∈
10241:−
10233:∈
10224:∀
10221:∣
10215:∈
10194:∀
10150:ϕ
10146:∧
10122:ψ
10111:∃
10108:∣
10102:∈
10047:∈
10014:−
10006:∈
9997:∃
9994:∣
9988:∈
9967:∃
9912:∈
9886:∩
9863:−
9812:∃
9769:∗
9741:⊆
9695:−
9687:⊆
9661:∈
9615:−
9607:∈
9581:∈
9544:−
9536:⊆
9510:−
9502:⊆
9491:. We see
9476:⊆
9456:Consider
9423:−
9415:⊆
9409:↔
9403:⊆
9391:∃
9353:∗
9331:≅
9310:∃
9276:⊆
9247:⊆
9230:−
9076:×
9061:∗
8979:×
8912:⊂
8821:∀
8794:∃
8764:×
8737:→
8652:⟶
8630:∗
8600:→
8538:∧
8518:∩
8492:ϕ
8465:ψ
8362:ϕ
8358:∧
8334:ψ
8323:∃
8320:∣
8314:∈
8220:ψ
8181:φ
8178:∧
8160:ϕ
8138:ϕ
8074:⊂
8033:ϕ
8029:∣
7991:ϕ
7739:(cf. the
7723:Kaplansky
7423:on a set
7302:from the
6994:⊔
6968:semigroup
6920:, by the
6825:, and if
6797: :
6780:cokernels
6758:adjoints.
6367:ε
6323:η
6271:ε
6229:η
6184:η
6181:∘
6178:ε
6111:ε
6057:η
6000:ε
5964:η
5917:η
5911:∘
5905:ε
5863:⊣
5851:η
5845:ε
5758:→
5737:ε
5691:ε
5647:→
5626:ε
5559:→
5544:η
5494:→
5479:η
5305:⊗
5302:−
5199:η
5196:∘
5193:ε
5009:η
5005:∘
4993:ε
4950:η
4940:∘
4928:ε
4884:and each
4854:η
4851:∘
4848:ε
4821:η
4815:∘
4809:ε
4748:⊣
4722:⊣
4710:η
4704:ε
4628:ε
4599:η
4558:ε
4535:η
4474:→
4452:η
4433:→
4417:ε
4328:for each
4225:morphisms
4068:→
4014:Φ
3983:−
3974:−
3950:→
3944:−
3938:−
3908:Φ
3671:→
3639:∘
3625:η
3612:η
3608:∘
3564:→
3492:η
3488:∘
3453:→
3403:→
3313:→
3298:η
3097:→
2998:→
2953:ϵ
2949:∘
2919:∘
2910:ϵ
2886:→
2801:∘
2792:ϵ
2759:→
2727:→
2646:→
2613:ϵ
2412:→
2141:formulaic
2042:) where S
1974:morphisms
1904:formulaic
1902:, and is
1889:formulaic
1861:formulaic
1806:⟩
1791:⟨
1785:⟩
1770:⟨
1694:⊣
1345:φ
1203:φ
1142:≅
1089:φ
927:⊣
752:→
736:−
686:→
670:−
569:→
544:−
496:→
474:−
356:≅
305:bijection
193:→
153:→
127:covariant
16889:Category
16864:Monoidal
16834:Enriched
16829:Diagonal
16809:Additive
16755:Glossary
16735:Category
16709:n-monoid
16662:concepts
16318:Colimits
16286:Products
16239:Morphism
16182:Concrete
16177:Additive
16167:Category
16093:functors
16081:WildCats
16025:(1998).
15832:and the
15494: :
15454: :
15253: :
15237: :
15178:colimits
15092:→
15074:′
15063:→
15054:′
15046:′
15030:′
15016:′
15005:′
14994:′
14986:→
14980:′
14972:′
14961:′
14953:→
14878:′
14835:′
14663:′
14623:′
14522:β², then
14506: :
14422:∈
14409: :
14351: :
14338:, not a
14306: :
14254: :
13321: :
13309: :
12721:, define
12713: :
12701: :
12639: :
12627: :
12442: :
12424: :
12347: :
12197:and get
12139: :
12112: :
12065: :
12049: :
12010:) is an
11289: :
11259: :
11243: :
11213: :
11105: :
11087: :
10773:and any
10498:solution
10484:powerset
9721:implies
8870:back to
7967:currying
7897: :
7881: :
7834:division
7750:is that
7719:antitone
7700:antitone
7581:between
7573: :
7540: :
7520: :
7391:Topology
7334:K-theory
7294: :
7222:algebras
7141: :
7125: :
7080: :
7056: :
7016:Suppose
6849:), then
6829: :
6721: :
6717:and let
6705: :
6621: :
6600: :
6587:Kernels.
6459:Products
6436:Products
6259:→
6223:→
5994:→
5952:→
5424: :
5396: :
5376:Examples
5290:, where
5270:) = hom(
5228:terminal
4763: .
4620:→
4594:→
4553:→
4527:→
4503:and the
4399:and two
4391: :
4379: :
4374:functors
4267:commutes
4249: :
4231: :
3885: :
3872: :
3867:functors
3678:′
3633:′
2994:′
2961:′
2375: :
2257:solves.
1980:between
1939:supremum
1830:β
1719:idea of
1075:Mac Lane
1067:cognates
997:between
869:, while
44:functors
16859:Logical
16824:Derived
16814:Adjoint
16797:Functor
16745:Outline
16704:n-group
16669:2-group
16624:Strict
16614:β-topos
16410:Modules
16348:Pushout
16296:Kernels
16229:Functor
16172:Abelian
16071:YouTube
15905:1993102
15819:comonad
15549:as the
15466:, then
15394:, then
15261:, then
15189:product
14553:β² then
13396: ,
12098:) from
11982:) is a
11082:functor
9026:Example
7949:. In a
7842:adjoint
7715:duality
7659:duality
7594:sheaves
7532:. Then
7511:compact
7479:to the
7378:in the
7195:monoids
7161:) = hom
7072:. Then
7036:. Then
6934:Algebra
6918:modules
6784:colimit
6504:be the
6448:kernels
5432:be the
5350:natural
5328:), and
5242:History
5224:initial
4344:). Hom(
4300:) β Hom
3853:hom-set
2294:adjoint
2268:, then
2237:is the
2187:is the
2183:. Then
2163:. Let
2128:* is a
1968:, with
1958:objects
1908:functor
1875:. The
1362:adjunct
1062:adjunct
1056:adjoint
411:natural
16874:Smooth
16691:Traced
16674:2-ring
16404:Fields
16390:Groups
16385:Magmas
16273:Limits
16049:
16039:
16013:
16003:
15970:
15903:
15626:Monads
15245:is an
15166:limits
14545:, and
14530:β² are
14156:) for
14053:) for
12438:, any
12420:, any
12309:, as (
12188:with Ξ·
12153:, as (
11162:counit
9803:under
9137:along
7889:is an
7838:invert
7786:, let
7770:, let
7737:monads
7717:(i.e.
7674:Every
7670:Posets
7622:, the
7604:, the
7566:Every
7548:, the
7445:Given
7382:: see
7262:. Let
7203:groups
6978:monoid
6778:, and
6715:kernel
6701:. Let
6446:, and
6302:
6207:
6154:
5884:
5342:equals
5236:monads
4501:counit
4332:in Hom
3894:and a
1956:whose
1612:, and
1428:, and
1219:or an
16844:Exact
16799:types
16685:-ring
16572:Weak
16556:Topos
16400:Rings
15997:(PDF)
15901:JSTOR
15879:(PDF)
15855:arXiv
15840:Notes
15809:, Ξ΅,
15640:monad
14798:and γ
14734:Here
14437:. If
14266:is a
14147:GX, X
14140:and Ξ΅
14031:and Ξ·
12250:is a
12014:from
11986:from
9956:. So
9676:. So
9197:into
9117:into
9028:: In
8109:into
7546:KHaus
7522:KHaus
7507:KHaus
7368:, or
7340:on a
7332:. In
7243:Field
7211:field
7048:with
7032:is a
6499:Grp β
6452:limit
6082:into
5356:from
5324:with
4656:and 1
4215:(the
4201:from
4181:β, β)
3468:with
3160:from
3112:is a
2783:with
2475:from
2427:is a
2318:right
2253:that
2220:start
1762:with
1262:. If
70:of a
16375:Sets
16037:ISBN
16001:ISBN
15968:ISBN
15947:nLab
15586:and
15575:and
15434:and
15409:are
15405:and
15233:and
15229:are
15225:and
14921:G' G
14919:and
14917:F F'
14810:and
14794:and
14526:and
14518:and
14382:) β
14136:for
14027:for
14006:and
12359:) =
12355:) β
12317:), Ξ·
12290:-).
12216:) β
12164:), Ξ·
12092:), Ξ·
12033:and
11285:) =
11239:) =
11191:and
11178:unit
11072:and
9265:For
8812:and
8258:and
7909:and
7516:and
7505:Let
7452:and
7409:sets
7399:Let
7278:Ring
7271:Ring
7264:Ring
7189:The
7020:and
6916:and
6864:and
6752:β 0.
6723:Ab β
6614:and
6575:sets
6569:The
6545:and
6390:GFGX
6350:GFGX
6308:GFGX
6134:FGFY
6100:FGFY
6084:FGFY
6036:FGFY
5412:the
5392:Let
5258:hom(
5232:unit
4683:and
4666:and
4505:unit
4387:and
4368:and
4191:(β,
4183:and
4148:and
4127:and
3880:and
3861:and
3654:for
2324:and
2316:and
2314:left
2139:and
2069:and
2002:are
1991:and
1912:dual
1873:ring
1241:and
1059:and
1021:and
969:and
709:and
526:and
433:and
303:, a
259:and
101:and
16219:End
16209:CCC
16047:Zbl
16011:Zbl
15924:",
15920:, "
15891:doi
15821:in
15561:of
15553:of
15532:of
15482:).
15221:If
15180:).
14802:β²,
14600:β²,
14398:in
14333:set
14290:of
14274:of
14144:= Ξ¦
14037:= Ξ¦
13838:in
13834:to
13824:, Ξ·
13816:in
13812:to
13802:, Ξ΅
13719:in
13526:in
13345:(-,
12877:= Ξ΅
12689:(-,
12518:))
12416:in
12398:in
12305:in
12297:in
12286:(-,
12102:to
12080:in
12022:in
12018:to
12004:, Ξ·
11994:in
11990:to
11976:, Ξ΅
11297:in
11154:β 1
11139:(β,
11064:An
10330:of
9335:Hom
9302:Hom
9036:Set
8967:in
8879:Sub
8848:Sub
8706:of
8656:Sub
8639:Sub
8416:of
7982:If
7877:If
7856:in
7766:in
7682:to
7542:Top
7526:Top
7487:of
7471:to
7467:of
7435:on
7407:to
7306:to
7300:Grp
7247:Dom
7236:Dom
7151:Mod
7131:Mod
7105:If
7094:Mod
7086:Mod
7070:Mod
7062:Mod
6942:rng
6912:of
6899:of
6793:If
6646:to
6573:of
6538:to
6522:Grp
6502:Grp
6467:Grp
6463:Grp
6398:FGX
6392:to
6312:FGX
6136:to
6090:of
5828:).
5815:to
5777:to
5724:to
5716:to
5664:FGX
5607:FGX
5584:to
5519:to
5460:GFY
5430:Set
5426:Grp
5405:Grp
5399:Set
5364:to
5266:),
5215:or
4888:in
4880:in
4662:on
4306:FYβ²
4261:in
4240:in
4213:Set
4211:to
4185:hom
4171:hom
4131:in
4123:in
3735:to
3366:in
3269:in
3220:in
3180:to
3136:in
3057:to
3013:in
2681:in
2584:in
2535:in
2495:to
2451:in
2191:of
2054:of
2046:β S
1976:in
1925:of
1869:rng
1672:If
1452:to
1448:is
1408:to
1404:is
1364:of
1215:an
893:or
846:or
795:in
619:in
413:in
279:in
235:in
30:In
16891::
16697:)
16693:)(
16099:,
16095:,
16045:.
16031:.
16009:.
15945:.
15899:.
15887:87
15885:.
15881:.
15825:.
15807:FG
15702:GF
15700:=
15634:,
15606:FG
15513:,
15498:β
15458:β
15257:β
15241:β
15170:is
15158:is
14786:,
14588:β
14577:β
14565:,
14510:β
14418:β
14374:=
14355:β
14310:β
14278::
14258:β
14153:GX
14150:(1
14134:GX
14050:FY
14047:(1
14044:FY
14042:,
14025:FY
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14014:β
14010::
14002:β
14000:FY
13998::
13822:FY
13800:GX
13325:β
13317:,
13313:β
13110:GX
13103:)
13097:(Ξ΅
13093:=
13090:GX
12889:(Ξ·
12880:FY
12874:FY
12719:GX
12717:β
12705:β
12703:FY
12643:β
12635:,
12631:β
12601:)
12589:,
12577:)
12569:=
12550:)
12539:)
12531:=
12503:)
12464:,
12449:β
12431:β
12409:,
12391:,
12367:)
12338:,
12328:,
12240:β
12146:β
12116:β
12069:β
12053:β
12002:FY
11974:GX
11293:β
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11265:GX
11263:β
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11119:A
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10870:.
10642:.
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7628:.
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6903:).
6845:,
6835:Ab
6833:β
6831:Ab
6814:,
6805:Ab
6799:Ab
6774:,
6770:,
6711:Ab
6709:β
6686:=
6662:,
6628:β
6607:β
6516:,
6476:,
6465:β
6442:,
6438:,
6408:.
6406:GX
6394:GX
6346:GX
6148:.
6146:FY
6142:FY
6138:FY
6092:FY
6080:FY
6040:FY
5813:FY
5809:GX
5797:FY
5779:GX
5722:FZ
5611:GX
5595:.
5582:FY
5529:GW
5464:FY
5448::
5428:β
5402:β
5372:.
5360:Γ
5282:))
5274:,
5156:.
5154:FY
5091:,
4892:,
4395:β
4383:β
4360:A
4350:Gf
4348:,
4340:,
4338:FY
4326:Fg
4322:h
4318:f
4316:β
4310:Xβ²
4308:,
4296:,
4294:FY
4284:,
4282:Fg
4269::
4256:β
4237:Xβ²
4235:β
4206:Γ
4195:β)
4158:.
4135:.
3889:β
3876:β
3850:A
3838:.
3755:.
3711:;
3516:.
3077:.
3033:;
2838::
2831:.
2379:β
2363:,
2359:,
2347:,
2343:,
2276:.
2208:R*
2206:β
2200:R*
2195:.
2179:)=
2161:R*
2159:)=
2114:R*
2112:β
2073:β
2062:β
2035:β
2028:,
2021:β
2017:,
2010:β
1995:β
1984:β
1964:β
1742:,
1669:.
1337:,
1073:,
942:.
819:.
584:op
511:op
129:)
38:,
16789:e
16782:t
16775:v
16689:(
16682:n
16680:E
16642:)
16638:(
16626:n
16590:)
16586:(
16574:n
16416:)
16412:(
16406:)
16402:(
16130:e
16123:t
16116:v
16103:.
16053:.
16017:.
15949:.
15907:.
15893::
15863:.
15857::
15823:C
15815:G
15813:Ξ·
15811:F
15803:F
15801:Ξ΅
15799:G
15781:T
15773:2
15769:T
15765::
15736:T
15727:D
15722:1
15718::
15698:T
15679:D
15669:D
15664::
15661:T
15648:D
15644:T
15642:γ
15636:G
15632:F
15616:F
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15544:D
15539:X
15534:C
15530:X
15526:C
15522:1
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15515:G
15511:F
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15480:C
15476:D
15468:G
15464:D
15460:D
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15452:G
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15436:D
15432:C
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15403:C
15396:F
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15350:D
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15315:(
15309:C
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15300:o
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15255:C
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15235:F
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15223:C
15212:.
15205:;
15142:C
15114:.
15108:C
15103:1
15087:G
15084:F
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15067:F
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15039:F
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15023:F
15020:G
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15002:F
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14977:F
14969:G
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14940:1
14900:.
14897:E
14891:C
14888::
14885:G
14875:G
14848:C
14842:E
14839::
14832:F
14825:F
14812:E
14808:D
14804:G
14800:F
14796:D
14792:C
14788:G
14784:F
14715:.
14712:)
14707:1
14691:1
14680:(
14671:=
14646:)
14634:(
14631:=
14602:G
14598:F
14592:β²
14590:G
14586:G
14581:β²
14579:F
14575:F
14567:G
14563:F
14559:G
14555:F
14551:G
14547:G
14543:G
14539:F
14528:G
14524:G
14520:G
14516:G
14512:C
14508:D
14504:F
14488:F
14484:F
14478:F
14474:F
14457:D
14451:C
14448::
14445:F
14426:.
14424:C
14420:X
14415:i
14411:X
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14400:I
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14384:f
14380:t
14378:(
14376:G
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14357:G
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14320:i
14316:X
14314:(
14312:G
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14299:f
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14280:G
14264:C
14260:D
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14230:.
14216:X
14213:G
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14183:=
14178:X
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14171:1
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14129:,
14117:)
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14104:(
14101:F
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14090:F
14082:=
14077:Y
14074:F
14070:1
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14040:Y
14034:Y
14029:X
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14004:X
13996:f
13976:)
13973:g
13970:(
13967:F
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13951:=
13948:)
13945:g
13942:(
13937:1
13929:X
13926:,
13923:Y
13909:Y
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13895:f
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13889:G
13886:=
13883:)
13880:f
13877:(
13872:X
13869:,
13866:Y
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13840:D
13836:G
13832:Y
13827:Y
13818:C
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13768:F
13765:(
13760:C
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13752:o
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13717:Y
13703:)
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13680:D
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13669:h
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13640:Y
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13634:,
13631:Y
13623:=
13618:Y
13590:)
13587:X
13584:G
13581:,
13578:X
13575:G
13572:(
13567:D
13562:m
13559:o
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13524:X
13510:)
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13501:X
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13495:F
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13487:C
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13468:)
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13447:1
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13436:,
13433:X
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13422:=
13417:X
13384:G
13378:F
13375::
13372:)
13366:,
13360:(
13347:G
13342:D
13337:F
13335:(
13332:C
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13315:C
13311:D
13307:F
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13275:=
13272:g
13264:X
13261:G
13257:1
13253:=
13243:g
13235:X
13232:G
13221:)
13216:X
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13205:G
13202:=
13190:Y
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13145:G
13142:=
13135:g
13107:Ξ·
13105:o
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13052:Y
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13025:)
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13009:F
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12974:)
12969:Y
12961:(
12958:F
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12924:=
12917:f
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12835:F
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12816:)
12813:g
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12765:G
12762:=
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12369:o
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12311:F
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12118:G
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580:t
576:e
573:S
564:D
559::
556:)
553:X
550:G
547:,
541:(
536:D
507:t
503:e
500:S
491:D
486::
483:)
480:X
477:,
471:F
468:(
463:C
441:Y
421:X
394:)
391:X
388:G
385:,
382:Y
379:(
373:D
367:m
364:o
361:h
353:)
350:X
347:,
344:Y
341:F
338:(
332:C
326:m
323:o
320:h
289:D
267:Y
245:C
223:X
198:D
188:C
183::
180:G
158:C
148:D
143::
140:F
111:D
87:C
27:.
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