12507:
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6100:
11331:
8501:
5747:
11697:
1579:
237:
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11572:
12502:{\displaystyle {\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}}
1682:
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5237:
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8453:
5638:
4897:
The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:
10134:
Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above)
8428:
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9447:
5564:
7564:
7492:
9537:
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9114:
7603:
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4357:
is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are
2073:
476:
6679:
5348:
107:, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed
11156:
1964:
1913:
1266:
3205:
275:
10692:. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each
1808:
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7877:
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3462:
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8713:
8676:
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11247:
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2504:
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8448:
8005:
6884:
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1409:
8815:
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169:
137:
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2140:
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One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares
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11365:
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9240:
8269:
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4015:
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2100:
543:
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3303:
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1444:
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Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
11208:
11118:
3533:
1192:
1004:
888:
862:
684:
658:
13194:: 113β34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
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10579:
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388:
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3422:
as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section
12604:
3116:. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.
12371:{\displaystyle {\begin{matrix}\bullet &\xrightarrow {a} &\bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}}
12296:{\displaystyle {\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \end{matrix}}}
10792:
10784:
3874:
12512:
which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.
11454:
13297:
10688:
The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not
4132:. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is
5413:. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the
5409:
notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called
1595:
13288:
6145:
12051:
is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids
12634:
7043:
for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the
2157:
The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let
13061:
13042:
5910:{\displaystyle {\mathcal {G}}_{n}={\mathcal {G}}_{1}\times _{{\mathcal {G}}_{0}}\cdots \times _{{\mathcal {G}}_{0}}{\mathcal {G}}_{1}}
9574:
6095:{\displaystyle {\begin{matrix}U_{ijk}&\to &U_{ij}\\\downarrow &&\downarrow \\U_{ik}&\to &U_{i}\end{matrix}}}
5169:
12551:
Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in
12105:
7608:
12171:
6229:
13231:
9161:
13384:
with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of
12054:
13077:
Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of
11802:
1278:
11396:
13199:
8376:
5576:
5443:
13323:
12998:
9386:
5522:
12926:
12668:
7521:
7458:
13397:
9455:
19:
This article is about groupoids in category theory. For the algebraic structure with a single binary operation, see
13365:", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115β122.
10772:
does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of
10463:
of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets
2936:
13108:
Cegarra, Antonio M.; Heredia, BenjamΓn A.; Remedios, JosuΓ© (2010-03-19). "Double groupoids and homotopy 2-types".
7569:
5758:
5355:
2038:
443:
6635:
5293:
8894:
11123:
4082:
1918:
1867:
11210:
are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids
8991:
1197:
13467:
13347:
13083:
13027:
12660:
9055:
3156:
242:
12043:
There is an additional structure which can be derived from groupoids internal to the category of groupoids,
1769:
1725:
13210:
Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in
11387:
11326:{\displaystyle \operatorname {Grpd} (G\times H,K)\cong \operatorname {Grpd} (G,\operatorname {GPD} (H,K)).}
5440:
A Δech groupoid is a special kind of groupoid associated to an equivalence relation given by an open cover
8496:{\displaystyle {\begin{aligned}&&X\\&&\downarrow \\Y&\rightarrow &Z\end{aligned}}}
7837:
3470:
3441:
13342:
12655:
10177:
choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object
8681:
8644:
3419:
11748:
8718:
7194:
6595:
5991:-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since
5943:
2466:
12650:
11742:
10633:
9245:
7948:
6821:
6351:
3901:
is unique. The covering morphisms of groupoids are especially useful because they can be used to model
2776:
1382:
306:
10725:, one would have to make a coherent choice of paths (or equivalence classes of paths) from each point
8788:
8277:
5742:{\displaystyle {\begin{aligned}s=\phi _{j}:U_{ij}\to U_{j}\\t=\phi _{i}:U_{ij}\to U_{i}\end{aligned}}}
142:
110:
13422:
R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In
10689:
8820:
8371:
4358:
2884:
2832:
2279:
2188:
1116:
396:
13240:
Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial
Representations and Partial Group Algebras".
10459:) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a
8857:
8184:
7918:
2310:
2251:
2219:
2105:
1059:
1009:
13477:
13472:
12529:
10535:
8573:
7701:
7274:
7125:
5159:
4439:
4371:
4235:
3538:
3372:
2645:
2345:
13361:
Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an
11071:
The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the
9119:
5119:
2732:
11996:
11797:
11692:{\displaystyle \hom _{\mathbf {Cat} }(i(G),C)\cong \hom _{\mathbf {Grpd} }(G,\mathrm {Core} (C))}
8606:
8431:
8039:
8010:
7048:
6220:
4481:
4300:
4199:
4163:
2691:
1815:
6795:
6769:
5041:
4413:
2570:
937:
9688:
6725:
4968:
1439:
894:
13337:
11857:
as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a
10230:
10119:. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of
8541:
8509:
6684:
3636:
3594:
901:
13312:
10456:
10452:
9632:
9617:
9563:
7433:
4771:
2397:
1574:{\displaystyle \mathrm {comp} _{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto gf}
1106:. (The previous two axioms already show that these expressions are defined and unambiguous.)
803:
765:
727:
689:
232:{\displaystyle \circ :(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow A\rightarrow C}
13377:
11161:
8102:
7329:
3841:
3130:
2606:
2544:
13389:
13385:
13058:
13039:
12970:
Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection".
12590:
12525:
11983:{\displaystyle \hom _{\mathbf {Grpd} }(\pi _{1}(X),G)\cong \hom _{\mathbf {sSet} }(X,N(G))}
11338:
11213:
10677:
10584:
10353:
10306:
10180:
10167:
10137:
9637:
9622:
9359:
9312:
9285:
9218:
8242:
7766:
7302:
7044:
6961:
6557:
6108:
5266:
5083:
4864:
4832:
4800:
4739:
4592:
4587:
4354:
4137:
3993:
3962:
3029:
2517:
2446:
2377:
2078:
528:
294:
10695:
6993:
6889:
6528:
3752:
3279:
3250:
8:
13303:
12560:
12556:
11379:
11187:
11097:
10174:
10070:
8303:(since this is the group of automorphisms). Then, a quotient groupoid can be of the form
7495:
6300:
5435:
5431:
5156:
3512:
1171:
983:
867:
841:
663:
637:
59:
51:
13406:
Groupoids: unifying internal and external symmetry — A tour through some examples.
11043:, and this is a useful way of obtaining information about presentations of the subgroup
10486:
13275:
13249:
13184:
13173:
13109:
12971:
12384:
11705:
11046:
11026:
11006:
10986:
10962:
10942:
10922:
10902:
10882:
10862:
10838:
10818:
10798:
10788:
10748:
10728:
10659:
10639:
10615:
10564:
10541:
10513:
10466:
10431:
10411:
10380:
10333:
10286:
10266:
10210:
10128:
10048:
9590:
9339:
8971:
8768:
8349:
8222:
7890:
7817:
7793:
7746:
7726:
7681:
7661:
7501:
7438:
7415:
7395:
7375:
7355:
7254:
7234:
7174:
7154:
7057:
7047:
of the group action, and the groupoid is transitive if and only if the group action is
7026:
6941:
6921:
6567:
6508:
6488:
6468:
6448:
6424:
6404:
6384:
6361:
6325:
6305:
5974:
5923:
5499:
5242:
5021:
4997:
4948:
4924:
4904:
4719:
4699:
4679:
4659:
4639:
4616:
4565:
4537:
4517:
4336:
4277:
4143:
4111:
4091:
4060:
4040:
4020:
3973:
3931:
3911:
3884:
3821:
3801:
3781:
3732:
3712:
3692:
3672:
3352:
3332:
3312:
3230:
3210:
3065:
608:
588:
568:
548:
505:
485:
373:
312:
10779:
Morphisms of groupoids come in more kinds than those of groups: we have, for example,
8306:
8158:
8068:
7080:
6137:
366:
A groupoid can be viewed as an algebraic structure consisting of a set with a binary
13433:
13267:
13227:
13177:
12943:
12922:
12664:
12564:
12521:
11375:
10606:
10531:
10205:
9209:
7884:
7020:
5260:
4609:, then a groupoid "representing" this equivalence relation can be formed as follows:
4078:
3968:
3306:
290:
13279:
11063:. For further information, see the books by Higgins and by Brown in the References.
5807:
giving the structure of a groupoid. In fact, this can be further extended by setting
13405:
13319:
13259:
13165:
12600:
12595:
12568:
12552:
11445:
11091:
9644:
9627:
9586:
9543:
9212:
7811:
7075:
6561:
5163:
3506:
3502:
976:
479:
438:
367:
350:
68:
64:
13130:
10721:
in terms of a single group, and this choice can be arbitrary. In the example from
13381:
13355:
13211:
13065:
13046:
12714:). Substituting the first into the second and applying 3. two more times yields (
12038:
10116:
9778:
9755:
9559:
5414:
4085:
3207:
containing every point that can be joined to x by a morphism in G. If two points
2146:
391:
342:
84:
75:
35:
31:
20:
11567:{\displaystyle \hom _{\mathbf {Grpd} }(C,G)\cong \hom _{\mathbf {Cat} }(C,i(G))}
13409:
13307:
13015:
An
Introduction to Groups, Groupoids and Their Representations: An Introduction
12580:
11790:
10401:
9913:
9890:
9570:
6133:
2162:
838:
is defined, then they are both defined (and they are equal to each other), and
100:
13461:
13293:
13271:
12537:
10602:
10558:, but an isomorphism requires specifying the set of points in each component;
10092:
9594:
9593:
acting on 13 points such that the elements fixing a point form a copy of the
6818:
for a right action). Multiplication (or composition) in the groupoid is then
5418:
5406:
630:
4368:
An important extension of this idea is to consider the fundamental groupoid
3908:
It is also true that the category of covering morphisms of a given groupoid
13263:
12991:
12533:
10773:
10769:
10120:
9868:
9845:
9666:
8272:
3902:
92:
13362:
12610:
12545:
11858:
10768:
As a more illuminating example, the classification of groupoids with one
10681:
10115:
If a groupoid has only one object, then the set of its morphisms forms a
7148:
4133:
3465:
1286:
334:
83:
is invertible. A category of this sort can be viewed as augmented with a
27:
13207:
10609:, but an isomorphism requires specifying what each equivalence class is:
10173:
Note that the isomorphism just mentioned is not unique, and there is no
13215:
13169:
12520:
When studying geometrical objects, the arising groupoids often carry a
10451:
In category-theoretic terms, each connected component of a groupoid is
10004:
9981:
9823:
9800:
6132:
are the target maps. This construction can be seen as a model for some
3633:
Particular kinds of morphisms of groupoids are of interest. A morphism
2335:, which gives a groupoid in the algebraic sense. Explicit reference to
1677:{\displaystyle \mathrm {inv} :G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}}
13224:
The group fixed by a family of injective endomorphisms of a free group
13156:
12636:
The Group Fixed by a Family of
Injective Endomorphisms of a Free Group
12536:. These last objects can be also studied in terms of their associated
9573:
form a groupoid (not a group, as not all moves can be composed). This
16:
Category where every morphism is invertible; generalization of a group
13445:
13416:
13254:
13226:, Mathematical Surveys and Monographs, vol. 195, AMS Bookstore,
12541:
10780:
9958:
9935:
7297:
6209:{\displaystyle \in {\check {H}}^{k}({\mathcal {U}},{\underline {A}})}
4129:
3666:
12454:
12401:
the same morphism, they can be vertically conjoined giving a diagram
12323:
12278:
11184:
and whose arrows are the natural equivalences of morphisms. Thus if
8366:
at the origin. Examples like these form the basis for the theory of
13078:
10722:
10460:
9547:
8367:
7880:
6442:
1282:
482:
because it is not necessarily defined for all pairs of elements of
338:
80:
13114:
12976:
4554:
may be chosen according to the geometry of the situation at hand.
12585:
10124:
3630:
is simply a functor between two (category-theoretic) groupoids.
10026:
4583:
3414:
Orbits form a partition of the set X, and a groupoid is called
285:
104:
5232:{\displaystyle X_{0}\times _{Y}X_{0}\subset X_{0}\times X_{0}}
10856:
9309:
while the target morphism is the addition of projection onto
5627:
The source and target maps are then given by the induced maps
5290:
under the surjective map of topological spaces. If we write,
1110:
Two easy and convenient properties follow from these axioms:
301:
95:. A groupoid where there is only one object is a usual group.
12155:{\displaystyle s,t:{\mathcal {G}}_{1}\to {\mathcal {G}}_{0}}
10400:
If a groupoid is not transitive, then it is isomorphic to a
13449:
13437:
7651:{\displaystyle \mathrm {Hom} (\mathrm {Gr} ,\mathrm {Gr} )}
4514:, where one considers only paths whose endpoints belong to
522:
is defined are not articulated here and vary by situation.
13135:
SΓ©minaire
Ehresmann. Topologie et gΓ©omΓ©trie diffΓ©rentielle
12215:{\displaystyle i:{\mathcal {G}}_{0}\to {\mathcal {G}}_{1}}
12029:
denotes the fundamental groupoid of the simplicial set X.
9546:
on a scheme, then this construction can be used to form a
9215:
can be used to form a groupoid. It has as objects the set
6281:{\displaystyle \sigma :\coprod U_{i_{1}\cdots i_{k}}\to A}
2149:
in an arbitrary category admitting finite fiber products.
8099:
is the set of equivalence classes from this group action
6291:
giving an explicit representation of cohomology classes.
3928:
is equivalent to the category of actions of the groupoid
54:
in several equivalent ways. A groupoid can be seen as a:
5080:, and which is completely intransitive (every singleton
2829:. To see that this is well defined, observe that since
2374:
in the algebraic sense, define an equivalence relation
545:
and have the following axiomatic properties: For all
13324:
Nonlinear dynamics of networks: the groupoid formalism
13107:
13017:; Alberto Ibort, Miguel A. Rodriguez; CRC Press, 2019.
12563:. Similarly, one can have groupoids with a compatible
12515:
12412:
12311:
12236:
9542:
Of course, if the abelian category is the category of
9198:{\displaystyle C_{1}{\overset {d}{\rightarrow }}C_{0}}
6002:
2276:
will in fact be defined everywhere. We define β to be
13239:
12410:
12387:
12309:
12234:
12174:
12108:
12092:{\displaystyle {\mathcal {G}}_{1},{\mathcal {G}}_{0}}
12057:
11999:
11873:
11805:
11751:
11708:
11581:
11457:
11399:
11341:
11335:
This result is of interest even if all the groupoids
11250:
11216:
11190:
11164:
11126:
11100:
11049:
11029:
11009:
10989:
10965:
10945:
10925:
10905:
10885:
10865:
10841:
10821:
10801:
10751:
10731:
10698:
10662:
10642:
10618:
10587:
10567:
10544:
10516:
10489:
10469:
10434:
10414:
10383:
10356:
10336:
10309:
10289:
10269:
10233:
10213:
10183:
10140:
9458:
9389:
9362:
9342:
9315:
9288:
9248:
9221:
9164:
9122:
9058:
8994:
8974:
8897:
8860:
8823:
8791:
8771:
8721:
8684:
8647:
8609:
8576:
8544:
8512:
8451:
8442:
Given a diagram of groupoids with groupoid morphisms
8379:
8352:
8309:
8280:
8245:
8225:
8187:
8161:
8105:
8071:
8042:
8013:
7951:
7921:
7893:
7840:
7820:
7796:
7769:
7749:
7729:
7704:
7684:
7664:
7611:
7572:
7524:
7504:
7461:
7441:
7418:
7398:
7378:
7358:
7332:
7305:
7277:
7257:
7237:
7197:
7177:
7157:
7128:
7083:
7060:
7029:
6996:
6964:
6944:
6924:
6892:
6824:
6798:
6772:
6728:
6687:
6638:
6598:
6570:
6531:
6511:
6491:
6471:
6451:
6427:
6407:
6387:
6364:
6328:
6308:
6232:
6148:
6111:
6000:
5977:
5946:
5926:
5816:
5761:
5636:
5579:
5525:
5502:
5446:
5401:
If we relax the reflexivity requirement and consider
5358:
5296:
5269:
5245:
5172:
5122:
5086:
5044:
5024:
5000:
4971:
4951:
4927:
4907:
4867:
4835:
4803:
4774:
4742:
4722:
4702:
4682:
4662:
4642:
4619:
4595:
4568:
4540:
4520:
4484:
4442:
4416:
4374:
4339:
4303:
4280:
4238:
4202:
4166:
4146:
4114:
4094:
4063:
4043:
4023:
3996:
3976:
3934:
3914:
3887:
3844:
3824:
3804:
3784:
3755:
3735:
3715:
3695:
3675:
3639:
3597:
3541:
3515:
3473:
3444:
3375:
3355:
3335:
3315:
3282:
3253:
3233:
3213:
3159:
3133:
3032:
2939:
2887:
2835:
2779:
2735:
2694:
2648:
2609:
2573:
2547:
2520:
2469:
2449:
2400:
2380:
2348:
2313:
2282:
2254:
2222:
2191:
2108:
2081:
2041:
1921:
1870:
1818:
1772:
1728:
1598:
1447:
1385:
1200:
1174:
1119:
1062:
1012:
986:
940:
904:
870:
844:
806:
768:
730:
692:
666:
640:
611:
591:
571:
551:
531:
508:
488:
446:
399:
376:
315:
245:
177:
145:
113:
13430:., 305β324. Amer. Math. Soc., Providence, RI (2006)
13292:
Cambridge Univ. Press. Shows how generalisations of
12559:, which is a Lie groupoid endowed with a compatible
11846:{\displaystyle N:\mathbf {Grpd} \to \mathbf {sSet} }
9566:), certain puzzles are better modeled as groupoids.
13398:
General theory of Lie groupoids and Lie algebroids.
12632:
11437:{\displaystyle i:\mathbf {Grpd} \to \mathbf {Cat} }
8370:. Another commonly studied family of orbifolds are
3064:Sets in the definitions above may be replaced with
13412:, Notices of the AMS, July 1996, pp. 744β752.
13372:Van Nostrand Notes in Mathematics. Republished in
12682:Proof of first property: from 2. and 3. we obtain
12628:
12626:
12501:
12393:
12370:
12295:
12214:
12154:
12091:
12021:
11982:
11845:
11777:
11733:
11691:
11566:
11436:
11359:
11325:
11234:
11202:
11176:
11150:
11112:
11055:
11035:
11015:
10995:
10971:
10951:
10931:
10911:
10891:
10871:
10847:
10827:
10807:
10757:
10737:
10713:
10668:
10648:
10624:
10593:
10573:
10550:
10522:
10498:
10475:
10440:
10420:
10389:
10369:
10342:
10322:
10295:
10275:
10255:
10219:
10196:
10158:
9531:
9441:
9375:
9348:
9328:
9301:
9282:; the source morphism is just the projection onto
9274:
9234:
9197:
9139:
9108:
9044:
8980:
8960:
8883:
8846:
8809:
8785:. Morphisms can be defined as a pair of morphisms
8777:
8757:
8707:
8670:
8633:
8595:
8562:
8530:
8495:
8422:
8358:
8338:
8295:
8263:
8231:
8203:
8173:
8147:
8091:
8057:
8028:
7999:
7937:
7899:
7871:
7826:
7802:
7782:
7755:
7735:
7715:
7690:
7670:
7650:
7597:
7558:
7510:
7486:
7447:
7424:
7404:
7384:
7364:
7344:
7318:
7288:
7263:
7243:
7223:
7183:
7163:
7139:
7114:
7066:
7035:
7011:
6982:
6950:
6930:
6907:
6878:
6810:
6784:
6758:
6714:
6673:
6624:
6576:
6546:
6517:
6497:
6477:
6457:
6433:
6413:
6393:
6370:
6334:
6314:
6280:
6208:
6124:
6094:
5983:
5963:
5932:
5909:
5796:
5741:
5615:
5558:
5508:
5488:
5384:
5342:
5282:
5251:
5231:
5147:
5098:
5072:
5030:
5006:
4983:
4957:
4933:
4913:
4885:
4853:
4821:
4786:
4760:
4728:
4708:
4688:
4668:
4648:
4625:
4601:
4574:
4546:
4526:
4506:
4470:
4428:
4402:
4345:
4325:
4286:
4266:
4224:
4188:
4152:
4120:
4100:
4069:
4049:
4029:
4009:
3982:
3940:
3920:
3893:
3865:
3830:
3810:
3790:
3770:
3741:
3721:
3701:
3681:
3657:
3615:
3583:
3527:
3485:
3456:
3403:
3361:
3341:
3321:
3297:
3268:
3239:
3219:
3199:
3145:
3045:
3014:
2925:
2873:
2821:
2765:
2721:
2680:
2630:
2592:
2559:
2533:
2498:
2455:
2412:
2386:
2359:
2327:
2299:
2268:
2236:
2208:
2134:
2094:
2067:
1958:
1907:
1854:
1802:
1758:
1676:
1573:
1403:
1260:
1186:
1157:
1098:
1048:
998:
964:
926:
882:
856:
830:
792:
762:are defined and are equal. Conversely, if one of
754:
716:
678:
652:
617:
597:
577:
557:
537:
514:
494:
470:
429:
382:
321:
269:
231:
163:
131:
13374:Reprints in Theory and Applications of Categories
11023:can be "lifted" to presentations of the groupoid
10601:is equivalent (as a groupoid) to one copy of the
8423:{\displaystyle \mathbb {P} (n_{1},\ldots ,n_{k})}
6136:. Also, another artifact of this construction is
5616:{\displaystyle {\mathcal {G}}_{1}=\coprod U_{ij}}
5489:{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}
5014:is only in relation with itself, one obtains the
3418:if it has only one orbit (equivalently, if it is
2492:
2491:
103:, a category in general can be viewed as a typed
13459:
10404:of groupoids of the above type, also called its
9442:{\displaystyle c_{1}+c_{0}\in C_{1}\oplus C_{0}}
7147:is the groupoid (category) with one element and
5559:{\displaystyle {\mathcal {G}}_{0}=\coprod U_{i}}
4613:The objects of the groupoid are the elements of
2102:is the set of all morphisms, and the two arrows
1289:, i.e., invertible. More explicitly, a groupoid
13354:Higgins, P. J., "The fundamental groupoid of a
12623:
7559:{\displaystyle \mathrm {Hom} (\mathrm {Gr} ,-)}
7487:{\displaystyle \mathrm {Gr} \to \mathrm {Set} }
5398:of a surjective submersion of smooth manifolds.
3369:, then the isomorphism is given by the mapping
3068:, as is generally the case in category theory.
13358:", J. London Math. Soc. (2) 13 (1976) 145β149.
9532:{\displaystyle t(c_{1}+c_{0})=d(c_{1})+c_{0}.}
5516:. Its objects are given by the disjoint union
4128:, with two paths being equivalent if they are
13313:Geometric Models for Noncommutative Algebras.
12165:and an embedding given by an identity functor
11785:denotes the subcategory of all isomorphisms.
10980:
10656:is equivalent (as a groupoid) to one copy of
8437:
4353:. Accordingly, the fundamental groupoid of a
3429:
3015:{\displaystyle (1_{x}*f*1_{y})*(g*1_{z})=f*g}
171:, say. Composition is then a total function:
13221:
12969:
10123:generalize to groupoids, with the notion of
8142:
8106:
8007:which takes each number to its negative, so
7994:
7958:
7866:
7841:
7598:{\displaystyle \mathrm {ob} (\mathrm {Gr} )}
5797:{\displaystyle \varepsilon :U_{i}\to U_{ii}}
5471:
5457:
5385:{\displaystyle X_{1}\rightrightarrows X_{0}}
5093:
5087:
3247:are in the same orbits, their vertex groups
2152:
2068:{\displaystyle G_{1}\rightrightarrows G_{0}}
471:{\displaystyle *:G\times G\rightharpoonup G}
12992:"Localization and Gromov-Witten Invariants"
11090:is, like the category of small categories,
10684:requires specifying what set each orbit is.
7352:. The categorical structure of the functor
6674:{\displaystyle \mathrm {hom} (C)=G\times X}
5343:{\displaystyle X_{1}=X_{0}\times _{Y}X_{0}}
4921:is in relation with every other element of
3071:
13131:"CatΓ©gories et structures : extraits"
11003:. In this way, presentations of the group
10166:. By transitivity, there will only be one
8961:{\displaystyle (x,\phi ,y),(x',\phi ',y')}
4991:as set of arrows, and which is transitive.
13253:
13128:
13113:
12975:
11151:{\displaystyle \operatorname {GPD} (H,K)}
8381:
8315:
8283:
8189:
7923:
7518:. In fact, this functor is isomorphic to
6105:is a cartesian diagram where the maps to
3509:as a subcategory, i.e., respectively, if
1959:{\displaystyle f^{-1}f=\mathrm {id} _{x}}
1908:{\displaystyle ff^{-1}=\mathrm {id} _{y}}
333:Groupoids are often used to reason about
13185:From groups to groupoids: a brief survey
12921:, 1999, The University of Chicago Press
11066:
9045:{\displaystyle f(\alpha ):f(x)\to f(x')}
3493:that is itself a groupoid. It is called
3053:, and the category-theoretic inverse of
1261:{\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}}
13004:from the original on February 12, 2020.
10530:is equivalent to the collection of the
9562:can be modeled using group theory (see
9109:{\displaystyle g(\beta ):g(y)\to g(y')}
5405:, then it becomes possible to consider
4557:
4436:is a chosen set of "base points". Here
4362:
4297:The orbits of the fundamental groupoid
4274:is then the vertex group for the point
3200:{\displaystyle s(t^{-1}(x))\subseteq X}
270:{\displaystyle h\circ g:A\rightarrow C}
13460:
13222:Dicks, Warren; Ventura, Enric (1996),
13155:
12919:A Concise Course in Algebraic Topology
10765:in the same path-connected component.
9150:
8346:, which has one point with stabilizer
1803:{\displaystyle \mathrm {id} _{y}\ f=f}
1759:{\displaystyle f\ \mathrm {id} _{x}=f}
502:. The precise conditions under which
349:) introduced groupoids implicitly via
346:
13424:Algebraic and geometric combinatorics
12852:is also defined. From 3. we obtain (
12540:, in analogy to the relation between
10776:with one endomorphism is nontrivial.
9604:
6958:, the vertex group consists of those
5570:and its arrows are the intersections
4333:are the path-connected components of
2443:be the set of equivalence classes of
2142:represent the source and the target.
1272:
12965:
12963:
12032:
8968:, there is a commutative diagram in
7872:{\displaystyle \{F_{g}\mid g\in G\}}
3486:{\displaystyle H\rightrightarrows Y}
3457:{\displaystyle G\rightrightarrows X}
12516:Groupoids with geometric structures
11788:
9580:
8708:{\displaystyle y\in {\text{Ob}}(Y)}
8671:{\displaystyle x\in {\text{Ob}}(X)}
8214:
2177:) (i.e. the sets of morphisms from
2145:More generally, one can consider a
370:. Precisely, it is a non-empty set
13:
12201:
12184:
12141:
12124:
12078:
12061:
11778:{\displaystyle \mathrm {Core} (C)}
11762:
11759:
11756:
11753:
11673:
11670:
11667:
11664:
11385:
8758:{\displaystyle \phi :f(x)\to g(y)}
7709:
7706:
7641:
7638:
7630:
7627:
7619:
7616:
7613:
7588:
7585:
7577:
7574:
7543:
7540:
7532:
7529:
7526:
7480:
7477:
7474:
7466:
7463:
7282:
7279:
7224:{\displaystyle X=F(\mathrm {Gr} )}
7214:
7211:
7133:
7130:
7105:
7102:
7099:
7091:
7088:
6646:
6643:
6640:
6625:{\displaystyle \mathrm {ob} (C)=X}
6603:
6600:
6185:
5964:{\displaystyle {\mathcal {G}}_{n}}
5950:
5940:-iterated fiber product where the
5896:
5880:
5856:
5837:
5820:
5583:
5529:
5449:
4696:, there is a single morphism from
4204:
2499:{\displaystyle G_{0}:=G/\!\!\sim }
2353:
2350:
2321:
2318:
2315:
2293:
2290:
2287:
2284:
2262:
2259:
2256:
2230:
2227:
2224:
2202:
2199:
2196:
2193:
1946:
1943:
1895:
1892:
1778:
1775:
1740:
1737:
1606:
1603:
1600:
1459:
1456:
1453:
1450:
1391:
1388:
14:
13489:
12960:
12812:is also defined. Moreover since
11745:that inverts every morphism, and
9275:{\displaystyle C_{1}\oplus C_{0}}
8000:{\displaystyle X=\{-2,-1,0,1,2\}}
7658:which is by definition the "set"
6879:{\displaystyle (h,y)(g,x)=(hg,x)}
5259:has a topology isomorphic to the
5239:is an equivalence relation since
2822:{\displaystyle gf:=f*g\in G(x,z)}
1404:{\displaystyle \mathrm {id} _{x}}
13286:F. Borceux, G. Janelidze, 2001,
13208:Higher dimensional group theory.
12756:Proof of second property: since
11947:
11944:
11941:
11938:
11889:
11886:
11883:
11880:
11839:
11836:
11833:
11830:
11822:
11819:
11816:
11813:
11645:
11642:
11639:
11636:
11594:
11591:
11588:
11531:
11528:
11525:
11473:
11470:
11467:
11464:
11430:
11427:
11424:
11416:
11413:
11410:
11407:
11158:whose objects are the morphisms
10408:(possibly with different groups
8810:{\displaystyle (\alpha ,\beta )}
8296:{\displaystyle \mathbb {A} ^{n}}
6681:and with source and target maps
6358:The objects are the elements of
6223:can be represented as a function
5425:
164:{\displaystyle h:B\rightarrow C}
132:{\displaystyle g:A\rightarrow B}
13376:, No. 7 (2005) pp. 1β195;
13122:
13101:
13071:
13052:
13033:
10448:for each connected component).
8847:{\displaystyle \alpha :x\to x'}
8430:and subspaces of them, such as
7191:of this category defines a set
6294:
4361:as categories (see the section
4232:). The usual fundamental group
4037:. The morphisms from the point
2926:{\displaystyle 1_{y}*(g*1_{z})}
2874:{\displaystyle (1_{x}*f)*1_{y}}
2300:{\displaystyle \mathrm {comp} }
2209:{\displaystyle \mathrm {comp} }
1158:{\displaystyle (a^{-1})^{-1}=a}
430:{\displaystyle {}^{-1}:G\to G,}
13380:. Substantial introduction to
13028:Puzzles, Groups, and Groupoids
13020:
13008:
12984:
12944:"fundamental groupoid in nLab"
12936:
12911:
12676:
12643:
12487:
12475:
12469:
12438:
12432:
12420:
12356:
12344:
12338:
12262:
12256:
12244:
12195:
12135:
12016:
12010:
11977:
11974:
11968:
11956:
11926:
11917:
11911:
11898:
11826:
11772:
11766:
11728:
11712:
11686:
11683:
11677:
11654:
11624:
11615:
11609:
11603:
11561:
11558:
11552:
11540:
11513:
11504:
11488:
11482:
11420:
11370:Another important property of
11317:
11314:
11302:
11287:
11275:
11257:
11168:
11145:
11133:
10899:and hence a covering morphism
10708:
10702:
10581:with the equivalence relation
10250:
10237:
10153:
10141:
9510:
9497:
9488:
9462:
9177:
9103:
9092:
9086:
9083:
9077:
9068:
9062:
9039:
9028:
9022:
9019:
9013:
9004:
8998:
8955:
8922:
8916:
8898:
8884:{\displaystyle \beta :y\to y'}
8870:
8833:
8804:
8792:
8752:
8746:
8740:
8737:
8731:
8702:
8696:
8665:
8659:
8628:
8610:
8554:
8522:
8481:
8467:
8417:
8385:
8333:
8310:
8258:
8252:
8204:{\displaystyle \mathbb {Z} /2}
8168:
8162:
8139:
8133:
8127:
8121:
8115:
8109:
8086:
8072:
8046:
8020:
7938:{\displaystyle \mathbb {Z} /2}
7645:
7623:
7592:
7581:
7553:
7536:
7470:
7336:
7218:
7207:
7109:
7084:
6977:
6965:
6873:
6858:
6852:
6840:
6837:
6825:
6744:
6732:
6703:
6691:
6656:
6650:
6613:
6607:
6272:
6203:
6180:
6168:
6155:
6149:
6073:
6051:
6045:
6023:
5778:
5722:
5673:
5394:which is sometimes called the
5369:
5139:
4880:
4868:
4848:
4836:
4816:
4804:
4755:
4743:
4501:
4495:
4465:
4453:
4397:
4385:
4320:
4314:
4261:
4249:
4219:
4213:
4183:
4177:
4136:. This groupoid is called the
3854:
3848:
3765:
3759:
3649:
3578:
3566:
3557:
3545:
3477:
3448:
3379:
3292:
3286:
3263:
3257:
3188:
3185:
3179:
3163:
2997:
2978:
2972:
2940:
2920:
2901:
2855:
2836:
2816:
2804:
2773:their composite is defined as
2757:
2745:
2716:
2704:
2625:
2613:
2328:{\displaystyle \mathrm {inv} }
2269:{\displaystyle \mathrm {inv} }
2237:{\displaystyle \mathrm {inv} }
2135:{\displaystyle G_{1}\to G_{0}}
2119:
2052:
1849:
1840:
1828:
1819:
1658:
1649:
1637:
1631:
1628:
1616:
1562:
1559:
1547:
1541:
1529:
1523:
1520:
1508:
1499:
1487:
1214:
1201:
1137:
1120:
1099:{\displaystyle {a^{-1}}*a*b=b}
1049:{\displaystyle a*b*{b^{-1}}=a}
825:
813:
781:
769:
749:
737:
705:
693:
462:
418:
356:
261:
223:
217:
214:
208:
202:
199:
196:
190:
184:
155:
123:
1:
13149:
11864:The nerve has a left adjoint
11242:there is a natural bijection
8596:{\displaystyle X\times _{Z}Y}
7915:Consider the group action of
7910:
7716:{\displaystyle \mathrm {Gr} }
7289:{\displaystyle \mathrm {Gr} }
7140:{\displaystyle \mathrm {Gr} }
5403:partial equivalence relations
5110:
4471:{\displaystyle \pi _{1}(X,A)}
4403:{\displaystyle \pi _{1}(X,A)}
4267:{\displaystyle \pi _{1}(X,x)}
3584:{\displaystyle G(x,y)=H(x,y)}
3423:
3404:{\displaystyle g\to fgf^{-1}}
2681:{\displaystyle 1_{x}*f*1_{y}}
2370:Conversely, given a groupoid
2360:{\displaystyle \mathrm {id} }
2244:become partial operations on
12633:Dicks & Ventura (1996).
11444:has both a left and a right
11120:we can construct a groupoid
10510:The fundamental groupoid of
10003:Commutative-and-associative
9589:is a groupoid introduced by
9140:{\displaystyle \phi ,\phi '}
8271:gives a group action on the
7271:(i.e. for every morphism in
6560:of morphisms interprets the
5148:{\displaystyle f:X_{0}\to Y}
3096:are the subsets of the form
2766:{\displaystyle g\in G(y,z),}
361:
50:) generalises the notion of
7:
13395:Mackenzie, K. C. H., 2005.
13343:Encyclopedia of Mathematics
13129:Ehresmann, Charles (1964).
12656:Encyclopedia of Mathematics
12574:
12022:{\displaystyle \pi _{1}(X)}
9569:The transformations of the
8634:{\displaystyle (x,\phi ,y)}
8570:, we can form the groupoid
8058:{\displaystyle 1\mapsto -1}
8029:{\displaystyle -2\mapsto 2}
7834:is isomorphic to the group
6485:correspond to the elements
4507:{\displaystyle \pi _{1}(X)}
4478:is a (wide) subgroupoid of
4326:{\displaystyle \pi _{1}(X)}
4225:{\displaystyle \Pi _{1}(X)}
4189:{\displaystyle \pi _{1}(X)}
3956:
3951:
3022:. The identity morphism on
2722:{\displaystyle f\in G(x,y)}
2638:as the set of all elements
1855:{\displaystyle (hg)f=h(gf)}
1426:for each triple of objects
10:
13494:
13059:The 15-puzzle groupoid (2)
13040:The 15-puzzle groupoid (1)
12036:
11743:localization of a category
9558:While puzzles such as the
9553:
8603:whose objects are triples
8438:Fiber product of groupoids
8372:weighted projective spaces
6886:which is defined provided
6811:{\displaystyle X\rtimes G}
6785:{\displaystyle G\ltimes X}
5429:
5073:{\displaystyle s=t=id_{X}}
4429:{\displaystyle A\subset X}
3960:
3873:. A fibration is called a
3430:Subgroupoids and morphisms
2593:{\displaystyle x\in G_{0}}
965:{\displaystyle a*{a^{-1}}}
18:
13370:Categories and groupoids.
12603:(not to be confused with
6759:{\displaystyle t(g,x)=gx}
6592:is a small category with
4984:{\displaystyle X\times X}
4365:for the general theory).
3665:of groupoids is called a
2153:Comparing the definitions
1584:for each pair of objects
1308:for each pair of objects
87:on the morphisms, called
13417:The Geometry of Momentum
13388:, and in topology, e.g.
13030:, The Everything Seminar
12616:
12555:one has the notion of a
12530:differentiable structure
10536:path-connected component
10256:{\displaystyle G(x_{0})}
8563:{\displaystyle g:Y\to Z}
8531:{\displaystyle f:X\to Z}
8065:. The quotient groupoid
7171:. Indeed, every functor
7054:Another way to describe
6715:{\displaystyle s(g,x)=x}
3658:{\displaystyle p:E\to B}
3616:{\displaystyle x,y\in Y}
3072:Vertex groups and orbits
2035:is sometimes denoted as
927:{\displaystyle a^{-1}*a}
305:: sets equipped with an
13368:Higgins, P. J., 1971.
13200:Topology and groupoids.
13189:Bull. London Math. Soc.
6348:transformation groupoid
6342:, then we can form the
6221:sheaf of abelian groups
4965:, which has the entire
4787:{\displaystyle x\sim y}
2413:{\displaystyle a\sim b}
1316:a (possibly empty) set
831:{\displaystyle a*(b*c)}
793:{\displaystyle (a*b)*c}
755:{\displaystyle a*(b*c)}
717:{\displaystyle (a*b)*c}
280:Special cases include:
13401:Cambridge Univ. Press.
13328:Bull. Amer. Math. Soc.
13322:, Ian Stewart, 2006, "
13264:10.1006/jabr.1999.8204
13183:Brown, Ronald, 1987, "
12510:
12503:
12395:
12379:
12372:
12297:
12223:
12216:
12163:
12156:
12093:
12023:
11984:
11847:
11779:
11735:
11693:
11568:
11438:
11361:
11327:
11236:
11204:
11178:
11177:{\displaystyle H\to K}
11152:
11114:
11057:
11037:
11017:
10997:
10973:
10953:
10933:
10913:
10893:
10873:
10849:
10829:
10809:
10759:
10739:
10715:
10680:of the action, but an
10670:
10650:
10626:
10595:
10575:
10552:
10524:
10500:
10483:, but only the groups
10477:
10442:
10422:
10391:
10371:
10344:
10324:
10297:
10277:
10257:
10221:
10198:
10160:
9610:Group-like structures
9533:
9443:
9377:
9350:
9330:
9303:
9276:
9242:and as arrows the set
9236:
9199:
9141:
9110:
9046:
8982:
8962:
8891:such that for triples
8885:
8848:
8811:
8779:
8759:
8709:
8672:
8635:
8597:
8564:
8532:
8497:
8424:
8360:
8340:
8297:
8265:
8233:
8205:
8181:has a group action of
8175:
8149:
8148:{\displaystyle \{,,\}}
8093:
8059:
8030:
8001:
7939:
7901:
7873:
7828:
7804:
7784:
7757:
7737:
7717:
7692:
7672:
7652:
7599:
7560:
7512:
7488:
7449:
7426:
7406:
7386:
7366:
7346:
7345:{\displaystyle X\to X}
7320:
7290:
7265:
7245:
7225:
7185:
7165:
7141:
7116:
7068:
7037:
7013:
6984:
6952:
6932:
6909:
6880:
6812:
6786:
6766:. It is often denoted
6760:
6716:
6675:
6626:
6578:
6548:
6519:
6499:
6479:
6459:
6435:
6415:
6395:
6372:
6336:
6316:
6289:
6282:
6217:
6210:
6126:
6103:
6096:
5985:
5965:
5934:
5918:
5911:
5805:
5798:
5750:
5743:
5625:
5617:
5568:
5560:
5510:
5490:
5393:
5386:
5350:then we get a groupoid
5344:
5284:
5253:
5233:
5149:
5100:
5074:
5032:
5008:
4985:
4959:
4935:
4915:
4887:
4855:
4823:
4788:
4762:
4730:
4710:
4690:
4670:
4650:
4627:
4603:
4576:
4548:
4528:
4508:
4472:
4430:
4404:
4347:
4327:
4288:
4268:
4226:
4190:
4154:
4122:
4102:
4071:
4051:
4031:
4011:
3984:
3942:
3922:
3895:
3867:
3866:{\displaystyle p(e)=b}
3832:
3812:
3792:
3772:
3743:
3723:
3703:
3683:
3659:
3617:
3585:
3529:
3487:
3458:
3426:for counterexamples).
3405:
3363:
3343:
3323:
3299:
3270:
3241:
3221:
3201:
3147:
3146:{\displaystyle x\in X}
3047:
3016:
2927:
2875:
2823:
2767:
2723:
2682:
2632:
2631:{\displaystyle G(x,y)}
2594:
2561:
2560:{\displaystyle a\in G}
2535:
2500:
2457:
2414:
2388:
2361:
2329:
2301:
2270:
2238:
2210:
2136:
2096:
2069:
1960:
1909:
1856:
1804:
1760:
1678:
1575:
1405:
1262:
1188:
1159:
1100:
1050:
1000:
966:
928:
884:
858:
832:
794:
756:
718:
680:
654:
619:
599:
579:
559:
539:
516:
496:
472:
431:
384:
323:
271:
233:
165:
133:
13248:. Elsevier: 505β532.
13158:Mathematische Annalen
12526:topological groupoids
12504:
12403:
12396:
12373:
12298:
12227:
12217:
12167:
12157:
12101:
12094:
12024:
11985:
11848:
11780:
11736:
11694:
11569:
11439:
11362:
11360:{\displaystyle G,H,K}
11328:
11237:
11235:{\displaystyle G,H,K}
11205:
11179:
11153:
11115:
11077:category of groupoids
11067:Category of groupoids
11058:
11038:
11018:
10998:
10974:
10954:
10934:
10914:
10894:
10874:
10850:
10830:
10810:
10760:
10740:
10716:
10671:
10651:
10627:
10596:
10594:{\displaystyle \sim }
10576:
10553:
10525:
10501:
10478:
10443:
10423:
10392:
10372:
10370:{\displaystyle x_{0}}
10345:
10325:
10323:{\displaystyle x_{0}}
10298:
10278:
10258:
10222:
10199:
10197:{\displaystyle x_{0}}
10161:
10159:{\displaystyle (G,X)}
9534:
9444:
9378:
9376:{\displaystyle C_{0}}
9351:
9331:
9329:{\displaystyle C_{1}}
9304:
9302:{\displaystyle C_{0}}
9277:
9237:
9235:{\displaystyle C_{0}}
9200:
9142:
9111:
9047:
8983:
8963:
8886:
8849:
8812:
8780:
8760:
8710:
8673:
8636:
8598:
8565:
8533:
8498:
8425:
8361:
8341:
8298:
8266:
8264:{\displaystyle GL(n)}
8234:
8206:
8176:
8150:
8094:
8060:
8031:
8002:
7940:
7902:
7874:
7829:
7810:. We deduce from the
7805:
7785:
7783:{\displaystyle F_{g}}
7763:) to the permutation
7758:
7738:
7718:
7693:
7673:
7653:
7600:
7561:
7513:
7496:Cayley representation
7489:
7450:
7434:representable functor
7427:
7407:
7387:
7367:
7347:
7321:
7319:{\displaystyle F_{g}}
7291:
7266:
7246:
7226:
7186:
7166:
7142:
7117:
7069:
7038:
7014:
6985:
6983:{\displaystyle (g,x)}
6953:
6933:
6910:
6881:
6813:
6787:
6761:
6717:
6676:
6627:
6588:More explicitly, the
6579:
6549:
6520:
6500:
6480:
6460:
6436:
6416:
6396:
6381:For any two elements
6373:
6337:
6317:
6283:
6225:
6211:
6141:
6127:
6125:{\displaystyle U_{i}}
6097:
5993:
5986:
5966:
5935:
5912:
5809:
5799:
5754:
5752:and the inclusion map
5744:
5629:
5618:
5572:
5561:
5518:
5511:
5491:
5387:
5351:
5345:
5285:
5283:{\displaystyle X_{0}}
5254:
5234:
5150:
5101:
5099:{\displaystyle \{x\}}
5075:
5033:
5009:
4986:
4960:
4936:
4916:
4888:
4886:{\displaystyle (z,x)}
4856:
4854:{\displaystyle (y,x)}
4824:
4822:{\displaystyle (z,y)}
4789:
4763:
4761:{\displaystyle (y,x)}
4731:
4711:
4691:
4671:
4651:
4636:For any two elements
4628:
4604:
4602:{\displaystyle \sim }
4586:, i.e. a set with an
4577:
4549:
4529:
4509:
4473:
4431:
4405:
4348:
4328:
4289:
4269:
4227:
4191:
4155:
4123:
4103:
4072:
4052:
4032:
4012:
4010:{\displaystyle G_{0}}
3985:
3943:
3923:
3896:
3879:covering of groupoids
3868:
3833:
3813:
3793:
3773:
3744:
3724:
3704:
3684:
3660:
3618:
3586:
3530:
3488:
3459:
3406:
3364:
3344:
3329:is any morphism from
3324:
3300:
3271:
3242:
3222:
3202:
3148:
3048:
3046:{\displaystyle 1_{x}}
3017:
2928:
2876:
2824:
2768:
2724:
2683:
2633:
2595:
2562:
2536:
2534:{\displaystyle 1_{x}}
2501:
2458:
2456:{\displaystyle \sim }
2415:
2389:
2387:{\displaystyle \sim }
2362:
2330:
2302:
2271:
2239:
2211:
2137:
2097:
2095:{\displaystyle G_{1}}
2070:
1961:
1910:
1857:
1805:
1761:
1679:
1576:
1406:
1379:a designated element
1263:
1189:
1160:
1101:
1051:
1001:
967:
929:
885:
859:
833:
795:
757:
719:
681:
655:
620:
600:
580:
560:
540:
538:{\displaystyle \ast }
517:
497:
473:
432:
385:
324:
272:
234:
166:
134:
13468:Algebraic structures
13434:fundamental groupoid
13408:" Also available in
13390:fundamental groupoid
13068:, Never Ending Books
13049:, Never Ending Books
12591:Homotopy type theory
12532:, turning them into
12524:, turning them into
12408:
12385:
12307:
12232:
12172:
12106:
12055:
11997:
11871:
11803:
11749:
11706:
11579:
11455:
11397:
11339:
11248:
11214:
11188:
11162:
11124:
11098:
11094:: for any groupoids
11079:, and is denoted by
11047:
11027:
11007:
10987:
10963:
10943:
10923:
10903:
10883:
10863:
10839:
10835:yields an action of
10819:
10799:
10749:
10729:
10714:{\displaystyle G(x)}
10696:
10660:
10640:
10616:
10585:
10565:
10542:
10514:
10487:
10467:
10432:
10412:
10406:connected components
10381:
10354:
10334:
10307:
10287:
10267:
10231:
10211:
10181:
10138:
9456:
9387:
9360:
9356:and projection onto
9340:
9313:
9286:
9246:
9219:
9162:
9120:
9056:
8992:
8972:
8895:
8858:
8821:
8789:
8769:
8719:
8682:
8645:
8607:
8574:
8542:
8510:
8449:
8432:CalabiβYau orbifolds
8377:
8350:
8307:
8278:
8243:
8223:
8185:
8159:
8103:
8069:
8040:
8011:
7949:
7919:
7891:
7838:
7818:
7794:
7767:
7747:
7727:
7702:
7682:
7662:
7609:
7570:
7522:
7502:
7459:
7439:
7416:
7396:
7376:
7356:
7330:
7303:
7275:
7255:
7235:
7195:
7175:
7155:
7126:
7081:
7058:
7027:
7019:, which is just the
7012:{\displaystyle gx=x}
6994:
6962:
6942:
6922:
6908:{\displaystyle y=gx}
6890:
6822:
6796:
6770:
6726:
6685:
6636:
6596:
6568:
6547:{\displaystyle gx=y}
6529:
6509:
6489:
6469:
6449:
6425:
6405:
6385:
6362:
6350:) representing this
6326:
6306:
6230:
6146:
6109:
5998:
5975:
5944:
5924:
5814:
5759:
5634:
5577:
5523:
5500:
5444:
5356:
5294:
5267:
5243:
5170:
5120:
5084:
5042:
5022:
4998:
4994:If every element of
4969:
4949:
4925:
4905:
4901:If every element of
4865:
4833:
4801:
4772:
4740:
4720:
4700:
4680:
4660:
4640:
4617:
4593:
4588:equivalence relation
4566:
4558:Equivalence relation
4538:
4518:
4482:
4440:
4414:
4372:
4355:path-connected space
4337:
4301:
4278:
4236:
4200:
4164:
4144:
4138:fundamental groupoid
4112:
4092:
4061:
4041:
4021:
3994:
3974:
3963:Fundamental groupoid
3932:
3912:
3885:
3842:
3822:
3802:
3782:
3778:there is a morphism
3771:{\displaystyle p(x)}
3753:
3733:
3713:
3693:
3673:
3637:
3595:
3539:
3513:
3471:
3442:
3373:
3353:
3333:
3313:
3298:{\displaystyle G(y)}
3280:
3269:{\displaystyle G(x)}
3251:
3231:
3211:
3157:
3153:is given by the set
3131:
3030:
2937:
2885:
2833:
2777:
2733:
2692:
2646:
2607:
2571:
2545:
2518:
2467:
2447:
2398:
2378:
2346:
2311:
2280:
2252:
2220:
2189:
2106:
2079:
2039:
1919:
1868:
1816:
1770:
1726:
1684:satisfying, for any
1596:
1445:
1383:
1198:
1172:
1117:
1060:
1010:
984:
938:
902:
868:
842:
804:
766:
728:
690:
664:
638:
609:
589:
569:
549:
529:
506:
486:
444:
397:
374:
313:
295:equivalence relation
243:
175:
143:
111:
13378:freely downloadable
13316:Especially Part VI.
13304:Cannas da Silva, A.
13084:"delooping in nLab"
12764:is defined, so is (
12651:"Brandt semi-group"
12557:symplectic groupoid
12458:
12327:
12282:
11374:is that it is both
11203:{\displaystyle H,K}
11113:{\displaystyle H,K}
10979:is a groupoid with
10789:universal morphisms
9611:
9577:on configurations.
9155:A two term complex
9151:Homological algebra
7412:-action on the set
5436:Nerve of a covering
5432:Simplicial manifold
4797:The composition of
4079:equivalence classes
3881:if further such an
3669:if for each object
3528:{\displaystyle X=Y}
2394:on its elements by
2165:of all of the sets
1187:{\displaystyle a*b}
999:{\displaystyle a*b}
972:are always defined.
883:{\displaystyle b*c}
857:{\displaystyle a*b}
679:{\displaystyle b*c}
653:{\displaystyle a*b}
478:. Here * is not a
343:Heinrich Brandt
99:In the presence of
13415:Weinstein, Alan, "
13404:Weinstein, Alan, "
13242:Journal of Algebra
13218:. Many references.
13170:10.1007/BF01209171
13064:2015-12-25 at the
13045:2015-12-25 at the
12820:is defined, so is
12605:algebraic groupoid
12499:
12497:
12391:
12368:
12366:
12293:
12291:
12212:
12152:
12089:
12019:
11980:
11843:
11775:
11731:
11689:
11564:
11434:
11357:
11323:
11232:
11200:
11174:
11148:
11110:
11053:
11033:
11013:
10993:
10969:
10949:
10929:
10909:
10889:
10869:
10845:
10825:
10805:
10795:. Thus a subgroup
10793:quotient morphisms
10785:covering morphisms
10755:
10735:
10711:
10666:
10646:
10622:
10591:
10571:
10548:
10532:fundamental groups
10520:
10499:{\displaystyle G.}
10496:
10473:
10438:
10418:
10387:
10367:
10340:
10320:
10293:
10273:
10253:
10217:
10194:
10170:under the action.
10156:
10129:group homomorphism
10127:replacing that of
10049:Commutative monoid
9609:
9605:Relation to groups
9591:John Horton Conway
9564:Rubik's Cube group
9529:
9439:
9373:
9346:
9326:
9299:
9272:
9232:
9195:
9137:
9106:
9042:
8978:
8958:
8881:
8844:
8807:
8775:
8755:
8705:
8668:
8631:
8593:
8560:
8528:
8493:
8491:
8420:
8356:
8336:
8293:
8261:
8229:
8201:
8171:
8145:
8089:
8055:
8026:
7997:
7945:on the finite set
7935:
7897:
7869:
7824:
7800:
7780:
7753:
7733:
7723:(i.e. the element
7713:
7688:
7668:
7648:
7595:
7556:
7508:
7484:
7445:
7422:
7402:
7382:
7362:
7342:
7316:
7286:
7261:
7241:
7221:
7181:
7161:
7137:
7112:
7064:
7033:
7009:
6980:
6948:
6928:
6905:
6876:
6808:
6782:
6756:
6712:
6671:
6622:
6574:
6544:
6515:
6495:
6475:
6455:
6431:
6411:
6391:
6368:
6332:
6312:
6278:
6219:for some constant
6206:
6201:
6122:
6092:
6090:
5981:
5961:
5930:
5907:
5794:
5739:
5737:
5613:
5556:
5506:
5486:
5382:
5340:
5280:
5249:
5229:
5145:
5096:
5070:
5038:as set of arrows,
5028:
5004:
4981:
4955:
4931:
4911:
4883:
4851:
4819:
4784:
4758:
4726:
4706:
4686:
4666:
4646:
4623:
4599:
4572:
4544:
4524:
4504:
4468:
4426:
4400:
4343:
4323:
4284:
4264:
4222:
4186:
4150:
4118:
4098:
4067:
4047:
4027:
4007:
3980:
3938:
3918:
3891:
3863:
3828:
3808:
3788:
3768:
3739:
3719:
3709:and each morphism
3699:
3679:
3655:
3613:
3581:
3525:
3483:
3454:
3401:
3359:
3339:
3319:
3295:
3266:
3237:
3217:
3197:
3143:
3043:
3012:
2923:
2871:
2819:
2763:
2719:
2678:
2628:
2590:
2557:
2531:
2496:
2453:
2410:
2384:
2367:) can be dropped.
2357:
2325:
2297:
2266:
2234:
2206:
2132:
2092:
2065:
1956:
1905:
1852:
1800:
1756:
1674:
1571:
1401:
1273:Category theoretic
1258:
1184:
1155:
1096:
1046:
996:
962:
924:
880:
854:
828:
790:
752:
714:
686:are defined, then
676:
650:
615:
595:
575:
555:
535:
512:
492:
468:
427:
380:
319:
293:that come with an
267:
229:
161:
129:
13426:, volume 423 of
13386:Grushko's theorem
13233:978-0-8218-0564-0
12569:complex structure
12565:Riemannian metric
12459:
12394:{\displaystyle a}
12328:
12283:
12033:Groupoids in Grpd
11734:{\displaystyle C}
11367:are just groups.
11073:groupoid category
11056:{\displaystyle H}
11036:{\displaystyle K}
11016:{\displaystyle G}
10996:{\displaystyle H}
10972:{\displaystyle K}
10952:{\displaystyle G}
10932:{\displaystyle K}
10912:{\displaystyle p}
10892:{\displaystyle G}
10872:{\displaystyle H}
10848:{\displaystyle G}
10828:{\displaystyle G}
10808:{\displaystyle H}
10758:{\displaystyle q}
10738:{\displaystyle p}
10669:{\displaystyle G}
10649:{\displaystyle G}
10632:equipped with an
10625:{\displaystyle X}
10607:equivalence class
10574:{\displaystyle X}
10551:{\displaystyle X}
10523:{\displaystyle X}
10476:{\displaystyle X}
10441:{\displaystyle X}
10421:{\displaystyle G}
10390:{\displaystyle x}
10343:{\displaystyle G}
10296:{\displaystyle x}
10276:{\displaystyle G}
10220:{\displaystyle h}
10206:group isomorphism
10113:
10112:
9383:. That is, given
9349:{\displaystyle d}
9183:
8981:{\displaystyle Z}
8778:{\displaystyle Z}
8694:
8657:
8359:{\displaystyle G}
8232:{\displaystyle G}
8219:Any finite group
7900:{\displaystyle G}
7827:{\displaystyle G}
7803:{\displaystyle G}
7756:{\displaystyle G}
7736:{\displaystyle g}
7691:{\displaystyle g}
7678:and the morphism
7671:{\displaystyle G}
7511:{\displaystyle G}
7448:{\displaystyle F}
7425:{\displaystyle G}
7405:{\displaystyle G}
7385:{\displaystyle F}
7365:{\displaystyle F}
7264:{\displaystyle G}
7244:{\displaystyle g}
7184:{\displaystyle F}
7164:{\displaystyle G}
7067:{\displaystyle G}
7036:{\displaystyle x}
7021:isotropy subgroup
6951:{\displaystyle X}
6931:{\displaystyle x}
6577:{\displaystyle G}
6518:{\displaystyle G}
6498:{\displaystyle g}
6478:{\displaystyle y}
6458:{\displaystyle x}
6434:{\displaystyle X}
6414:{\displaystyle y}
6394:{\displaystyle x}
6371:{\displaystyle X}
6335:{\displaystyle X}
6315:{\displaystyle G}
6194:
6171:
5984:{\displaystyle n}
5933:{\displaystyle n}
5509:{\displaystyle X}
5496:of some manifold
5261:quotient topology
5252:{\displaystyle Y}
5031:{\displaystyle X}
5007:{\displaystyle X}
4958:{\displaystyle X}
4934:{\displaystyle X}
4914:{\displaystyle X}
4768:) if and only if
4729:{\displaystyle y}
4709:{\displaystyle x}
4689:{\displaystyle X}
4669:{\displaystyle y}
4649:{\displaystyle x}
4626:{\displaystyle X}
4575:{\displaystyle X}
4547:{\displaystyle A}
4527:{\displaystyle A}
4346:{\displaystyle X}
4287:{\displaystyle x}
4153:{\displaystyle X}
4121:{\displaystyle q}
4101:{\displaystyle p}
4070:{\displaystyle q}
4050:{\displaystyle p}
4030:{\displaystyle X}
3983:{\displaystyle X}
3969:topological space
3941:{\displaystyle B}
3921:{\displaystyle B}
3894:{\displaystyle e}
3875:covering morphism
3831:{\displaystyle x}
3811:{\displaystyle E}
3791:{\displaystyle e}
3742:{\displaystyle B}
3722:{\displaystyle b}
3702:{\displaystyle E}
3682:{\displaystyle x}
3628:groupoid morphism
3362:{\displaystyle y}
3342:{\displaystyle x}
3322:{\displaystyle f}
3240:{\displaystyle y}
3220:{\displaystyle x}
3112:is any object of
3076:Given a groupoid
1976:is an element of
1790:
1734:
1375:for every object
1360:is an element of
1356:to indicate that
1194:is defined, then
1006:is defined, then
890:are also defined.
618:{\displaystyle G}
598:{\displaystyle c}
578:{\displaystyle b}
558:{\displaystyle a}
515:{\displaystyle *}
495:{\displaystyle G}
383:{\displaystyle G}
351:Brandt semigroups
322:{\displaystyle G}
13485:
13351:
13298:Galois groupoids
13289:Galois theories.
13283:
13257:
13236:
13180:
13143:
13142:
13126:
13120:
13119:
13117:
13105:
13099:
13097:
13095:
13094:
13075:
13069:
13056:
13050:
13037:
13031:
13026:Jim Belk (2008)
13024:
13018:
13012:
13006:
13005:
13003:
12996:
12988:
12982:
12981:
12979:
12967:
12958:
12957:
12955:
12954:
12940:
12934:
12915:
12909:
12680:
12674:
12673:
12647:
12641:
12640:
12630:
12601:Groupoid algebra
12596:Inverse category
12553:Poisson geometry
12508:
12506:
12505:
12500:
12498:
12473:
12450:
12436:
12400:
12398:
12397:
12392:
12377:
12375:
12374:
12369:
12367:
12342:
12319:
12302:
12300:
12299:
12294:
12292:
12274:
12260:
12221:
12219:
12218:
12213:
12211:
12210:
12205:
12204:
12194:
12193:
12188:
12187:
12161:
12159:
12158:
12153:
12151:
12150:
12145:
12144:
12134:
12133:
12128:
12127:
12098:
12096:
12095:
12090:
12088:
12087:
12082:
12081:
12071:
12070:
12065:
12064:
12045:double-groupoids
12028:
12026:
12025:
12020:
12009:
12008:
11989:
11987:
11986:
11981:
11952:
11951:
11950:
11910:
11909:
11894:
11893:
11892:
11852:
11850:
11849:
11844:
11842:
11825:
11784:
11782:
11781:
11776:
11765:
11740:
11738:
11737:
11732:
11727:
11726:
11698:
11696:
11695:
11690:
11676:
11650:
11649:
11648:
11599:
11598:
11597:
11573:
11571:
11570:
11565:
11536:
11535:
11534:
11503:
11502:
11478:
11477:
11476:
11443:
11441:
11440:
11435:
11433:
11419:
11366:
11364:
11363:
11358:
11332:
11330:
11329:
11324:
11241:
11239:
11238:
11233:
11209:
11207:
11206:
11201:
11183:
11181:
11180:
11175:
11157:
11155:
11154:
11149:
11119:
11117:
11116:
11111:
11092:Cartesian closed
11062:
11060:
11059:
11054:
11042:
11040:
11039:
11034:
11022:
11020:
11019:
11014:
11002:
11000:
10999:
10994:
10978:
10976:
10975:
10970:
10958:
10956:
10955:
10950:
10938:
10936:
10935:
10930:
10918:
10916:
10915:
10910:
10898:
10896:
10895:
10890:
10878:
10876:
10875:
10870:
10854:
10852:
10851:
10846:
10834:
10832:
10831:
10826:
10814:
10812:
10811:
10806:
10764:
10762:
10761:
10756:
10744:
10742:
10741:
10736:
10720:
10718:
10717:
10712:
10675:
10673:
10672:
10667:
10655:
10653:
10652:
10647:
10631:
10629:
10628:
10623:
10600:
10598:
10597:
10592:
10580:
10578:
10577:
10572:
10557:
10555:
10554:
10549:
10529:
10527:
10526:
10521:
10505:
10503:
10502:
10497:
10482:
10480:
10479:
10474:
10447:
10445:
10444:
10439:
10427:
10425:
10424:
10419:
10396:
10394:
10393:
10388:
10376:
10374:
10373:
10368:
10366:
10365:
10349:
10347:
10346:
10341:
10330:, a morphism in
10329:
10327:
10326:
10321:
10319:
10318:
10302:
10300:
10299:
10294:
10282:
10280:
10279:
10274:
10262:
10260:
10259:
10254:
10249:
10248:
10226:
10224:
10223:
10218:
10203:
10201:
10200:
10195:
10193:
10192:
10165:
10163:
10162:
10157:
9612:
9608:
9587:Mathieu groupoid
9581:Mathieu groupoid
9544:coherent sheaves
9538:
9536:
9535:
9530:
9525:
9524:
9509:
9508:
9487:
9486:
9474:
9473:
9448:
9446:
9445:
9440:
9438:
9437:
9425:
9424:
9412:
9411:
9399:
9398:
9382:
9380:
9379:
9374:
9372:
9371:
9355:
9353:
9352:
9347:
9335:
9333:
9332:
9327:
9325:
9324:
9308:
9306:
9305:
9300:
9298:
9297:
9281:
9279:
9278:
9273:
9271:
9270:
9258:
9257:
9241:
9239:
9238:
9233:
9231:
9230:
9213:Abelian category
9208:of objects in a
9204:
9202:
9201:
9196:
9194:
9193:
9184:
9176:
9174:
9173:
9146:
9144:
9143:
9138:
9136:
9115:
9113:
9112:
9107:
9102:
9051:
9049:
9048:
9043:
9038:
8987:
8985:
8984:
8979:
8967:
8965:
8964:
8959:
8954:
8943:
8932:
8890:
8888:
8887:
8882:
8880:
8853:
8851:
8850:
8845:
8843:
8816:
8814:
8813:
8808:
8784:
8782:
8781:
8776:
8764:
8762:
8761:
8756:
8714:
8712:
8711:
8706:
8695:
8692:
8677:
8675:
8674:
8669:
8658:
8655:
8640:
8638:
8637:
8632:
8602:
8600:
8599:
8594:
8589:
8588:
8569:
8567:
8566:
8561:
8537:
8535:
8534:
8529:
8502:
8500:
8499:
8494:
8492:
8465:
8464:
8456:
8455:
8429:
8427:
8426:
8421:
8416:
8415:
8397:
8396:
8384:
8365:
8363:
8362:
8357:
8345:
8343:
8342:
8339:{\displaystyle }
8337:
8329:
8324:
8323:
8318:
8302:
8300:
8299:
8294:
8292:
8291:
8286:
8270:
8268:
8267:
8262:
8238:
8236:
8235:
8230:
8215:Quotient variety
8210:
8208:
8207:
8202:
8197:
8192:
8180:
8178:
8177:
8174:{\displaystyle }
8172:
8154:
8152:
8151:
8146:
8098:
8096:
8095:
8092:{\displaystyle }
8090:
8082:
8064:
8062:
8061:
8056:
8035:
8033:
8032:
8027:
8006:
8004:
8003:
7998:
7944:
7942:
7941:
7936:
7931:
7926:
7906:
7904:
7903:
7898:
7883:of the group of
7878:
7876:
7875:
7870:
7853:
7852:
7833:
7831:
7830:
7825:
7812:Yoneda embedding
7809:
7807:
7806:
7801:
7789:
7787:
7786:
7781:
7779:
7778:
7762:
7760:
7759:
7754:
7742:
7740:
7739:
7734:
7722:
7720:
7719:
7714:
7712:
7697:
7695:
7694:
7689:
7677:
7675:
7674:
7669:
7657:
7655:
7654:
7649:
7644:
7633:
7622:
7604:
7602:
7601:
7596:
7591:
7580:
7565:
7563:
7562:
7557:
7546:
7535:
7517:
7515:
7514:
7509:
7493:
7491:
7490:
7485:
7483:
7469:
7454:
7452:
7451:
7446:
7431:
7429:
7428:
7423:
7411:
7409:
7408:
7403:
7391:
7389:
7388:
7383:
7372:assures us that
7371:
7369:
7368:
7363:
7351:
7349:
7348:
7343:
7325:
7323:
7322:
7317:
7315:
7314:
7295:
7293:
7292:
7287:
7285:
7270:
7268:
7267:
7262:
7250:
7248:
7247:
7242:
7230:
7228:
7227:
7222:
7217:
7190:
7188:
7187:
7182:
7170:
7168:
7167:
7162:
7146:
7144:
7143:
7138:
7136:
7121:
7119:
7118:
7115:{\displaystyle }
7113:
7108:
7094:
7076:functor category
7073:
7071:
7070:
7065:
7042:
7040:
7039:
7034:
7018:
7016:
7015:
7010:
6989:
6987:
6986:
6981:
6957:
6955:
6954:
6949:
6937:
6935:
6934:
6929:
6914:
6912:
6911:
6906:
6885:
6883:
6882:
6877:
6817:
6815:
6814:
6809:
6791:
6789:
6788:
6783:
6765:
6763:
6762:
6757:
6721:
6719:
6718:
6713:
6680:
6678:
6677:
6672:
6649:
6631:
6629:
6628:
6623:
6606:
6583:
6581:
6580:
6575:
6562:binary operation
6553:
6551:
6550:
6545:
6524:
6522:
6521:
6516:
6504:
6502:
6501:
6496:
6484:
6482:
6481:
6476:
6464:
6462:
6461:
6456:
6440:
6438:
6437:
6432:
6420:
6418:
6417:
6412:
6400:
6398:
6397:
6392:
6377:
6375:
6374:
6369:
6341:
6339:
6338:
6333:
6322:acts on the set
6321:
6319:
6318:
6313:
6287:
6285:
6284:
6279:
6271:
6270:
6269:
6268:
6256:
6255:
6215:
6213:
6212:
6207:
6202:
6189:
6188:
6179:
6178:
6173:
6172:
6164:
6131:
6129:
6128:
6123:
6121:
6120:
6101:
6099:
6098:
6093:
6091:
6087:
6086:
6070:
6069:
6049:
6040:
6039:
6020:
6019:
5990:
5988:
5987:
5982:
5970:
5968:
5967:
5962:
5960:
5959:
5954:
5953:
5939:
5937:
5936:
5931:
5916:
5914:
5913:
5908:
5906:
5905:
5900:
5899:
5892:
5891:
5890:
5889:
5884:
5883:
5868:
5867:
5866:
5865:
5860:
5859:
5847:
5846:
5841:
5840:
5830:
5829:
5824:
5823:
5803:
5801:
5800:
5795:
5793:
5792:
5777:
5776:
5748:
5746:
5745:
5740:
5738:
5734:
5733:
5721:
5720:
5705:
5704:
5685:
5684:
5672:
5671:
5656:
5655:
5622:
5620:
5619:
5614:
5612:
5611:
5593:
5592:
5587:
5586:
5565:
5563:
5562:
5557:
5555:
5554:
5539:
5538:
5533:
5532:
5515:
5513:
5512:
5507:
5495:
5493:
5492:
5487:
5485:
5484:
5469:
5468:
5453:
5452:
5391:
5389:
5388:
5383:
5381:
5380:
5368:
5367:
5349:
5347:
5346:
5341:
5339:
5338:
5329:
5328:
5319:
5318:
5306:
5305:
5289:
5287:
5286:
5281:
5279:
5278:
5258:
5256:
5255:
5250:
5238:
5236:
5235:
5230:
5228:
5227:
5215:
5214:
5202:
5201:
5192:
5191:
5182:
5181:
5164:smooth manifolds
5154:
5152:
5151:
5146:
5138:
5137:
5105:
5103:
5102:
5097:
5079:
5077:
5076:
5071:
5069:
5068:
5037:
5035:
5034:
5029:
5013:
5011:
5010:
5005:
4990:
4988:
4987:
4982:
4964:
4962:
4961:
4956:
4941:, we obtain the
4940:
4938:
4937:
4932:
4920:
4918:
4917:
4912:
4892:
4890:
4889:
4884:
4860:
4858:
4857:
4852:
4828:
4826:
4825:
4820:
4793:
4791:
4790:
4785:
4767:
4765:
4764:
4759:
4735:
4733:
4732:
4727:
4715:
4713:
4712:
4707:
4695:
4693:
4692:
4687:
4675:
4673:
4672:
4667:
4655:
4653:
4652:
4647:
4632:
4630:
4629:
4624:
4608:
4606:
4605:
4600:
4581:
4579:
4578:
4573:
4553:
4551:
4550:
4545:
4533:
4531:
4530:
4525:
4513:
4511:
4510:
4505:
4494:
4493:
4477:
4475:
4474:
4469:
4452:
4451:
4435:
4433:
4432:
4427:
4409:
4407:
4406:
4401:
4384:
4383:
4352:
4350:
4349:
4344:
4332:
4330:
4329:
4324:
4313:
4312:
4293:
4291:
4290:
4285:
4273:
4271:
4270:
4265:
4248:
4247:
4231:
4229:
4228:
4223:
4212:
4211:
4195:
4193:
4192:
4187:
4176:
4175:
4159:
4157:
4156:
4151:
4127:
4125:
4124:
4119:
4107:
4105:
4104:
4099:
4076:
4074:
4073:
4068:
4056:
4054:
4053:
4048:
4036:
4034:
4033:
4028:
4016:
4014:
4013:
4008:
4006:
4005:
3989:
3987:
3986:
3981:
3947:
3945:
3944:
3939:
3927:
3925:
3924:
3919:
3900:
3898:
3897:
3892:
3872:
3870:
3869:
3864:
3837:
3835:
3834:
3829:
3817:
3815:
3814:
3809:
3797:
3795:
3794:
3789:
3777:
3775:
3774:
3769:
3748:
3746:
3745:
3740:
3728:
3726:
3725:
3720:
3708:
3706:
3705:
3700:
3688:
3686:
3685:
3680:
3664:
3662:
3661:
3656:
3622:
3620:
3619:
3614:
3590:
3588:
3587:
3582:
3534:
3532:
3531:
3526:
3492:
3490:
3489:
3484:
3463:
3461:
3460:
3455:
3410:
3408:
3407:
3402:
3400:
3399:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3328:
3326:
3325:
3320:
3304:
3302:
3301:
3296:
3275:
3273:
3272:
3267:
3246:
3244:
3243:
3238:
3226:
3224:
3223:
3218:
3206:
3204:
3203:
3198:
3178:
3177:
3152:
3150:
3149:
3144:
3052:
3050:
3049:
3044:
3042:
3041:
3021:
3019:
3018:
3013:
2996:
2995:
2971:
2970:
2952:
2951:
2932:
2930:
2929:
2924:
2919:
2918:
2897:
2896:
2880:
2878:
2877:
2872:
2870:
2869:
2848:
2847:
2828:
2826:
2825:
2820:
2772:
2770:
2769:
2764:
2728:
2726:
2725:
2720:
2687:
2685:
2684:
2679:
2677:
2676:
2658:
2657:
2637:
2635:
2634:
2629:
2599:
2597:
2596:
2591:
2589:
2588:
2566:
2564:
2563:
2558:
2540:
2538:
2537:
2532:
2530:
2529:
2505:
2503:
2502:
2497:
2490:
2479:
2478:
2462:
2460:
2459:
2454:
2419:
2417:
2416:
2411:
2393:
2391:
2390:
2385:
2366:
2364:
2363:
2358:
2356:
2334:
2332:
2331:
2326:
2324:
2306:
2304:
2303:
2298:
2296:
2275:
2273:
2272:
2267:
2265:
2243:
2241:
2240:
2235:
2233:
2215:
2213:
2212:
2207:
2205:
2141:
2139:
2138:
2133:
2131:
2130:
2118:
2117:
2101:
2099:
2098:
2093:
2091:
2090:
2074:
2072:
2071:
2066:
2064:
2063:
2051:
2050:
1965:
1963:
1962:
1957:
1955:
1954:
1949:
1934:
1933:
1914:
1912:
1911:
1906:
1904:
1903:
1898:
1886:
1885:
1861:
1859:
1858:
1853:
1809:
1807:
1806:
1801:
1788:
1787:
1786:
1781:
1765:
1763:
1762:
1757:
1749:
1748:
1743:
1732:
1683:
1681:
1680:
1675:
1673:
1672:
1609:
1580:
1578:
1577:
1572:
1480:
1479:
1462:
1410:
1408:
1407:
1402:
1400:
1399:
1394:
1277:A groupoid is a
1267:
1265:
1264:
1259:
1257:
1256:
1241:
1240:
1225:
1224:
1193:
1191:
1190:
1185:
1164:
1162:
1161:
1156:
1148:
1147:
1135:
1134:
1105:
1103:
1102:
1097:
1077:
1076:
1075:
1055:
1053:
1052:
1047:
1039:
1038:
1037:
1005:
1003:
1002:
997:
971:
969:
968:
963:
961:
960:
959:
933:
931:
930:
925:
917:
916:
889:
887:
886:
881:
863:
861:
860:
855:
837:
835:
834:
829:
799:
797:
796:
791:
761:
759:
758:
753:
723:
721:
720:
715:
685:
683:
682:
677:
659:
657:
656:
651:
624:
622:
621:
616:
604:
602:
601:
596:
584:
582:
581:
576:
564:
562:
561:
556:
544:
542:
541:
536:
521:
519:
518:
513:
501:
499:
498:
493:
480:binary operation
477:
475:
474:
469:
439:partial function
436:
434:
433:
428:
411:
410:
402:
389:
387:
386:
381:
368:partial function
337:objects such as
328:
326:
325:
320:
276:
274:
273:
268:
238:
236:
235:
230:
170:
168:
167:
162:
138:
136:
135:
130:
101:dependent typing
91:by analogy with
69:binary operation
65:partial function
30:, especially in
13493:
13492:
13488:
13487:
13486:
13484:
13483:
13482:
13478:Homotopy theory
13473:Category theory
13458:
13457:
13382:category theory
13356:graph of groups
13336:
13234:
13212:homotopy theory
13197:—, 2006.
13152:
13147:
13146:
13127:
13123:
13106:
13102:
13092:
13090:
13082:
13079:homotopy theory
13076:
13072:
13066:Wayback Machine
13057:
13053:
13047:Wayback Machine
13038:
13034:
13025:
13021:
13013:
13009:
13001:
12994:
12990:
12989:
12985:
12968:
12961:
12952:
12950:
12942:
12941:
12937:
12916:
12912:
12755:
12681:
12677:
12671:
12649:
12648:
12644:
12631:
12624:
12619:
12577:
12561:symplectic form
12528:, or even some
12518:
12496:
12495:
12490:
12485:
12479:
12478:
12472:
12466:
12465:
12460:
12448:
12442:
12441:
12435:
12429:
12428:
12423:
12418:
12411:
12409:
12406:
12405:
12386:
12383:
12382:
12365:
12364:
12359:
12354:
12348:
12347:
12341:
12335:
12334:
12329:
12317:
12310:
12308:
12305:
12304:
12290:
12289:
12284:
12272:
12266:
12265:
12259:
12253:
12252:
12247:
12242:
12235:
12233:
12230:
12229:
12206:
12200:
12199:
12198:
12189:
12183:
12182:
12181:
12173:
12170:
12169:
12146:
12140:
12139:
12138:
12129:
12123:
12122:
12121:
12107:
12104:
12103:
12083:
12077:
12076:
12075:
12066:
12060:
12059:
12058:
12056:
12053:
12052:
12041:
12039:Double groupoid
12035:
12004:
12000:
11998:
11995:
11994:
11937:
11936:
11932:
11905:
11901:
11879:
11878:
11874:
11872:
11869:
11868:
11829:
11812:
11804:
11801:
11800:
11794:
11752:
11750:
11747:
11746:
11719:
11715:
11707:
11704:
11703:
11663:
11635:
11634:
11630:
11587:
11586:
11582:
11580:
11577:
11576:
11524:
11523:
11519:
11495:
11491:
11463:
11462:
11458:
11456:
11453:
11452:
11423:
11406:
11398:
11395:
11394:
11391:
11340:
11337:
11336:
11249:
11246:
11245:
11215:
11212:
11211:
11189:
11186:
11185:
11163:
11160:
11159:
11125:
11122:
11121:
11099:
11096:
11095:
11069:
11048:
11045:
11044:
11028:
11025:
11024:
11008:
11005:
11004:
10988:
10985:
10984:
10964:
10961:
10960:
10944:
10941:
10940:
10924:
10921:
10920:
10904:
10901:
10900:
10884:
10881:
10880:
10864:
10861:
10860:
10840:
10837:
10836:
10820:
10817:
10816:
10800:
10797:
10796:
10750:
10747:
10746:
10730:
10727:
10726:
10697:
10694:
10693:
10661:
10658:
10657:
10641:
10638:
10637:
10617:
10614:
10613:
10586:
10583:
10582:
10566:
10563:
10562:
10543:
10540:
10539:
10515:
10512:
10511:
10488:
10485:
10484:
10468:
10465:
10464:
10433:
10430:
10429:
10413:
10410:
10409:
10382:
10379:
10378:
10361:
10357:
10355:
10352:
10351:
10335:
10332:
10331:
10314:
10310:
10308:
10305:
10304:
10288:
10285:
10284:
10283:, and for each
10268:
10265:
10264:
10244:
10240:
10232:
10229:
10228:
10212:
10209:
10208:
10188:
10184:
10182:
10179:
10178:
10139:
10136:
10135:
9607:
9600:
9583:
9556:
9520:
9516:
9504:
9500:
9482:
9478:
9469:
9465:
9457:
9454:
9453:
9433:
9429:
9420:
9416:
9407:
9403:
9394:
9390:
9388:
9385:
9384:
9367:
9363:
9361:
9358:
9357:
9341:
9338:
9337:
9320:
9316:
9314:
9311:
9310:
9293:
9289:
9287:
9284:
9283:
9266:
9262:
9253:
9249:
9247:
9244:
9243:
9226:
9222:
9220:
9217:
9216:
9189:
9185:
9175:
9169:
9165:
9163:
9160:
9159:
9153:
9129:
9121:
9118:
9117:
9095:
9057:
9054:
9053:
9031:
8993:
8990:
8989:
8973:
8970:
8969:
8947:
8936:
8925:
8896:
8893:
8892:
8873:
8859:
8856:
8855:
8836:
8822:
8819:
8818:
8790:
8787:
8786:
8770:
8767:
8766:
8720:
8717:
8716:
8691:
8683:
8680:
8679:
8654:
8646:
8643:
8642:
8608:
8605:
8604:
8584:
8580:
8575:
8572:
8571:
8543:
8540:
8539:
8511:
8508:
8507:
8490:
8489:
8484:
8477:
8471:
8470:
8462:
8461:
8452:
8450:
8447:
8446:
8440:
8411:
8407:
8392:
8388:
8380:
8378:
8375:
8374:
8351:
8348:
8347:
8325:
8319:
8314:
8313:
8308:
8305:
8304:
8287:
8282:
8281:
8279:
8276:
8275:
8244:
8241:
8240:
8224:
8221:
8220:
8217:
8193:
8188:
8186:
8183:
8182:
8160:
8157:
8156:
8104:
8101:
8100:
8078:
8070:
8067:
8066:
8041:
8038:
8037:
8012:
8009:
8008:
7950:
7947:
7946:
7927:
7922:
7920:
7917:
7916:
7913:
7892:
7889:
7888:
7848:
7844:
7839:
7836:
7835:
7819:
7816:
7815:
7814:that the group
7795:
7792:
7791:
7774:
7770:
7768:
7765:
7764:
7748:
7745:
7744:
7728:
7725:
7724:
7705:
7703:
7700:
7699:
7683:
7680:
7679:
7663:
7660:
7659:
7637:
7626:
7612:
7610:
7607:
7606:
7584:
7573:
7571:
7568:
7567:
7539:
7525:
7523:
7520:
7519:
7503:
7500:
7499:
7473:
7462:
7460:
7457:
7456:
7440:
7437:
7436:
7432:. The (unique)
7417:
7414:
7413:
7397:
7394:
7393:
7377:
7374:
7373:
7357:
7354:
7353:
7331:
7328:
7327:
7310:
7306:
7304:
7301:
7300:
7278:
7276:
7273:
7272:
7256:
7253:
7252:
7236:
7233:
7232:
7210:
7196:
7193:
7192:
7176:
7173:
7172:
7156:
7153:
7152:
7129:
7127:
7124:
7123:
7098:
7087:
7082:
7079:
7078:
7059:
7056:
7055:
7028:
7025:
7024:
6995:
6992:
6991:
6963:
6960:
6959:
6943:
6940:
6939:
6923:
6920:
6919:
6891:
6888:
6887:
6823:
6820:
6819:
6797:
6794:
6793:
6771:
6768:
6767:
6727:
6724:
6723:
6686:
6683:
6682:
6639:
6637:
6634:
6633:
6599:
6597:
6594:
6593:
6590:action groupoid
6569:
6566:
6565:
6530:
6527:
6526:
6510:
6507:
6506:
6490:
6487:
6486:
6470:
6467:
6466:
6450:
6447:
6446:
6426:
6423:
6422:
6406:
6403:
6402:
6386:
6383:
6382:
6363:
6360:
6359:
6344:action groupoid
6327:
6324:
6323:
6307:
6304:
6303:
6297:
6264:
6260:
6251:
6247:
6246:
6242:
6231:
6228:
6227:
6193:
6184:
6183:
6174:
6163:
6162:
6161:
6147:
6144:
6143:
6116:
6112:
6110:
6107:
6106:
6089:
6088:
6082:
6078:
6076:
6071:
6062:
6058:
6055:
6054:
6048:
6042:
6041:
6032:
6028:
6026:
6021:
6009:
6005:
6001:
5999:
5996:
5995:
5976:
5973:
5972:
5955:
5949:
5948:
5947:
5945:
5942:
5941:
5925:
5922:
5921:
5901:
5895:
5894:
5893:
5885:
5879:
5878:
5877:
5876:
5872:
5861:
5855:
5854:
5853:
5852:
5848:
5842:
5836:
5835:
5834:
5825:
5819:
5818:
5817:
5815:
5812:
5811:
5785:
5781:
5772:
5768:
5760:
5757:
5756:
5736:
5735:
5729:
5725:
5713:
5709:
5700:
5696:
5687:
5686:
5680:
5676:
5664:
5660:
5651:
5647:
5637:
5635:
5632:
5631:
5604:
5600:
5588:
5582:
5581:
5580:
5578:
5575:
5574:
5550:
5546:
5534:
5528:
5527:
5526:
5524:
5521:
5520:
5501:
5498:
5497:
5474:
5470:
5464:
5460:
5448:
5447:
5445:
5442:
5441:
5438:
5428:
5415:effective topos
5376:
5372:
5363:
5359:
5357:
5354:
5353:
5334:
5330:
5324:
5320:
5314:
5310:
5301:
5297:
5295:
5292:
5291:
5274:
5270:
5268:
5265:
5264:
5244:
5241:
5240:
5223:
5219:
5210:
5206:
5197:
5193:
5187:
5183:
5177:
5173:
5171:
5168:
5167:
5133:
5129:
5121:
5118:
5117:
5113:
5085:
5082:
5081:
5064:
5060:
5043:
5040:
5039:
5023:
5020:
5019:
4999:
4996:
4995:
4970:
4967:
4966:
4950:
4947:
4946:
4926:
4923:
4922:
4906:
4903:
4902:
4866:
4863:
4862:
4834:
4831:
4830:
4802:
4799:
4798:
4773:
4770:
4769:
4741:
4738:
4737:
4721:
4718:
4717:
4701:
4698:
4697:
4681:
4678:
4677:
4661:
4658:
4657:
4641:
4638:
4637:
4618:
4615:
4614:
4594:
4591:
4590:
4567:
4564:
4563:
4560:
4539:
4536:
4535:
4519:
4516:
4515:
4489:
4485:
4483:
4480:
4479:
4447:
4443:
4441:
4438:
4437:
4415:
4412:
4411:
4379:
4375:
4373:
4370:
4369:
4338:
4335:
4334:
4308:
4304:
4302:
4299:
4298:
4279:
4276:
4275:
4243:
4239:
4237:
4234:
4233:
4207:
4203:
4201:
4198:
4197:
4196:(or sometimes,
4171:
4167:
4165:
4162:
4161:
4145:
4142:
4141:
4113:
4110:
4109:
4093:
4090:
4089:
4062:
4059:
4058:
4042:
4039:
4038:
4022:
4019:
4018:
4001:
3997:
3995:
3992:
3991:
3975:
3972:
3971:
3965:
3959:
3954:
3933:
3930:
3929:
3913:
3910:
3909:
3886:
3883:
3882:
3843:
3840:
3839:
3823:
3820:
3819:
3803:
3800:
3799:
3783:
3780:
3779:
3754:
3751:
3750:
3734:
3731:
3730:
3714:
3711:
3710:
3694:
3691:
3690:
3674:
3671:
3670:
3638:
3635:
3634:
3596:
3593:
3592:
3540:
3537:
3536:
3514:
3511:
3510:
3472:
3469:
3468:
3443:
3440:
3439:
3432:
3392:
3388:
3374:
3371:
3370:
3354:
3351:
3350:
3334:
3331:
3330:
3314:
3311:
3310:
3281:
3278:
3277:
3252:
3249:
3248:
3232:
3229:
3228:
3212:
3209:
3208:
3170:
3166:
3158:
3155:
3154:
3132:
3129:
3128:
3086:isotropy groups
3074:
3037:
3033:
3031:
3028:
3027:
2991:
2987:
2966:
2962:
2947:
2943:
2938:
2935:
2934:
2933:exist, so does
2914:
2910:
2892:
2888:
2886:
2883:
2882:
2865:
2861:
2843:
2839:
2834:
2831:
2830:
2778:
2775:
2774:
2734:
2731:
2730:
2693:
2690:
2689:
2672:
2668:
2653:
2649:
2647:
2644:
2643:
2608:
2605:
2604:
2584:
2580:
2572:
2569:
2568:
2546:
2543:
2542:
2525:
2521:
2519:
2516:
2515:
2486:
2474:
2470:
2468:
2465:
2464:
2448:
2445:
2444:
2442:
2399:
2396:
2395:
2379:
2376:
2375:
2349:
2347:
2344:
2343:
2341:
2314:
2312:
2309:
2308:
2283:
2281:
2278:
2277:
2255:
2253:
2250:
2249:
2223:
2221:
2218:
2217:
2192:
2190:
2187:
2186:
2155:
2147:groupoid object
2126:
2122:
2113:
2109:
2107:
2104:
2103:
2086:
2082:
2080:
2077:
2076:
2059:
2055:
2046:
2042:
2040:
2037:
2036:
1950:
1942:
1941:
1926:
1922:
1920:
1917:
1916:
1899:
1891:
1890:
1878:
1874:
1869:
1866:
1865:
1817:
1814:
1813:
1782:
1774:
1773:
1771:
1768:
1767:
1744:
1736:
1735:
1727:
1724:
1723:
1665:
1661:
1599:
1597:
1594:
1593:
1463:
1449:
1448:
1446:
1443:
1442:
1395:
1387:
1386:
1384:
1381:
1380:
1299:
1281:in which every
1275:
1249:
1245:
1233:
1229:
1217:
1213:
1199:
1196:
1195:
1173:
1170:
1169:
1140:
1136:
1127:
1123:
1118:
1115:
1114:
1068:
1064:
1063:
1061:
1058:
1057:
1030:
1026:
1025:
1011:
1008:
1007:
985:
982:
981:
952:
948:
947:
939:
936:
935:
909:
905:
903:
900:
899:
869:
866:
865:
843:
840:
839:
805:
802:
801:
767:
764:
763:
729:
726:
725:
691:
688:
687:
665:
662:
661:
639:
636:
635:
610:
607:
606:
590:
587:
586:
570:
567:
566:
550:
547:
546:
530:
527:
526:
525:The operations
507:
504:
503:
487:
484:
483:
445:
442:
441:
403:
401:
400:
398:
395:
394:
392:unary operation
375:
372:
371:
364:
359:
314:
311:
310:
244:
241:
240:
176:
173:
172:
144:
141:
140:
112:
109:
108:
85:unary operation
79:in which every
44:Brandt groupoid
36:homotopy theory
32:category theory
24:
21:magma (algebra)
17:
12:
11:
5:
13491:
13481:
13480:
13475:
13470:
13456:
13455:
13443:
13431:
13420:
13413:
13402:
13393:
13366:
13359:
13352:
13334:
13320:Golubitsky, M.
13317:
13301:
13284:
13237:
13232:
13219:
13204:
13195:
13181:
13164:(1): 360β366,
13151:
13148:
13145:
13144:
13121:
13100:
13070:
13051:
13032:
13019:
13007:
12983:
12959:
12935:
12910:
12780:. Therefore (
12675:
12669:
12642:
12621:
12620:
12618:
12615:
12614:
12613:
12608:
12598:
12593:
12588:
12583:
12576:
12573:
12538:Lie algebroids
12517:
12514:
12494:
12491:
12489:
12486:
12484:
12481:
12480:
12477:
12474:
12471:
12468:
12467:
12464:
12461:
12457:
12453:
12449:
12447:
12444:
12443:
12440:
12437:
12434:
12431:
12430:
12427:
12424:
12422:
12419:
12417:
12414:
12413:
12390:
12363:
12360:
12358:
12355:
12353:
12350:
12349:
12346:
12343:
12340:
12337:
12336:
12333:
12330:
12326:
12322:
12318:
12316:
12313:
12312:
12288:
12285:
12281:
12277:
12273:
12271:
12268:
12267:
12264:
12261:
12258:
12255:
12254:
12251:
12248:
12246:
12243:
12241:
12238:
12237:
12209:
12203:
12197:
12192:
12186:
12180:
12177:
12149:
12143:
12137:
12132:
12126:
12120:
12117:
12114:
12111:
12086:
12080:
12074:
12069:
12063:
12037:Main article:
12034:
12031:
12018:
12015:
12012:
12007:
12003:
11991:
11990:
11979:
11976:
11973:
11970:
11967:
11964:
11961:
11958:
11955:
11949:
11946:
11943:
11940:
11935:
11931:
11928:
11925:
11922:
11919:
11916:
11913:
11908:
11904:
11900:
11897:
11891:
11888:
11885:
11882:
11877:
11841:
11838:
11835:
11832:
11828:
11824:
11821:
11818:
11815:
11811:
11808:
11793:
11787:
11774:
11771:
11768:
11764:
11761:
11758:
11755:
11730:
11725:
11722:
11718:
11714:
11711:
11700:
11699:
11688:
11685:
11682:
11679:
11675:
11672:
11669:
11666:
11662:
11659:
11656:
11653:
11647:
11644:
11641:
11638:
11633:
11629:
11626:
11623:
11620:
11617:
11614:
11611:
11608:
11605:
11602:
11596:
11593:
11590:
11585:
11574:
11563:
11560:
11557:
11554:
11551:
11548:
11545:
11542:
11539:
11533:
11530:
11527:
11522:
11518:
11515:
11512:
11509:
11506:
11501:
11498:
11494:
11490:
11487:
11484:
11481:
11475:
11472:
11469:
11466:
11461:
11432:
11429:
11426:
11422:
11418:
11415:
11412:
11409:
11405:
11402:
11393:The inclusion
11390:
11384:
11356:
11353:
11350:
11347:
11344:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11301:
11298:
11295:
11292:
11289:
11286:
11283:
11280:
11277:
11274:
11271:
11268:
11265:
11262:
11259:
11256:
11253:
11231:
11228:
11225:
11222:
11219:
11199:
11196:
11193:
11173:
11170:
11167:
11147:
11144:
11141:
11138:
11135:
11132:
11129:
11109:
11106:
11103:
11068:
11065:
11052:
11032:
11012:
10992:
10983:isomorphic to
10968:
10948:
10928:
10908:
10888:
10868:
10855:on the set of
10844:
10824:
10804:
10754:
10745:to each point
10734:
10710:
10707:
10704:
10701:
10686:
10685:
10665:
10645:
10621:
10610:
10590:
10570:
10559:
10547:
10519:
10495:
10492:
10472:
10437:
10417:
10402:disjoint union
10386:
10364:
10360:
10339:
10317:
10313:
10292:
10272:
10252:
10247:
10243:
10239:
10236:
10216:
10191:
10187:
10155:
10152:
10149:
10146:
10143:
10111:
10110:
10107:
10104:
10101:
10098:
10095:
10089:
10088:
10085:
10082:
10079:
10076:
10073:
10067:
10066:
10063:
10060:
10057:
10054:
10051:
10045:
10044:
10041:
10038:
10035:
10032:
10029:
10023:
10022:
10019:
10016:
10013:
10010:
10007:
10000:
9999:
9996:
9993:
9990:
9987:
9984:
9977:
9976:
9973:
9970:
9967:
9964:
9961:
9954:
9953:
9950:
9947:
9944:
9941:
9938:
9932:
9931:
9928:
9925:
9922:
9919:
9916:
9909:
9908:
9905:
9902:
9899:
9896:
9893:
9887:
9886:
9883:
9880:
9877:
9874:
9871:
9864:
9863:
9860:
9857:
9854:
9851:
9848:
9842:
9841:
9838:
9835:
9832:
9829:
9826:
9819:
9818:
9815:
9812:
9809:
9806:
9803:
9797:
9796:
9793:
9790:
9787:
9784:
9781:
9774:
9773:
9770:
9767:
9764:
9761:
9758:
9752:
9751:
9748:
9745:
9742:
9739:
9736:
9729:
9728:
9725:
9722:
9719:
9716:
9713:
9707:
9706:
9703:
9700:
9697:
9694:
9691:
9689:Small category
9685:
9684:
9681:
9678:
9675:
9672:
9669:
9663:
9662:
9659:
9656:
9653:
9650:
9647:
9641:
9640:
9635:
9630:
9625:
9620:
9615:
9606:
9603:
9598:
9582:
9579:
9571:fifteen puzzle
9555:
9552:
9550:of groupoids.
9540:
9539:
9528:
9523:
9519:
9515:
9512:
9507:
9503:
9499:
9496:
9493:
9490:
9485:
9481:
9477:
9472:
9468:
9464:
9461:
9436:
9432:
9428:
9423:
9419:
9415:
9410:
9406:
9402:
9397:
9393:
9370:
9366:
9345:
9336:composed with
9323:
9319:
9296:
9292:
9269:
9265:
9261:
9256:
9252:
9229:
9225:
9206:
9205:
9192:
9188:
9182:
9179:
9172:
9168:
9152:
9149:
9135:
9132:
9128:
9125:
9105:
9101:
9098:
9094:
9091:
9088:
9085:
9082:
9079:
9076:
9073:
9070:
9067:
9064:
9061:
9041:
9037:
9034:
9030:
9027:
9024:
9021:
9018:
9015:
9012:
9009:
9006:
9003:
9000:
8997:
8977:
8957:
8953:
8950:
8946:
8942:
8939:
8935:
8931:
8928:
8924:
8921:
8918:
8915:
8912:
8909:
8906:
8903:
8900:
8879:
8876:
8872:
8869:
8866:
8863:
8842:
8839:
8835:
8832:
8829:
8826:
8806:
8803:
8800:
8797:
8794:
8774:
8754:
8751:
8748:
8745:
8742:
8739:
8736:
8733:
8730:
8727:
8724:
8704:
8701:
8698:
8690:
8687:
8667:
8664:
8661:
8653:
8650:
8630:
8627:
8624:
8621:
8618:
8615:
8612:
8592:
8587:
8583:
8579:
8559:
8556:
8553:
8550:
8547:
8527:
8524:
8521:
8518:
8515:
8504:
8503:
8488:
8485:
8483:
8480:
8478:
8476:
8473:
8472:
8469:
8466:
8463:
8460:
8457:
8454:
8439:
8436:
8419:
8414:
8410:
8406:
8403:
8400:
8395:
8391:
8387:
8383:
8355:
8335:
8332:
8328:
8322:
8317:
8312:
8290:
8285:
8260:
8257:
8254:
8251:
8248:
8228:
8216:
8213:
8200:
8196:
8191:
8170:
8167:
8164:
8144:
8141:
8138:
8135:
8132:
8129:
8126:
8123:
8120:
8117:
8114:
8111:
8108:
8088:
8085:
8081:
8077:
8074:
8054:
8051:
8048:
8045:
8025:
8022:
8019:
8016:
7996:
7993:
7990:
7987:
7984:
7981:
7978:
7975:
7972:
7969:
7966:
7963:
7960:
7957:
7954:
7934:
7930:
7925:
7912:
7909:
7896:
7868:
7865:
7862:
7859:
7856:
7851:
7847:
7843:
7823:
7799:
7777:
7773:
7752:
7732:
7711:
7708:
7687:
7667:
7647:
7643:
7640:
7636:
7632:
7629:
7625:
7621:
7618:
7615:
7594:
7590:
7587:
7583:
7579:
7576:
7555:
7552:
7549:
7545:
7542:
7538:
7534:
7531:
7528:
7507:
7482:
7479:
7476:
7472:
7468:
7465:
7444:
7421:
7401:
7381:
7361:
7341:
7338:
7335:
7313:
7309:
7284:
7281:
7260:
7240:
7231:and for every
7220:
7216:
7213:
7209:
7206:
7203:
7200:
7180:
7160:
7135:
7132:
7111:
7107:
7104:
7101:
7097:
7093:
7090:
7086:
7063:
7032:
7008:
7005:
7002:
6999:
6979:
6976:
6973:
6970:
6967:
6947:
6927:
6904:
6901:
6898:
6895:
6875:
6872:
6869:
6866:
6863:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6807:
6804:
6801:
6781:
6778:
6775:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6711:
6708:
6705:
6702:
6699:
6696:
6693:
6690:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6648:
6645:
6642:
6621:
6618:
6615:
6612:
6609:
6605:
6602:
6586:
6585:
6573:
6555:
6543:
6540:
6537:
6534:
6514:
6494:
6474:
6454:
6430:
6410:
6390:
6379:
6367:
6331:
6311:
6296:
6293:
6277:
6274:
6267:
6263:
6259:
6254:
6250:
6245:
6241:
6238:
6235:
6205:
6200:
6197:
6192:
6187:
6182:
6177:
6170:
6167:
6160:
6157:
6154:
6151:
6119:
6115:
6085:
6081:
6077:
6075:
6072:
6068:
6065:
6061:
6057:
6056:
6053:
6050:
6047:
6044:
6043:
6038:
6035:
6031:
6027:
6025:
6022:
6018:
6015:
6012:
6008:
6004:
6003:
5980:
5958:
5952:
5929:
5904:
5898:
5888:
5882:
5875:
5871:
5864:
5858:
5851:
5845:
5839:
5833:
5828:
5822:
5791:
5788:
5784:
5780:
5775:
5771:
5767:
5764:
5732:
5728:
5724:
5719:
5716:
5712:
5708:
5703:
5699:
5695:
5692:
5689:
5688:
5683:
5679:
5675:
5670:
5667:
5663:
5659:
5654:
5650:
5646:
5643:
5640:
5639:
5610:
5607:
5603:
5599:
5596:
5591:
5585:
5553:
5549:
5545:
5542:
5537:
5531:
5505:
5483:
5480:
5477:
5473:
5467:
5463:
5459:
5456:
5451:
5427:
5424:
5423:
5422:
5417:introduced by
5399:
5396:banal groupoid
5379:
5375:
5371:
5366:
5362:
5337:
5333:
5327:
5323:
5317:
5313:
5309:
5304:
5300:
5277:
5273:
5248:
5226:
5222:
5218:
5213:
5209:
5205:
5200:
5196:
5190:
5186:
5180:
5176:
5144:
5141:
5136:
5132:
5128:
5125:
5112:
5109:
5108:
5107:
5095:
5092:
5089:
5067:
5063:
5059:
5056:
5053:
5050:
5047:
5027:
5003:
4992:
4980:
4977:
4974:
4954:
4930:
4910:
4895:
4894:
4882:
4879:
4876:
4873:
4870:
4850:
4847:
4844:
4841:
4838:
4818:
4815:
4812:
4809:
4806:
4795:
4783:
4780:
4777:
4757:
4754:
4751:
4748:
4745:
4725:
4705:
4685:
4665:
4645:
4634:
4622:
4598:
4571:
4559:
4556:
4543:
4523:
4503:
4500:
4497:
4492:
4488:
4467:
4464:
4461:
4458:
4455:
4450:
4446:
4425:
4422:
4419:
4399:
4396:
4393:
4390:
4387:
4382:
4378:
4342:
4322:
4319:
4316:
4311:
4307:
4283:
4263:
4260:
4257:
4254:
4251:
4246:
4242:
4221:
4218:
4215:
4210:
4206:
4185:
4182:
4179:
4174:
4170:
4149:
4117:
4097:
4066:
4046:
4026:
4004:
4000:
3979:
3961:Main article:
3958:
3955:
3953:
3950:
3937:
3917:
3890:
3862:
3859:
3856:
3853:
3850:
3847:
3827:
3807:
3787:
3767:
3764:
3761:
3758:
3738:
3718:
3698:
3678:
3654:
3651:
3648:
3645:
3642:
3612:
3609:
3606:
3603:
3600:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3524:
3521:
3518:
3482:
3479:
3476:
3453:
3450:
3447:
3431:
3428:
3398:
3395:
3391:
3387:
3384:
3381:
3378:
3358:
3338:
3318:
3294:
3291:
3288:
3285:
3265:
3262:
3259:
3256:
3236:
3216:
3196:
3193:
3190:
3187:
3184:
3181:
3176:
3173:
3169:
3165:
3162:
3142:
3139:
3136:
3123:of a groupoid
3073:
3070:
3040:
3036:
3011:
3008:
3005:
3002:
2999:
2994:
2990:
2986:
2983:
2980:
2977:
2974:
2969:
2965:
2961:
2958:
2955:
2950:
2946:
2942:
2922:
2917:
2913:
2909:
2906:
2903:
2900:
2895:
2891:
2868:
2864:
2860:
2857:
2854:
2851:
2846:
2842:
2838:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2688:exists. Given
2675:
2671:
2667:
2664:
2661:
2656:
2652:
2627:
2624:
2621:
2618:
2615:
2612:
2587:
2583:
2579:
2576:
2556:
2553:
2550:
2528:
2524:
2495:
2489:
2485:
2482:
2477:
2473:
2452:
2440:
2409:
2406:
2403:
2383:
2355:
2352:
2342:(and hence to
2339:
2323:
2320:
2317:
2295:
2292:
2289:
2286:
2264:
2261:
2258:
2232:
2229:
2226:
2204:
2201:
2198:
2195:
2163:disjoint union
2154:
2151:
2129:
2125:
2121:
2116:
2112:
2089:
2085:
2062:
2058:
2054:
2049:
2045:
2012:is called the
1992:is called the
1970:
1969:
1968:
1967:
1953:
1948:
1945:
1940:
1937:
1932:
1929:
1925:
1902:
1897:
1894:
1889:
1884:
1881:
1877:
1873:
1863:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1811:
1799:
1796:
1793:
1785:
1780:
1777:
1755:
1752:
1747:
1742:
1739:
1731:
1671:
1668:
1664:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1608:
1605:
1602:
1582:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1478:
1475:
1472:
1469:
1466:
1461:
1458:
1455:
1452:
1424:
1398:
1393:
1390:
1373:
1297:
1279:small category
1274:
1271:
1270:
1269:
1255:
1252:
1248:
1244:
1239:
1236:
1232:
1228:
1223:
1220:
1216:
1212:
1209:
1206:
1203:
1183:
1180:
1177:
1166:
1154:
1151:
1146:
1143:
1139:
1133:
1130:
1126:
1122:
1108:
1107:
1095:
1092:
1089:
1086:
1083:
1080:
1074:
1071:
1067:
1045:
1042:
1036:
1033:
1029:
1024:
1021:
1018:
1015:
995:
992:
989:
973:
958:
955:
951:
946:
943:
923:
920:
915:
912:
908:
891:
879:
876:
873:
853:
850:
847:
827:
824:
821:
818:
815:
812:
809:
789:
786:
783:
780:
777:
774:
771:
751:
748:
745:
742:
739:
736:
733:
713:
710:
707:
704:
701:
698:
695:
675:
672:
669:
649:
646:
643:
614:
594:
574:
554:
534:
511:
491:
467:
464:
461:
458:
455:
452:
449:
426:
423:
420:
417:
414:
409:
406:
379:
363:
360:
358:
355:
331:
330:
318:
298:
266:
263:
260:
257:
254:
251:
248:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
160:
157:
154:
151:
148:
128:
125:
122:
119:
116:
97:
96:
72:
67:replacing the
15:
9:
6:
4:
3:
2:
13490:
13479:
13476:
13474:
13471:
13469:
13466:
13465:
13463:
13454:
13452:
13447:
13444:
13442:
13440:
13435:
13432:
13429:
13428:Contemp. Math
13425:
13421:
13418:
13414:
13411:
13407:
13403:
13400:
13399:
13394:
13391:
13387:
13383:
13379:
13375:
13371:
13367:
13364:
13360:
13357:
13353:
13349:
13345:
13344:
13339:
13335:
13332:
13329:
13325:
13321:
13318:
13315:
13314:
13309:
13305:
13302:
13299:
13295:
13294:Galois theory
13291:
13290:
13285:
13281:
13277:
13273:
13269:
13265:
13261:
13256:
13251:
13247:
13243:
13238:
13235:
13229:
13225:
13220:
13217:
13214:and in group
13213:
13209:
13205:
13202:
13201:
13196:
13193:
13190:
13186:
13182:
13179:
13175:
13171:
13167:
13163:
13159:
13154:
13153:
13140:
13136:
13132:
13125:
13116:
13111:
13104:
13089:
13085:
13080:
13074:
13067:
13063:
13060:
13055:
13048:
13044:
13041:
13036:
13029:
13023:
13016:
13011:
13000:
12997:. p. 9.
12993:
12987:
12978:
12973:
12966:
12964:
12949:
12945:
12939:
12932:
12931:see chapter 2
12928:
12927:0-226-51183-9
12924:
12920:
12914:
12907:
12903:
12899:
12895:
12891:
12887:
12883:
12879:
12875:
12871:
12867:
12863:
12859:
12855:
12851:
12847:
12843:
12839:
12836:. Therefore
12835:
12831:
12827:
12823:
12819:
12815:
12811:
12807:
12803:
12799:
12795:
12791:
12787:
12783:
12779:
12775:
12771:
12767:
12763:
12759:
12753:
12749:
12745:
12741:
12737:
12733:
12729:
12725:
12721:
12717:
12713:
12709:
12705:
12701:
12697:
12693:
12689:
12685:
12679:
12672:
12670:1-4020-0609-8
12666:
12662:
12658:
12657:
12652:
12646:
12638:
12637:
12629:
12627:
12622:
12612:
12609:
12606:
12602:
12599:
12597:
12594:
12592:
12589:
12587:
12584:
12582:
12579:
12578:
12572:
12570:
12566:
12562:
12558:
12554:
12549:
12547:
12543:
12539:
12535:
12534:Lie groupoids
12531:
12527:
12523:
12513:
12509:
12492:
12482:
12462:
12455:
12451:
12445:
12425:
12415:
12402:
12388:
12378:
12361:
12351:
12331:
12324:
12320:
12314:
12286:
12279:
12275:
12269:
12249:
12239:
12226:
12222:
12207:
12190:
12178:
12175:
12166:
12162:
12147:
12130:
12118:
12115:
12112:
12109:
12100:
12099:with functors
12084:
12072:
12067:
12050:
12046:
12040:
12030:
12013:
12005:
12001:
11971:
11965:
11962:
11959:
11953:
11933:
11929:
11923:
11920:
11914:
11906:
11902:
11895:
11875:
11867:
11866:
11865:
11862:
11860:
11856:
11809:
11806:
11799:
11798:nerve functor
11792:
11786:
11769:
11744:
11723:
11720:
11716:
11709:
11680:
11660:
11657:
11651:
11631:
11627:
11621:
11618:
11612:
11606:
11600:
11583:
11575:
11555:
11549:
11546:
11543:
11537:
11520:
11516:
11510:
11507:
11499:
11496:
11492:
11485:
11479:
11459:
11451:
11450:
11449:
11447:
11403:
11400:
11389:
11383:
11381:
11377:
11373:
11368:
11354:
11351:
11348:
11345:
11342:
11333:
11320:
11311:
11308:
11305:
11299:
11296:
11293:
11290:
11284:
11281:
11278:
11272:
11269:
11266:
11263:
11260:
11254:
11251:
11243:
11229:
11226:
11223:
11220:
11217:
11197:
11194:
11191:
11171:
11165:
11142:
11139:
11136:
11130:
11127:
11107:
11104:
11101:
11093:
11089:
11086:The category
11084:
11082:
11078:
11074:
11064:
11050:
11030:
11010:
10990:
10982:
10981:vertex groups
10966:
10946:
10926:
10906:
10886:
10866:
10858:
10842:
10822:
10802:
10794:
10790:
10786:
10782:
10777:
10775:
10774:vector spaces
10771:
10766:
10752:
10732:
10724:
10705:
10699:
10691:
10683:
10679:
10663:
10643:
10636:of the group
10635:
10619:
10611:
10608:
10604:
10603:trivial group
10588:
10568:
10560:
10545:
10537:
10533:
10517:
10509:
10508:
10507:
10506:For example,
10493:
10490:
10470:
10462:
10458:
10454:
10449:
10435:
10415:
10407:
10403:
10398:
10384:
10362:
10358:
10337:
10315:
10311:
10290:
10270:
10245:
10241:
10234:
10214:
10207:
10189:
10185:
10176:
10171:
10169:
10150:
10147:
10144:
10132:
10130:
10126:
10122:
10118:
10108:
10105:
10102:
10099:
10096:
10094:
10093:Abelian group
10091:
10090:
10086:
10083:
10080:
10077:
10074:
10072:
10069:
10068:
10064:
10061:
10058:
10055:
10052:
10050:
10047:
10046:
10042:
10039:
10036:
10033:
10030:
10028:
10025:
10024:
10020:
10017:
10014:
10011:
10008:
10006:
10002:
10001:
9997:
9994:
9991:
9988:
9985:
9983:
9979:
9978:
9974:
9971:
9968:
9965:
9962:
9960:
9956:
9955:
9951:
9948:
9945:
9942:
9939:
9937:
9934:
9933:
9929:
9926:
9923:
9920:
9917:
9915:
9911:
9910:
9906:
9903:
9900:
9897:
9894:
9892:
9889:
9888:
9884:
9881:
9878:
9875:
9872:
9870:
9866:
9865:
9861:
9858:
9855:
9852:
9849:
9847:
9844:
9843:
9839:
9836:
9833:
9830:
9827:
9825:
9821:
9820:
9816:
9813:
9810:
9807:
9804:
9802:
9799:
9798:
9794:
9791:
9788:
9785:
9782:
9780:
9776:
9775:
9771:
9768:
9765:
9762:
9759:
9757:
9754:
9753:
9749:
9746:
9743:
9740:
9737:
9735:
9731:
9730:
9726:
9723:
9720:
9717:
9714:
9712:
9709:
9708:
9704:
9701:
9698:
9695:
9692:
9690:
9687:
9686:
9682:
9679:
9676:
9673:
9670:
9668:
9665:
9664:
9660:
9657:
9654:
9651:
9648:
9646:
9645:Partial magma
9643:
9642:
9639:
9636:
9634:
9631:
9629:
9626:
9624:
9621:
9619:
9616:
9614:
9613:
9602:
9596:
9595:Mathieu group
9592:
9588:
9578:
9576:
9575:groupoid acts
9572:
9567:
9565:
9561:
9551:
9549:
9545:
9526:
9521:
9517:
9513:
9505:
9501:
9494:
9491:
9483:
9479:
9475:
9470:
9466:
9459:
9452:
9451:
9450:
9434:
9430:
9426:
9421:
9417:
9413:
9408:
9404:
9400:
9395:
9391:
9368:
9364:
9343:
9321:
9317:
9294:
9290:
9267:
9263:
9259:
9254:
9250:
9227:
9223:
9214:
9211:
9190:
9186:
9180:
9170:
9166:
9158:
9157:
9156:
9148:
9133:
9130:
9126:
9123:
9099:
9096:
9089:
9080:
9074:
9071:
9065:
9059:
9035:
9032:
9025:
9016:
9010:
9007:
9001:
8995:
8975:
8951:
8948:
8944:
8940:
8937:
8933:
8929:
8926:
8919:
8913:
8910:
8907:
8904:
8901:
8877:
8874:
8867:
8864:
8861:
8840:
8837:
8830:
8827:
8824:
8801:
8798:
8795:
8772:
8749:
8743:
8734:
8728:
8725:
8722:
8699:
8688:
8685:
8662:
8651:
8648:
8625:
8622:
8619:
8616:
8613:
8590:
8585:
8581:
8577:
8557:
8551:
8548:
8545:
8525:
8519:
8516:
8513:
8486:
8479:
8474:
8458:
8445:
8444:
8443:
8435:
8433:
8412:
8408:
8404:
8401:
8398:
8393:
8389:
8373:
8369:
8353:
8330:
8326:
8320:
8288:
8274:
8255:
8249:
8246:
8239:that maps to
8226:
8212:
8198:
8194:
8165:
8136:
8130:
8124:
8118:
8112:
8083:
8079:
8075:
8052:
8049:
8043:
8023:
8017:
8014:
7991:
7988:
7985:
7982:
7979:
7976:
7973:
7970:
7967:
7964:
7961:
7955:
7952:
7932:
7928:
7908:
7894:
7886:
7882:
7863:
7860:
7857:
7854:
7849:
7845:
7821:
7813:
7797:
7775:
7771:
7750:
7730:
7685:
7665:
7634:
7566:and so sends
7550:
7547:
7505:
7497:
7442:
7435:
7419:
7399:
7379:
7359:
7339:
7333:
7311:
7307:
7299:
7258:
7238:
7204:
7201:
7198:
7178:
7158:
7151:to the group
7150:
7095:
7077:
7074:-sets is the
7061:
7052:
7050:
7046:
7030:
7022:
7006:
7003:
7000:
6997:
6974:
6971:
6968:
6945:
6925:
6916:
6902:
6899:
6896:
6893:
6870:
6867:
6864:
6861:
6855:
6849:
6846:
6843:
6834:
6831:
6828:
6805:
6802:
6799:
6779:
6776:
6773:
6753:
6750:
6747:
6741:
6738:
6735:
6729:
6709:
6706:
6700:
6697:
6694:
6688:
6668:
6665:
6662:
6659:
6653:
6619:
6616:
6610:
6591:
6571:
6563:
6559:
6556:
6541:
6538:
6535:
6532:
6512:
6492:
6472:
6452:
6444:
6428:
6408:
6388:
6380:
6365:
6357:
6356:
6355:
6353:
6349:
6345:
6329:
6309:
6302:
6292:
6288:
6275:
6265:
6261:
6257:
6252:
6248:
6243:
6239:
6236:
6233:
6224:
6222:
6216:
6198:
6195:
6190:
6175:
6165:
6158:
6152:
6140:
6139:
6135:
6117:
6113:
6102:
6083:
6079:
6066:
6063:
6059:
6036:
6033:
6029:
6016:
6013:
6010:
6006:
5992:
5978:
5956:
5927:
5917:
5902:
5886:
5873:
5869:
5862:
5849:
5843:
5831:
5826:
5808:
5804:
5789:
5786:
5782:
5773:
5769:
5765:
5762:
5753:
5749:
5730:
5726:
5717:
5714:
5710:
5706:
5701:
5697:
5693:
5690:
5681:
5677:
5668:
5665:
5661:
5657:
5652:
5648:
5644:
5641:
5628:
5624:
5608:
5605:
5601:
5597:
5594:
5589:
5571:
5567:
5551:
5547:
5543:
5540:
5535:
5517:
5503:
5481:
5478:
5475:
5465:
5461:
5454:
5437:
5433:
5426:Δech groupoid
5420:
5419:Martin Hyland
5416:
5412:
5408:
5407:semidecidable
5404:
5400:
5397:
5392:
5377:
5373:
5364:
5360:
5335:
5331:
5325:
5321:
5315:
5311:
5307:
5302:
5298:
5275:
5271:
5262:
5246:
5224:
5220:
5216:
5211:
5207:
5203:
5198:
5194:
5188:
5184:
5178:
5174:
5165:
5161:
5158:
5142:
5134:
5130:
5126:
5123:
5115:
5114:
5106:is an orbit).
5090:
5065:
5061:
5057:
5054:
5051:
5048:
5045:
5025:
5017:
5016:unit groupoid
5001:
4993:
4978:
4975:
4972:
4952:
4944:
4943:pair groupoid
4928:
4908:
4900:
4899:
4898:
4877:
4874:
4871:
4845:
4842:
4839:
4813:
4810:
4807:
4796:
4781:
4778:
4775:
4752:
4749:
4746:
4723:
4703:
4683:
4663:
4643:
4635:
4620:
4612:
4611:
4610:
4596:
4589:
4585:
4569:
4555:
4541:
4521:
4498:
4490:
4486:
4462:
4459:
4456:
4448:
4444:
4423:
4420:
4417:
4394:
4391:
4388:
4380:
4376:
4366:
4364:
4360:
4356:
4340:
4317:
4309:
4305:
4295:
4281:
4258:
4255:
4252:
4244:
4240:
4216:
4208:
4180:
4172:
4168:
4147:
4139:
4135:
4131:
4115:
4095:
4087:
4084:
4080:
4064:
4057:to the point
4044:
4024:
4002:
3998:
3977:
3970:
3964:
3949:
3935:
3915:
3906:
3904:
3903:covering maps
3888:
3880:
3876:
3860:
3857:
3851:
3845:
3825:
3805:
3785:
3762:
3756:
3736:
3716:
3696:
3676:
3668:
3652:
3646:
3643:
3640:
3631:
3629:
3624:
3610:
3607:
3604:
3601:
3598:
3575:
3572:
3569:
3563:
3560:
3554:
3551:
3548:
3542:
3522:
3519:
3516:
3508:
3504:
3500:
3496:
3480:
3474:
3467:
3451:
3445:
3437:
3427:
3425:
3421:
3417:
3412:
3396:
3393:
3389:
3385:
3382:
3376:
3356:
3336:
3316:
3308:
3289:
3283:
3260:
3254:
3234:
3214:
3194:
3191:
3182:
3174:
3171:
3167:
3160:
3140:
3137:
3134:
3126:
3122:
3117:
3115:
3111:
3107:
3103:
3099:
3095:
3091:
3090:object groups
3087:
3083:
3082:vertex groups
3079:
3069:
3067:
3062:
3060:
3056:
3038:
3034:
3025:
3009:
3006:
3003:
3000:
2992:
2988:
2984:
2981:
2975:
2967:
2963:
2959:
2956:
2953:
2948:
2944:
2915:
2911:
2907:
2904:
2898:
2893:
2889:
2866:
2862:
2858:
2852:
2849:
2844:
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2813:
2810:
2807:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2760:
2754:
2751:
2748:
2742:
2739:
2736:
2713:
2710:
2707:
2701:
2698:
2695:
2673:
2669:
2665:
2662:
2659:
2654:
2650:
2641:
2622:
2619:
2616:
2610:
2601:
2585:
2581:
2577:
2574:
2554:
2551:
2548:
2526:
2522:
2513:
2509:
2493:
2487:
2483:
2480:
2475:
2471:
2450:
2439:
2435:
2431:
2427:
2423:
2407:
2404:
2401:
2381:
2373:
2368:
2338:
2247:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2150:
2148:
2143:
2127:
2123:
2114:
2110:
2087:
2083:
2060:
2056:
2047:
2043:
2034:
2029:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1951:
1938:
1935:
1930:
1927:
1923:
1900:
1887:
1882:
1879:
1875:
1871:
1864:
1846:
1843:
1837:
1834:
1831:
1825:
1822:
1812:
1797:
1794:
1791:
1783:
1753:
1750:
1745:
1729:
1722:
1721:
1719:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1669:
1666:
1662:
1655:
1652:
1646:
1643:
1640:
1634:
1625:
1622:
1619:
1613:
1610:
1591:
1587:
1583:
1568:
1565:
1556:
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1550:
1544:
1538:
1535:
1532:
1526:
1517:
1514:
1511:
1505:
1502:
1496:
1493:
1490:
1484:
1481:
1476:
1473:
1470:
1467:
1464:
1441:
1437:
1433:
1429:
1425:
1422:
1418:
1414:
1396:
1378:
1374:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1307:
1306:
1305:
1303:
1296:
1292:
1288:
1284:
1280:
1253:
1250:
1246:
1242:
1237:
1234:
1230:
1226:
1221:
1218:
1210:
1207:
1204:
1181:
1178:
1175:
1167:
1152:
1149:
1144:
1141:
1131:
1128:
1124:
1113:
1112:
1111:
1093:
1090:
1087:
1084:
1081:
1078:
1072:
1069:
1065:
1043:
1040:
1034:
1031:
1027:
1022:
1019:
1016:
1013:
993:
990:
987:
979:
978:
974:
956:
953:
949:
944:
941:
921:
918:
913:
910:
906:
897:
896:
892:
877:
874:
871:
851:
848:
845:
822:
819:
816:
810:
807:
787:
784:
778:
775:
772:
746:
743:
740:
734:
731:
711:
708:
702:
699:
696:
673:
670:
667:
647:
644:
641:
633:
632:
631:Associativity
628:
627:
626:
612:
592:
572:
552:
532:
523:
509:
489:
481:
465:
459:
456:
453:
450:
447:
440:
424:
421:
415:
412:
407:
404:
393:
377:
369:
354:
352:
348:
344:
340:
336:
316:
308:
304:
303:
299:
296:
292:
288:
287:
283:
282:
281:
278:
264:
258:
255:
252:
249:
246:
226:
220:
211:
205:
193:
187:
181:
178:
158:
152:
149:
146:
126:
120:
117:
114:
106:
102:
94:
90:
86:
82:
78:
77:
73:
70:
66:
62:
61:
57:
56:
55:
53:
49:
48:virtual group
45:
41:
37:
33:
29:
22:
13450:
13438:
13427:
13423:
13396:
13373:
13369:
13341:
13330:
13327:
13311:
13308:A. Weinstein
13287:
13255:math/9903129
13245:
13241:
13223:
13198:
13191:
13188:
13161:
13157:
13138:
13134:
13124:
13103:
13091:. Retrieved
13087:
13073:
13054:
13035:
13022:
13010:
12986:
12951:. Retrieved
12947:
12938:
12930:
12918:
12913:
12905:
12901:
12897:
12893:
12889:
12885:
12881:
12877:
12873:
12869:
12865:
12861:
12857:
12853:
12849:
12845:
12841:
12837:
12833:
12829:
12825:
12821:
12817:
12813:
12809:
12805:
12801:
12797:
12793:
12789:
12785:
12781:
12777:
12773:
12769:
12765:
12761:
12757:
12751:
12747:
12743:
12739:
12735:
12731:
12727:
12723:
12719:
12715:
12711:
12707:
12703:
12699:
12695:
12691:
12687:
12683:
12678:
12654:
12645:
12639:. p. 6.
12635:
12550:
12546:Lie algebras
12519:
12511:
12404:
12380:
12228:
12224:
12168:
12164:
12102:
12048:
12044:
12042:
11992:
11863:
11854:
11795:
11789:Relation to
11741:denotes the
11701:
11392:
11386:Relation to
11371:
11369:
11334:
11244:
11087:
11085:
11080:
11076:
11072:
11070:
10778:
10770:endomorphism
10767:
10687:
10450:
10405:
10399:
10172:
10133:
10121:group theory
10114:
9980:Associative
9957:Commutative
9912:Commutative
9869:unital magma
9867:Commutative
9846:Unital magma
9822:Commutative
9777:Commutative
9733:
9732:Commutative
9710:
9667:Semigroupoid
9633:Cancellation
9584:
9568:
9560:Rubik's Cube
9557:
9541:
9207:
9154:
8505:
8441:
8273:affine space
8218:
7914:
7885:permutations
7296:) induces a
7053:
6917:
6589:
6587:
6354:as follows:
6352:group action
6347:
6343:
6298:
6295:Group action
6290:
6226:
6218:
6142:
6104:
5994:
5919:
5810:
5806:
5755:
5751:
5630:
5626:
5573:
5569:
5519:
5439:
5410:
5402:
5395:
5352:
5155:is a smooth
5018:, which has
5015:
4942:
4896:
4561:
4367:
4296:
3966:
3907:
3818:starting at
3749:starting at
3632:
3627:
3625:
3498:
3494:
3435:
3433:
3415:
3413:
3124:
3120:
3118:
3113:
3109:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3077:
3075:
3063:
3058:
3054:
3023:
2639:
2602:
2511:
2507:
2437:
2433:
2429:
2425:
2421:
2371:
2369:
2336:
2245:
2182:
2178:
2174:
2170:
2166:
2158:
2156:
2144:
2032:
2030:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1971:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1589:
1585:
1435:
1431:
1427:
1420:
1416:
1412:
1376:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1301:
1294:
1290:
1276:
1109:
975:
893:
629:
524:
365:
332:
300:
284:
279:
98:
93:group theory
88:
74:
58:
47:
43:
42:(less often
39:
25:
13410:Postscript.
13363:orbit space
13088:ncatlab.org
12948:ncatlab.org
12611:R-algebroid
11859:Kan complex
10919:from, say,
10815:of a group
10682:isomorphism
10303:other than
9638:Commutative
9623:Associative
7790:of the set
7605:to the set
6558:Composition
6134:β-groupoids
5971:represents
4736:(denote by
4134:associative
4017:be the set
3905:of spaces.
3466:subcategory
3436:subgroupoid
3127:at a point
2603:Now define
2307:and to be
2031:A groupoid
1592:a function
1344:; we write
1287:isomorphism
357:Definitions
335:geometrical
309:of a group
28:mathematics
13462:Categories
13338:"Groupoid"
13216:cohomology
13150:References
13093:2017-10-31
12953:2017-09-17
12917:J.P. May,
12581:β-groupoid
12542:Lie groups
12047:. Because
11380:cocomplete
10781:fibrations
10457:isomorphic
10453:equivalent
10005:quasigroup
9982:quasigroup
9824:quasigroup
9801:Quasigroup
9449:, we have
7911:Finite set
7392:defines a
7149:isomorphic
7049:transitive
6525:such that
6138:k-cocycles
5430:See also:
5411:PER models
5160:submersion
5157:surjective
4534:. The set
4359:equivalent
4160:, denoted
4083:continuous
3838:such that
3591:for every
3416:transitive
3307:isomorphic
2642:such that
2506:. Denote
2020:, written
2000:, written
239:, so that
13348:EMS Press
13272:0021-8693
13206:—,
13178:119597988
13115:1003.3820
12977:0803.1529
12663:, 2001 ,
12661:EMS Press
12493:∙
12488:→
12483:∙
12476:↓
12470:↓
12463:∙
12446:∙
12439:↓
12433:↓
12426:∙
12421:→
12416:∙
12362:∙
12357:→
12352:∙
12345:↓
12339:↓
12332:∙
12315:∙
12287:∙
12270:∙
12263:↓
12257:↓
12250:∙
12245:→
12240:∙
12196:→
12136:→
12002:π
11954:
11930:≅
11903:π
11896:
11827:→
11721:−
11652:
11628:≅
11601:
11538:
11517:≅
11497:−
11480:
11421:→
11300:
11285:
11279:≅
11264:×
11255:
11169:→
11131:
11075:, or the
10676:for each
10605:for each
10589:∼
10455:(but not
10428:and sets
10109:Required
10087:Unneeded
10065:Required
10043:Unneeded
10021:Required
9998:Unneeded
9975:Required
9959:semigroup
9952:Unneeded
9936:Semigroup
9930:Required
9907:Unneeded
9885:Required
9862:Unneeded
9840:Required
9817:Unneeded
9795:Required
9772:Unneeded
9750:Required
9727:Unneeded
9705:Unneeded
9683:Unneeded
9661:Unneeded
9427:⊕
9414:∈
9260:⊕
9178:→
9131:ϕ
9124:ϕ
9087:→
9066:β
9023:→
9002:α
8938:ϕ
8908:ϕ
8871:→
8862:β
8834:→
8825:α
8802:β
8796:α
8741:→
8723:ϕ
8689:∈
8652:∈
8620:ϕ
8582:×
8555:→
8523:→
8482:→
8468:↓
8402:…
8368:orbifolds
8050:−
8047:↦
8021:↦
8015:−
7971:−
7962:−
7861:∈
7855:∣
7551:−
7471:→
7337:→
7298:bijection
6803:⋊
6777:⋉
6666:×
6443:morphisms
6273:→
6258:⋯
6240:∐
6234:σ
6199:_
6169:ˇ
6159:∈
6153:σ
6074:→
6052:↓
6046:↓
6024:→
5874:×
5870:⋯
5850:×
5779:→
5763:ε
5723:→
5698:ϕ
5674:→
5649:ϕ
5598:∐
5544:∐
5479:∈
5370:⇉
5322:×
5217:×
5204:⊂
5185:×
5140:→
4976:×
4779:∼
4597:∼
4487:π
4445:π
4421:⊂
4377:π
4306:π
4241:π
4205:Π
4169:π
4130:homotopic
3948:on sets.
3667:fibration
3650:→
3608:∈
3501:if it is
3478:⇉
3449:⇉
3420:connected
3394:−
3380:→
3192:⊆
3172:−
3138:∈
3108:), where
3007:∗
2985:∗
2976:∗
2960:∗
2954:∗
2908:∗
2899:∗
2859:∗
2850:∗
2799:∈
2793:∗
2740:∈
2699:∈
2666:∗
2660:∗
2578:∈
2552:∈
2494:∼
2451:∼
2405:∼
2382:∼
2120:→
2053:⇉
1928:−
1880:−
1667:−
1659:↦
1632:→
1563:↦
1524:→
1503:×
1330:morphisms
1293:is a set
1251:−
1243:∗
1235:−
1219:−
1208:∗
1179:∗
1142:−
1129:−
1085:∗
1079:∗
1070:−
1032:−
1023:∗
1017:∗
991:∗
954:−
945:∗
919:∗
911:−
875:∗
849:∗
820:∗
811:∗
785:∗
776:∗
744:∗
735:∗
709:∗
700:∗
671:∗
645:∗
533:∗
510:∗
463:⇀
457:×
448:∗
419:→
405:−
362:Algebraic
339:manifolds
262:→
250:∘
224:→
218:→
209:→
200:→
191:→
179:∘
156:→
124:→
13419:" (2002)
13333:: 305-64
13296:lead to
13280:14622598
13062:Archived
13043:Archived
12999:Archived
12575:See also
12522:topology
12452:→
12321:→
12276:→
11376:complete
10959:, where
10723:topology
10612:The set
10561:The set
10534:of each
10461:multiset
10106:Required
10103:Required
10100:Required
10097:Required
10084:Required
10081:Required
10078:Required
10075:Required
10062:Unneeded
10059:Required
10056:Required
10053:Required
10040:Unneeded
10037:Required
10034:Required
10031:Required
10018:Required
10015:Unneeded
10012:Required
10009:Required
9995:Required
9992:Unneeded
9989:Required
9986:Required
9972:Unneeded
9969:Unneeded
9966:Required
9963:Required
9949:Unneeded
9946:Unneeded
9943:Required
9940:Required
9927:Required
9924:Required
9921:Unneeded
9918:Required
9904:Required
9901:Required
9898:Unneeded
9895:Required
9882:Unneeded
9879:Required
9876:Unneeded
9873:Required
9859:Unneeded
9856:Required
9853:Unneeded
9850:Required
9837:Required
9834:Unneeded
9831:Unneeded
9828:Required
9814:Required
9811:Unneeded
9808:Unneeded
9805:Required
9792:Unneeded
9789:Unneeded
9786:Unneeded
9783:Required
9769:Unneeded
9766:Unneeded
9763:Unneeded
9760:Required
9747:Required
9744:Required
9741:Required
9738:Unneeded
9734:Groupoid
9724:Required
9721:Required
9718:Required
9715:Unneeded
9711:Groupoid
9702:Unneeded
9699:Required
9696:Required
9693:Unneeded
9680:Unneeded
9677:Unneeded
9674:Required
9671:Unneeded
9658:Unneeded
9655:Unneeded
9652:Unneeded
9649:Unneeded
9628:Identity
9548:presheaf
9210:concrete
9134:′
9116:and the
9100:′
9036:′
8952:′
8941:′
8930:′
8878:′
8841:′
8641:, where
7881:subgroup
7455: :
7326: :
7122:, where
5111:Examples
3967:Given a
3957:Topology
3952:Examples
3026:is then
2185:). Then
2075:, where
1712: :
1700: :
1688: :
1440:function
1348: :
1283:morphism
977:Identity
81:morphism
76:Category
40:groupoid
13448:at the
13436:at the
13350:, 2001
13141:: 1β31.
12586:2-group
12571:, etc.
11853:embeds
11446:adjoint
10690:natural
10175:natural
10125:functor
9618:Closure
9554:Puzzles
8211:on it.
7494:is the
6299:If the
5920:as the
5166:, then
3066:classes
2463:, i.e.
2161:be the
2008:), and
1988:) then
1336:) from
1302:objects
895:Inverse
390:with a
345: (
286:Setoids
89:inverse
63:with a
13306:, and
13278:
13270:
13230:
13176:
13081:, see
12925:
12667:
11993:Here,
11702:Here,
10857:cosets
10791:, and
10634:action
10027:Monoid
8817:where
8715:, and
8506:where
8155:, and
6441:, the
4584:setoid
4410:where
3990:, let
3080:, the
2436:. Let
2248:, and
2014:target
1994:source
1789:
1733:
1708:, and
1434:, and
1334:arrows
1285:is an
1056:, and
585:, and
437:and a
307:action
302:G-sets
105:monoid
13276:S2CID
13250:arXiv
13174:S2CID
13110:arXiv
13002:(PDF)
12995:(PDF)
12972:arXiv
12860:) = (
12754:. β
12738:) = (
12718:) = (
12702:) = (
12698:and (
12617:Notes
12567:, or
12381:with
10678:orbit
10350:from
10227:from
10168:orbit
10117:group
10071:Group
9779:magma
9756:Magma
7045:orbit
6990:with
6445:from
6301:group
4582:is a
4363:below
4088:from
4086:paths
3464:is a
3424:below
3309:: if
3121:orbit
2567:with
1328:) of
1304:with
980:: If
634:: If
60:Group
52:group
13446:core
13268:ISSN
13228:ISBN
12923:ISBN
12908:. β
12884:) *
12868:) *
12808:) *
12788:) *
12772:) *
12742:) *
12722:) *
12706:) *
12665:ISBN
12544:and
12303:and
12049:Grpd
11855:Grpd
11796:The
11791:sSet
11378:and
11372:Grpd
11282:Grpd
11252:Grpd
11088:Grpd
11081:Grpd
10204:, a
9914:loop
9891:Loop
9585:The
8854:and
8538:and
8036:and
7879:, a
6918:For
6792:(or
6722:and
6632:and
6401:and
6346:(or
5434:and
4829:and
4656:and
4077:are
3507:full
3503:wide
3499:full
3495:wide
3305:are
3276:and
3227:and
3119:The
2881:and
2729:and
2420:iff
2216:and
2028:).
1915:and
1766:and
1332:(or
1312:and
934:and
864:and
724:and
660:and
347:1927
291:sets
38:, a
34:and
13453:Lab
13441:Lab
13326:",
13260:doi
13246:226
13187:,"
13166:doi
12876:= (
12800:= (
12734:* (
12710:* (
11934:hom
11876:hom
11632:hom
11584:hom
11521:hom
11460:hom
11388:Cat
11297:GPD
11128:GPD
10939:to
10879:in
10859:of
10538:of
10377:to
10263:to
8988:of
8765:in
7887:of
7743:of
7698:of
7498:of
7251:in
7023:at
6938:in
6564:of
6505:of
6465:to
6421:in
5263:of
5162:of
5116:If
4945:of
4861:is
4716:to
4676:in
4562:If
4140:of
4108:to
4081:of
3877:or
3798:of
3729:of
3689:of
3535:or
3505:or
3497:or
3438:of
3349:to
3092:in
3088:or
3084:or
3057:is
2541:if
2514:by
2181:to
2016:of
1996:of
1972:If
1411:of
1340:to
1300:of
1168:If
800:or
605:in
139:,
46:or
26:In
13464::
13346:,
13340:,
13331:43
13310:,
13274:.
13266:.
13258:.
13244:.
13192:19
13172:,
13162:96
13160:,
13137:.
13133:.
13086:.
12962:^
12946:.
12904:*
12900:=
12896:*
12892:*
12888:*
12880:*
12872:*
12864:*
12856:*
12848:*
12844:*
12840:*
12832:=
12828:*
12824:*
12816:*
12804:*
12796:*
12792:*
12784:*
12776:*
12768:*
12760:*
12750:=
12746:*
12730:*
12726:*
12694:*
12690:*
12686:=
12659:,
12653:,
12625:^
12548:.
11861:.
11448::
11382:.
11083:.
10787:,
10783:,
10397:.
10131:.
9601:.
9599:12
9147:.
9052:,
8693:Ob
8678:,
8656:Ob
8434:.
7907:.
7051:.
6915:.
4294:.
3626:A
3623:.
3434:A
3411:.
3061:.
2787::=
2600:.
2510:β
2481::=
2432:β
2428:=
2424:β
1720::
1716:β
1704:β
1696:,
1692:β
1588:,
1438:a
1430:,
1423:);
1372:);
1352:β
898::
625:,
565:,
353:.
341:.
289::
277:.
13451:n
13439:n
13392:.
13300:.
13282:.
13262::
13252::
13168::
13139:6
13118:.
13112::
13098:.
13096:.
12980:.
12974::
12956:.
12933:)
12929:(
12906:a
12902:b
12898:a
12894:b
12890:b
12886:a
12882:b
12878:a
12874:a
12870:a
12866:b
12862:a
12858:b
12854:a
12850:a
12846:b
12842:b
12838:a
12834:a
12830:b
12826:b
12822:a
12818:b
12814:a
12810:a
12806:b
12802:a
12798:b
12794:b
12790:a
12786:b
12782:a
12778:b
12774:a
12770:b
12766:a
12762:b
12758:a
12752:a
12748:a
12744:a
12740:a
12736:a
12732:a
12728:a
12724:a
12720:a
12716:a
12712:a
12708:a
12704:a
12700:a
12696:a
12692:a
12688:a
12684:a
12607:)
12456:a
12389:a
12325:a
12280:a
12208:1
12202:G
12191:0
12185:G
12179::
12176:i
12148:0
12142:G
12131:1
12125:G
12119::
12116:t
12113:,
12110:s
12085:0
12079:G
12073:,
12068:1
12062:G
12017:)
12014:X
12011:(
12006:1
11978:)
11975:)
11972:G
11969:(
11966:N
11963:,
11960:X
11957:(
11948:t
11945:e
11942:S
11939:s
11927:)
11924:G
11921:,
11918:)
11915:X
11912:(
11907:1
11899:(
11890:d
11887:p
11884:r
11881:G
11840:t
11837:e
11834:S
11831:s
11823:d
11820:p
11817:r
11814:G
11810::
11807:N
11773:)
11770:C
11767:(
11763:e
11760:r
11757:o
11754:C
11729:]
11724:1
11717:C
11713:[
11710:C
11687:)
11684:)
11681:C
11678:(
11674:e
11671:r
11668:o
11665:C
11661:,
11658:G
11655:(
11646:d
11643:p
11640:r
11637:G
11625:)
11622:C
11619:,
11616:)
11613:G
11610:(
11607:i
11604:(
11595:t
11592:a
11589:C
11562:)
11559:)
11556:G
11553:(
11550:i
11547:,
11544:C
11541:(
11532:t
11529:a
11526:C
11514:)
11511:G
11508:,
11505:]
11500:1
11493:C
11489:[
11486:C
11483:(
11474:d
11471:p
11468:r
11465:G
11431:t
11428:a
11425:C
11417:d
11414:p
11411:r
11408:G
11404::
11401:i
11355:K
11352:,
11349:H
11346:,
11343:G
11321:.
11318:)
11315:)
11312:K
11309:,
11306:H
11303:(
11294:,
11291:G
11288:(
11276:)
11273:K
11270:,
11267:H
11261:G
11258:(
11230:K
11227:,
11224:H
11221:,
11218:G
11198:K
11195:,
11192:H
11172:K
11166:H
11146:)
11143:K
11140:,
11137:H
11134:(
11108:K
11105:,
11102:H
11051:H
11031:K
11011:G
10991:H
10967:K
10947:G
10927:K
10907:p
10887:G
10867:H
10843:G
10823:G
10803:H
10753:q
10733:p
10709:)
10706:x
10703:(
10700:G
10664:G
10644:G
10620:X
10569:X
10546:X
10518:X
10494:.
10491:G
10471:X
10436:X
10416:G
10385:x
10363:0
10359:x
10338:G
10316:0
10312:x
10291:x
10271:G
10251:)
10246:0
10242:x
10238:(
10235:G
10215:h
10190:0
10186:x
10154:)
10151:X
10148:,
10145:G
10142:(
9597:M
9527:.
9522:0
9518:c
9514:+
9511:)
9506:1
9502:c
9498:(
9495:d
9492:=
9489:)
9484:0
9480:c
9476:+
9471:1
9467:c
9463:(
9460:t
9435:0
9431:C
9422:1
9418:C
9409:0
9405:c
9401:+
9396:1
9392:c
9369:0
9365:C
9344:d
9322:1
9318:C
9295:0
9291:C
9268:0
9264:C
9255:1
9251:C
9228:0
9224:C
9191:0
9187:C
9181:d
9171:1
9167:C
9127:,
9104:)
9097:y
9093:(
9090:g
9084:)
9081:y
9078:(
9075:g
9072::
9069:)
9063:(
9060:g
9040:)
9033:x
9029:(
9026:f
9020:)
9017:x
9014:(
9011:f
9008::
9005:)
8999:(
8996:f
8976:Z
8956:)
8949:y
8945:,
8934:,
8927:x
8923:(
8920:,
8917:)
8914:y
8911:,
8905:,
8902:x
8899:(
8875:y
8868:y
8865::
8838:x
8831:x
8828::
8805:)
8799:,
8793:(
8773:Z
8753:)
8750:y
8747:(
8744:g
8738:)
8735:x
8732:(
8729:f
8726::
8703:)
8700:Y
8697:(
8686:y
8666:)
8663:X
8660:(
8649:x
8629:)
8626:y
8623:,
8617:,
8614:x
8611:(
8591:Y
8586:Z
8578:X
8558:Z
8552:Y
8549::
8546:g
8526:Z
8520:X
8517::
8514:f
8487:Z
8475:Y
8459:X
8418:)
8413:k
8409:n
8405:,
8399:,
8394:1
8390:n
8386:(
8382:P
8354:G
8334:]
8331:G
8327:/
8321:n
8316:A
8311:[
8289:n
8284:A
8259:)
8256:n
8253:(
8250:L
8247:G
8227:G
8199:2
8195:/
8190:Z
8169:]
8166:0
8163:[
8143:}
8140:]
8137:2
8134:[
8131:,
8128:]
8125:1
8122:[
8119:,
8116:]
8113:0
8110:[
8107:{
8087:]
8084:G
8080:/
8076:X
8073:[
8053:1
8044:1
8024:2
8018:2
7995:}
7992:2
7989:,
7986:1
7983:,
7980:0
7977:,
7974:1
7968:,
7965:2
7959:{
7956:=
7953:X
7933:2
7929:/
7924:Z
7895:G
7867:}
7864:G
7858:g
7850:g
7846:F
7842:{
7822:G
7798:G
7776:g
7772:F
7751:G
7731:g
7710:r
7707:G
7686:g
7666:G
7646:)
7642:r
7639:G
7635:,
7631:r
7628:G
7624:(
7620:m
7617:o
7614:H
7593:)
7589:r
7586:G
7582:(
7578:b
7575:o
7554:)
7548:,
7544:r
7541:G
7537:(
7533:m
7530:o
7527:H
7506:G
7481:t
7478:e
7475:S
7467:r
7464:G
7443:F
7420:G
7400:G
7380:F
7360:F
7340:X
7334:X
7312:g
7308:F
7283:r
7280:G
7259:G
7239:g
7219:)
7215:r
7212:G
7208:(
7205:F
7202:=
7199:X
7179:F
7159:G
7134:r
7131:G
7110:]
7106:t
7103:e
7100:S
7096:,
7092:r
7089:G
7085:[
7062:G
7031:x
7007:x
7004:=
7001:x
6998:g
6978:)
6975:x
6972:,
6969:g
6966:(
6946:X
6926:x
6903:x
6900:g
6897:=
6894:y
6874:)
6871:x
6868:,
6865:g
6862:h
6859:(
6856:=
6853:)
6850:x
6847:,
6844:g
6841:(
6838:)
6835:y
6832:,
6829:h
6826:(
6806:G
6800:X
6780:X
6774:G
6754:x
6751:g
6748:=
6745:)
6742:x
6739:,
6736:g
6733:(
6730:t
6710:x
6707:=
6704:)
6701:x
6698:,
6695:g
6692:(
6689:s
6669:X
6663:G
6660:=
6657:)
6654:C
6651:(
6647:m
6644:o
6641:h
6620:X
6617:=
6614:)
6611:C
6608:(
6604:b
6601:o
6584:.
6572:G
6554:;
6542:y
6539:=
6536:x
6533:g
6513:G
6493:g
6473:y
6453:x
6429:X
6409:y
6389:x
6378:;
6366:X
6330:X
6310:G
6276:A
6266:k
6262:i
6253:1
6249:i
6244:U
6237::
6204:)
6196:A
6191:,
6186:U
6181:(
6176:k
6166:H
6156:]
6150:[
6118:i
6114:U
6084:i
6080:U
6067:k
6064:i
6060:U
6037:j
6034:i
6030:U
6017:k
6014:j
6011:i
6007:U
5979:n
5957:n
5951:G
5928:n
5903:1
5897:G
5887:0
5881:G
5863:0
5857:G
5844:1
5838:G
5832:=
5827:n
5821:G
5790:i
5787:i
5783:U
5774:i
5770:U
5766::
5731:i
5727:U
5718:j
5715:i
5711:U
5707::
5702:i
5694:=
5691:t
5682:j
5678:U
5669:j
5666:i
5662:U
5658::
5653:j
5645:=
5642:s
5623:.
5609:j
5606:i
5602:U
5595:=
5590:1
5584:G
5566:,
5552:i
5548:U
5541:=
5536:0
5530:G
5504:X
5482:I
5476:i
5472:}
5466:i
5462:U
5458:{
5455:=
5450:U
5421:.
5378:0
5374:X
5365:1
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5336:0
5332:X
5326:Y
5316:0
5312:X
5308:=
5303:1
5299:X
5276:0
5272:X
5247:Y
5225:0
5221:X
5212:0
5208:X
5199:0
5195:X
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5179:0
5175:X
5143:Y
5135:0
5131:X
5127::
5124:f
5094:}
5091:x
5088:{
5066:X
5062:d
5058:i
5055:=
5052:t
5049:=
5046:s
5026:X
5002:X
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4973:X
4953:X
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4909:X
4893:.
4881:)
4878:x
4875:,
4872:z
4869:(
4849:)
4846:x
4843:,
4840:y
4837:(
4817:)
4814:y
4811:,
4808:z
4805:(
4794:;
4782:y
4776:x
4756:)
4753:x
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4744:(
4724:y
4704:x
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4664:y
4644:x
4633:;
4621:X
4570:X
4542:A
4522:A
4502:)
4499:X
4496:(
4491:1
4466:)
4463:A
4460:,
4457:X
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4418:A
4398:)
4395:A
4392:,
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4381:1
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4321:)
4318:X
4315:(
4310:1
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4262:)
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4220:)
4217:X
4214:(
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4173:1
4148:X
4116:q
4096:p
4065:q
4045:p
4025:X
4003:0
3999:G
3978:X
3936:B
3916:B
3889:e
3861:b
3858:=
3855:)
3852:e
3849:(
3846:p
3826:x
3806:E
3786:e
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3763:x
3760:(
3757:p
3737:B
3717:b
3697:E
3677:x
3653:B
3647:E
3644::
3641:p
3611:Y
3605:y
3602:,
3599:x
3579:)
3576:y
3573:,
3570:x
3567:(
3564:H
3561:=
3558:)
3555:y
3552:,
3549:x
3546:(
3543:G
3523:Y
3520:=
3517:X
3481:Y
3475:H
3452:X
3446:G
3397:1
3390:f
3386:g
3383:f
3377:g
3357:y
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3317:f
3293:)
3290:y
3287:(
3284:G
3264:)
3261:x
3258:(
3255:G
3235:y
3215:x
3195:X
3189:)
3186:)
3183:x
3180:(
3175:1
3168:t
3164:(
3161:s
3141:X
3135:x
3125:G
3114:G
3110:x
3106:x
3104:,
3102:x
3100:(
3098:G
3094:G
3078:G
3059:f
3055:f
3039:x
3035:1
3024:x
3010:g
3004:f
3001:=
2998:)
2993:z
2989:1
2982:g
2979:(
2973:)
2968:y
2964:1
2957:f
2949:x
2945:1
2941:(
2921:)
2916:z
2912:1
2905:g
2902:(
2894:y
2890:1
2867:y
2863:1
2856:)
2853:f
2845:x
2841:1
2837:(
2817:)
2814:z
2811:,
2808:x
2805:(
2802:G
2796:g
2790:f
2784:f
2781:g
2761:,
2758:)
2755:z
2752:,
2749:y
2746:(
2743:G
2737:g
2717:)
2714:y
2711:,
2708:x
2705:(
2702:G
2696:f
2674:y
2670:1
2663:f
2655:x
2651:1
2640:f
2626:)
2623:y
2620:,
2617:x
2614:(
2611:G
2586:0
2582:G
2575:x
2555:G
2549:a
2527:x
2523:1
2512:a
2508:a
2488:/
2484:G
2476:0
2472:G
2441:0
2438:G
2434:b
2430:b
2426:a
2422:a
2408:b
2402:a
2372:G
2354:d
2351:i
2340:0
2337:G
2322:v
2319:n
2316:i
2294:p
2291:m
2288:o
2285:c
2263:v
2260:n
2257:i
2246:G
2231:v
2228:n
2225:i
2203:p
2200:m
2197:o
2194:c
2183:y
2179:x
2175:y
2173:,
2171:x
2169:(
2167:G
2159:G
2128:0
2124:G
2115:1
2111:G
2088:1
2084:G
2061:0
2057:G
2048:1
2044:G
2033:G
2026:f
2024:(
2022:t
2018:f
2010:y
2006:f
2004:(
2002:s
1998:f
1990:x
1986:y
1984:,
1982:x
1980:(
1978:G
1974:f
1966:.
1952:x
1947:d
1944:i
1939:=
1936:f
1931:1
1924:f
1901:y
1896:d
1893:i
1888:=
1883:1
1876:f
1872:f
1862:;
1850:)
1847:f
1844:g
1841:(
1838:h
1835:=
1832:f
1829:)
1826:g
1823:h
1820:(
1810:;
1798:f
1795:=
1792:f
1784:y
1779:d
1776:i
1754:f
1751:=
1746:x
1741:d
1738:i
1730:f
1718:w
1714:z
1710:h
1706:z
1702:y
1698:g
1694:y
1690:x
1686:f
1670:1
1663:f
1656:f
1653::
1650:)
1647:x
1644:,
1641:y
1638:(
1635:G
1629:)
1626:y
1623:,
1620:x
1617:(
1614:G
1611::
1607:v
1604:n
1601:i
1590:y
1586:x
1581:;
1569:f
1566:g
1560:)
1557:f
1554:,
1551:g
1548:(
1545::
1542:)
1539:z
1536:,
1533:x
1530:(
1527:G
1521:)
1518:y
1515:,
1512:x
1509:(
1506:G
1500:)
1497:z
1494:,
1491:y
1488:(
1485:G
1482::
1477:z
1474:,
1471:y
1468:,
1465:x
1460:p
1457:m
1454:o
1451:c
1436:z
1432:y
1428:x
1421:x
1419:,
1417:x
1415:(
1413:G
1397:x
1392:d
1389:i
1377:x
1370:y
1368:,
1366:x
1364:(
1362:G
1358:f
1354:y
1350:x
1346:f
1342:y
1338:x
1326:y
1324:,
1322:x
1320:(
1318:G
1314:y
1310:x
1298:0
1295:G
1291:G
1268:.
1254:1
1247:a
1238:1
1231:b
1227:=
1222:1
1215:)
1211:b
1205:a
1202:(
1182:b
1176:a
1165:,
1153:a
1150:=
1145:1
1138:)
1132:1
1125:a
1121:(
1094:b
1091:=
1088:b
1082:a
1073:1
1066:a
1044:a
1041:=
1035:1
1028:b
1020:b
1014:a
994:b
988:a
957:1
950:a
942:a
922:a
914:1
907:a
878:c
872:b
852:b
846:a
826:)
823:c
817:b
814:(
808:a
788:c
782:)
779:b
773:a
770:(
750:)
747:c
741:b
738:(
732:a
712:c
706:)
703:b
697:a
694:(
674:c
668:b
648:b
642:a
613:G
593:c
573:b
553:a
490:G
466:G
460:G
454:G
451::
425:,
422:G
416:G
413::
408:1
378:G
329:.
317:G
297:,
265:C
259:A
256::
253:g
247:h
227:C
221:A
215:)
212:B
206:A
203:(
197:)
194:C
188:B
185:(
182::
159:C
153:B
150::
147:h
127:B
121:A
118::
115:g
71:;
23:.
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