1663:. Originally published in: The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168. Republished in a free on-line journal: Reprints in Theory and Applications of Categories, No. 6 (2004), with the permission of Springer-Verlag.
1387:
of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist
606:
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802:
676:, and one morphism for each element of the group. This would not be counted as concrete according to the intuitive notion described at the top of this article. But every faithful
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that a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a category
1568:
39:(or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional
989:
47:
as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the
1375:
1364:
to the set containing its objects and morphisms. Functors can be simply viewed as functions acting on the objects and morphisms.
56:
214:, the homomorphisms of a concrete category may be formally identified with their underlying functions (i.e., their images under
420:
In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the "concrete category
1622:
462:
may map different objects to the same set and, if this occurs, it will also map different morphisms to the same function.
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whose objects are small categories and whose morphisms are functors can be made concrete by sending each category
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611:
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815:
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be faithful means that it maps different morphisms between the same objects to different functions. However,
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911:
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Similarly, any set with four elements can be given two non-isomorphic group structures: one isomorphic to
677:
28:
1505:
1690:
210:(e.g., group homomorphisms, ring homomorphisms, etc.) Because of the faithfulness of the functor
1404:. In the same article, Freyd cites an earlier result that the category of "small categories and
1460:
870:
218:); the homomorphisms then regain the usual interpretation as "structure-preserving" functions.
66:
A concrete category, when defined without reference to the notion of a category, consists of a
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861:
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and their sup-preserving maps. Conversely, starting from this equivalence we can recover
8:
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1504:. The category of models for this signature then contains a full subcategory which is
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It follows from the previous example that the opposite of any concretizable category
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545:, but they have the same underlying function, namely the identity function on
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is often equipped not with the "obvious" forgetful functor but the functor
601:{\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} }
1169:
1401:
1168:
is concretizable. The forgetful functor which arises in this way is the
1065:
under inclusion, those functions between them arising from some relation
55:, and trivially also the category of sets itself. On the other hand, the
20:
1448:
497:
of topological spaces and continuous maps, but mapped to the same set
1350:
656:
may be regarded as an "abstract" category with one arbitrary object,
63:, i.e. it does not admit a faithful functor to the category of sets.
1567:. For example, it may be useful to think of the models of a theory
1384:
266:
44:
1640:
AdΓ‘mek, JiΕΓ, Herrlich, Horst, & Strecker, George E.; (1990).
385:
section exhibits two large categories that are not concretizable.
16:
Category equipped with a faithful functor to the category of sets
707:
206:
It is customary to call the morphisms in a concrete category
1411:
1221:
is any small category, then there exists a faithful functor
1054:{\displaystyle \rho (A)=\{y\in Y\mid \exists \,x\in A:xRy\}}
713:
may be regarded as an abstract category with a unique arrow
703:
can be made into a concrete category in at least one way.
1497:
ranging over the class of all cardinal numbers, forms a
1408:-classes of functors" also fails to be concretizable.
1646:(4.2MB PDF). Originally publ. John Wiley & Sons.
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by representing each set as itself and each function
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is equivalent to a full subcategory of the category
1536:. For this reason, it makes sense to call a pair (
729:. This can be made concrete by defining a functor
405:. Hence there may be several concrete categories (
1283:{\displaystyle \coprod _{c\in \mathrm {ob} C}X(c)}
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978:
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381:in less elementary language via presheaves.) The
249:. All small categories are concretizable: define
102:, the identity function on the underlying set of
1682:
1605:
1520:In some parts of category theory, most notably
699:. Since every group acts faithfully on itself,
199:its "underlying set", and to every morphism in
473:are two different topologies on the same set
393:Contrary to intuition, concreteness is not a
114:, and the composition of a homomorphism from
1668:Concrete categories and infinitary languages
1048:
1008:
979:{\displaystyle \rho :2^{X}\rightarrow 2^{Y}}
791:
767:
293:), and its morphism part maps each morphism
237:); i.e., if there exists a faithful functor
1582:In this context, a concrete category over
1349:which maps a Banach space to its (closed)
636:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
179:(the category of sets and functions) is a
1412:Implicit structure of concrete categories
1026:
629:
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413:) all corresponding to the same category
401:may admit several faithful functors into
253:so that its object part maps each object
1515:
1485:-ary operations of a concrete category (
849:{\displaystyle D(x)\hookrightarrow D(y)}
424:". For example, "the concrete category
1524:, it is common to replace the category
797:{\displaystyle D(x)=\{a\in P:a\leq x\}}
683:(equivalently, every representation of
541:) are considered distinct morphisms in
493:) are distinct objects in the category
57:homotopy category of topological spaces
1683:
382:
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229:if there exists a concrete category (
1575:as forming a concrete category over
1316:For technical reasons, the category
933:{\displaystyle R\subseteq X\times Y}
509:. Moreover, the identity morphism (
1673:Journal of Pure and Applied Algebra
1368:
1202:may be equipped with the composite
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195:, which assigns to every object of
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1186:is again concretizable, since if
1643:Abstract and Concrete Categories
691:) determines a faithful functor
273:(i.e. all morphisms of the form
122:followed by a homomorphism from
98:. Furthermore, for every object
1305:one obtains a faithful functor
873:can be made concrete by taking
261:to the set of all morphisms of
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1290:. By composing this with the
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1271:
1170:contravariant powerset functor
1061:. Noting that power sets are
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996:
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843:
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1654:. (now free on-line edition).
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1617:(3rd ed.), AMS Chelsea,
1101:of the forgetful functor for
525:) and the identity morphism (
141:
90:, from the underlying set of
1144:formed as the set of pairs (
1069:in this way are exactly the
130:must be a homomorphism from
106:must be a homomorphism from
7:
1416:Given a concrete category (
203:its "underlying function".
86:a set of functions, called
10:
1707:
1528:with a different category
388:
78:; and for any two objects
501:by the forgetful functor
333:) which maps each member
191:is to be thought of as a
94:to the underlying set of
1666:RosickΓ½, JiΕΓ; (1981).
1660:Homotopy is not concrete
1593:
1379:, where the objects are
1071:supremum-preserving maps
869:and whose morphisms are
74:, each equipped with an
31:that is equipped with a
1105:with this embedding of
741:which maps each object
1657:Freyd, Peter; (1970).
1586:is sometimes called a
1562:concrete category over
1461:natural transformation
1392:faithful functor from
1383:and the morphisms are
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1233:which maps a presheaf
1190:is a faithful functor
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1676:, Volume 22, Issue 3.
1516:Relative concreteness
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1136:as the relation from
1120:can be embedded into
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903:
901:{\displaystyle 2^{X}}
851:
812:to the inclusion map
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689:group of permutations
671:
669:{\displaystyle \ast }
638:
603:
454:The requirement that
353:) to the composition
1481:-ary predicates and
1400:was first proven by
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990:
944:
912:
885:
816:
749:
660:
612:
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1552:a faithful functor
1406:natural equivalence
1329:linear contractions
377:expresses the same
1607:Mac Lane, Saunders
1548:is a category and
1381:topological spaces
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1267:
1051:
976:
930:
908:and each relation
898:
865:whose objects are
846:
794:
666:
633:
598:
428:" means the pair (
373:). (Item 6 under
165:is a category, and
53:category of groups
1624:978-0-8218-1646-2
1611:Birkhoff, Garrett
1532:, often called a
1477:The class of all
1244:
1237:to the coproduct
1089:as the composite
1083:complete lattices
1063:complete lattices
881:to its power set
193:forgetful functor
148:concrete category
25:concrete category
1698:
1628:
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1385:homotopy classes
1369:Counter-examples
1292:Yoneda embedding
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647:Further examples
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465:For example, if
442:identity functor
383:Counter-examples
375:Further examples
309:to the function
181:faithful functor
37:category of sets
33:faithful functor
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1691:Category theory
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1472:N-ary operation
1457:N-ary predicate
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1445:U(c) = (U(c))
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1373:The category
1363:
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1356:The category
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1326:
1325:Banach spaces
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1116:The category
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859:The category
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88:homomorphisms
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61:concretizable
58:
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43:, and of its
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22:
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1601:
1587:
1583:
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1576:
1570:
1564:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1522:topos theory
1519:
1509:
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1478:
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1471:
1467:
1463:
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1302:
1298:
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1234:
1230:
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1211:
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1199:
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684:
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653:
551:
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444:
440:denotes the
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188:
187:The functor
186:
176:
172:
168:
162:
158:) such that
155:
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24:
18:
1402:Peter Freyd
986:defined by
221:A category
150:is a pair (
21:mathematics
1634:References
1506:equivalent
1449:subfunctor
1156:) for all
652:Any group
142:Definition
1588:construct
1502:signature
1447:. Then a
1351:unit ball
1253:∈
1246:∐
1073:. Hence
1031:∈
1024:∃
1021:∣
1015:∈
994:ρ
964:→
948:ρ
925:×
919:⊆
871:relations
832:↪
786:≤
774:∈
721:whenever
664:∗
578:×
477:, then (
45:morphisms
41:structure
1685:Category
1613:(1999),
1544:) where
1493:), with
1424:) and a
1338: :
1225: :
1164:; hence
733: :
436:) where
395:property
267:codomain
171: :
51:and the
29:category
1615:Algebra
1540:,
1459:and a
1420:,
537:,
529:,
521:,
513:,
489:,
485:) and (
481:,
432:,
409:,
389:Remarks
241::
72:objects
59:is not
35:to the
1650:
1621:
1431:, let
265:whose
1594:Notes
1573:sorts
1569:with
1499:large
1198:then
708:poset
687:as a
533:) β (
517:) β (
68:class
27:is a
1648:ISBN
1619:ISBN
1394:hTop
1376:hTop
1327:and
867:sets
681:-set
469:and
325:) β
82:and
23:, a
1670:.
1584:Set
1577:Set
1526:Set
1508:to
1470:an
1451:of
1441:Set
1398:Set
1396:to
1390:any
1358:Cat
1347:Set
1340:Ban
1323:of
1318:Ban
1311:Set
1303:Set
1231:Set
1227:Set
1217:If
1212:Set
1208:Set
1196:Set
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