3395:
2657:
3642:
3662:
3652:
1652:): two loops (under equivalence relation of homotopy) may not have the same base point so they cannot multiply with each other. In the language of category, this means here two morphisms may not have the same source object (or target object, because in this case for any morphism the source object and the target object are same: the base point) so they can not compose with each other.
38:
1656:
1152:
definition in which each object is identified with the corresponding identity morphism. This stems from the idea that the fundamental data of categories are morphisms and not objects. In fact, categories can be defined without reference to objects at all using a partial binary operation with additional properties.
1151:
From these axioms, one can prove that there is exactly one identity morphism for every object. Often the map assigning each object its identity morphism is treated as an extra part of the structure of a category, namely a class function i: ob(C) β mor(C). Some authors use a slight variant of the
93:
is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities
98:
and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.
2643:
is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical
1759:
forms another concrete category, where morphisms are graph homomorphisms (i.e., mappings between graphs which send vertices to vertices and edges to edges in a way that preserves all adjacency and incidence relations).
1212:. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Since, in small categories, the objects form a set, a small category can be viewed as an
2094:
can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the
1445:. Actually, in the view of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider a topological space
1399:
is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories.
1642:
1527:
109:
Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two
2618:
if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. Examples include
1589:
1562:
1474:
1333:, there can be at most one morphism between any two objects. The existence of identity morphisms and the composability of the morphisms are guaranteed by the
700:
can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any
3039:
2952:
175:
2995:
2915:
2893:
2811:
2977:
1687:
The class of all preordered sets with order-preserving functions (i.e., monotone-increasing functions) as morphisms forms a category,
103:
102:
In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the
2961:
2937:
2856:
2792:
2765:
69:) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows
94:
between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to
2772:
2025:
2014:
150:
3691:
3032:
2876:
2825:
2685:
3236:
3191:
2821:
2069:
3665:
3605:
2871:
2048:
1800:
1689:
2495:
Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:
120:
Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include
3655:
3441:
3305:
3213:
2541:
1732:
1438:
1269:
1275:
Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called
3614:
3258:
3196:
3119:
2615:
2577:
2352:
2321:
1976:
114:
31:
1995:
1294:
as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category.
187:
below. The basic definitions in this article are contained within the first few chapters of any of these books.
3686:
3645:
3601:
3206:
3025:
2589:
3201:
3183:
2608:
1224:
properties. Large categories on the other hand can be used to create "structures" of algebraic structures.
3408:
3174:
3154:
2675:
2629:
1792:
1330:
1265:
1237:
971:
131:
82:
3290:
3129:
2032:
1853:
1671:
1342:
1252:
1221:
213:
198:
2656:
1598:
1483:
3102:
3097:
2573:
1917:
1713:
1645:
1442:
1346:
1241:
218:
203:
167:
712:
There are many equivalent definitions of a category. One commonly used definition is as follows. A
3446:
3394:
3324:
3320:
3124:
2527:
2478:
2006:
1956:
1930:
1908:
1895:
1840:
1815:
1678:
as needed) where composition of morphisms is concatenation of paths. Such a category is called the
1675:
1403:
1365:
1338:
1213:
733:
651:
1437:
is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups,
41:
This is a category with a collection of objects A, B, C and collection of morphisms denoted f, g,
3300:
3295:
3277:
3159:
3134:
2662:
2483:
1900:
1889:
1876:
1845:
1834:
1718:
1705:
1334:
1233:
724:
629:
141:
3609:
3546:
3534:
3436:
3361:
3356:
3314:
3310:
3092:
3087:
2991:
2957:
2947:
2933:
2911:
2889:
2866:
2852:
2829:
2807:
2788:
2761:
2670:
2604:
2585:
2095:
2020:
1881:
1870:
1787:
1694:
1530:
1280:
1276:
1257:
1177:
180:
163:
155:
145:
136:
127:
78:
66:
3570:
3456:
3431:
3366:
3351:
3346:
3285:
3114:
3082:
2633:
2620:
2597:
2596:, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an
1781:
1749:
1741:
1376:
1372:
1246:
225:
208:
122:
74:
2839:
1567:
1540:
1452:
3482:
3048:
2925:
2903:
2835:
2039:
1261:
359:
336:
117:" for purposes of category theory, even if they do not have precisely the same structure.
89:
45:, and the loops are the identity arrows. This category is typically denoted by a boldface
3519:
2611:
exist in it. The categories of sets, abelian groups and topological spaces are complete.
1250:. It is the most basic and the most commonly used category in mathematics. The category
3514:
3498:
3461:
3451:
3371:
2781:
1667:
1664:
1350:
494:
471:
159:
2834:, Reprints in Theory and Applications of Categories, vol. 12 (revised ed.),
2568:, and the composition of morphisms is compatible with these group structures; i.e. is
17:
3680:
3509:
3341:
3218:
3144:
2972:
2569:
2565:
2001:
1737:
1680:
901:
673:
70:
3263:
3164:
2461:
2424:
2184:
2057:
1982:
1952:
1859:
1756:
1181:
449:
426:
247:
3524:
1644:, then the set we get has only the structure of groupoid (which is called as the
73:
and the existence of an identity arrow for each object. A simple example is the
3504:
3376:
3246:
2383:
2247:
1701:, and requiring that morphisms are functions that respect this added structure.
1427:
1369:
1354:
54:
2752:
1132:
Some authors write the composite of morphisms in "diagrammatic order", writing
27:
Mathematical object that generalizes the standard notions of sets and functions
3556:
3494:
3107:
2956:, Graduate Texts in Mathematics, vol. 5 (2nd ed.), Springer-Verlag,
2680:
2652:
2312:
2061:
1968:
585:
562:
404:
381:
95:
2600:. A typical example of an abelian category is the category of abelian groups.
1240:
between them (as morphisms), where the composition of morphisms is the usual
3550:
3241:
2581:
1987:
1426:
under composition. A morphism that is invertible in this sense is called an
539:
516:
2530:, where the objects are represented as points and the morphisms as arrows.
1674:
of the graph, and the morphisms are the paths in the graph (augmented with
2469:
is both an endomorphism and an isomorphism. The class of automorphisms of
1406:
can be seen as a category with a single object in which every morphism is
37:
3619:
3251:
3149:
3004:
2593:
2164:
1709:
1564:, and as a set it has the structure of group; if then let the base point
1434:
1419:
1298:
746:
701:
314:
291:
1763:
Other examples of concrete categories are given by the following table.
3589:
3579:
3228:
3139:
2990:, Advanced studies in mathematics and logic, vol. 3, Polimetrica,
2152:
2074:
1821:
3584:
2546:
2453:
1940:
1826:
1361:
1217:
697:
607:
1655:
1395:
are precisely the elements of the monoid, the identity morphism of
3466:
3017:
2640:
2143:, and the morphisms are also pairs, consisting of one morphism in
1305:, β€) forms a small category, where the objects are the members of
36:
1697:, i.e. a category obtained by adding some type of structure onto
3008:
3406:
3059:
3021:
2806:, Oxford logic guides, vol. 49, Oxford University Press,
2847:
Borceux, Francis (1994), "Handbook of
Categorical Algebra",
2751:
AdΓ‘mek, JiΕΓ; Herrlich, Horst; Strecker, George E. (1990),
2325:
if it has a right inverse, i.e. if there exists a morphism
173:
The classic and still much used text on category theory is
2930:
Conceptual
Mathematics: A First Introduction to Categories
2851:, vol. 50β52, Cambridge: Cambridge University Press,
2356:
if it has a left inverse, i.e. if there exists a morphism
1109:) when there may be confusion about to which category hom(
2387:
if it has an inverse, i.e. if there exists a morphism
2135:: the objects are pairs consisting of one object from
1716:
as the composition operation forms a large category,
1601:
1570:
1543:
1486:
1455:
3569:
3533:
3481:
3474:
3425:
3334:
3276:
3227:
3182:
3173:
3070:
2780:
1636:
1583:
1556:
1521:
1468:
2849:Encyclopedia of Mathematics and its Applications
2316:if it is both a monomorphism and an epimorphism.
2576:. If, furthermore, the category has all finite
1290:is the small category that has the elements of
1264:between them (as morphisms). Abstracting from
1379:) forms a small category with a single object
3033:
2884:Herrlich, Horst; Strecker, George E. (2007),
8:
2526:) can most conveniently be represented with
2477:). For locally small categories, it forms a
1353:can be seen as a category when viewed as an
32:Category (disambiguation) Β§ Mathematics
1341:of the preorder. By the same argument, any
1236:of all sets (as objects) together with all
162:. All of the preceding categories have the
3661:
3651:
3478:
3422:
3403:
3179:
3067:
3056:
3040:
3026:
3018:
843:. Morphisms in this subclass are written
189:
2779:Asperti, Andrea; Longo, Giuseppe (1991),
2704:
1625:
1606:
1600:
1575:
1569:
1548:
1542:
1510:
1491:
1485:
1460:
1454:
1309:, the morphisms are arrows pointing from
2953:Categories for the Working Mathematician
1765:
1654:
1192:is a category such that for all objects
176:Categories for the Working Mathematician
170:as the associative operation on arrows.
2697:
2073:consists of all small categories, with
2064:between them form a concrete category.
2481:under morphism composition called the
1670:a small category: the objects are the
1387:is any fixed set.) The morphisms from
2444:). For locally small categories, end(
1730:is a concrete category. The category
1349:can be seen as a small category. Any
895:such that the following axioms hold:
7:
1740:and their group homomorphisms, is a
827:) denotes the subclass of morphisms
183:. Other references are given in the
2978:Stanford Encyclopedia of Philosophy
2518:Relations among morphisms (such as
2502:is a monomorphism and a retraction;
2255:) if it is right-cancellable, i.e.
776:class function cod: mor(C) β ob(C),
765:class function dom: mor(C) β ob(C),
113:categories may also be considered "
2192:) if it is left-cancellable, i.e.
104:semantics of programming languages
25:
2564:) are not just sets but actually
2124:are categories, one can form the
1637:{\displaystyle \pi _{1}(X,x_{0})}
1522:{\displaystyle \pi _{1}(X,x_{0})}
1272:, a special class of categories.
3660:
3650:
3641:
3640:
3393:
2783:Categories, Types and Structures
2754:Abstract and Concrete Categories
2655:
2508:is an epimorphism and a section;
2436:. The class of endomorphisms of
184:
2932:, Cambridge University Press,
2081:Construction of new categories
1809:monotone-increasing functions
1631:
1612:
1516:
1497:
1410:, that is, for every morphism
1:
2975:, in Zalta, Edward N. (ed.),
2971:Marquis, Jean-Pierre (2006),
2831:Toposes, Triples and Theories
2686:Table of mathematical symbols
2151:. Such pairs can be composed
1704:The class of all groups with
2572:. Such a category is called
1595:, and take the union of all
1268:instead of functions yields
584:Commutative-and-associative
3335:Constructions on categories
2986:Sica, Giandomenico (2006),
2872:Encyclopedia of Mathematics
2771:(now free on-line edition,
2456:under morphism composition.
2077:between them as morphisms.
2049:continuously differentiable
1007:, such that every morphism
980:, there exists a morphism 1
3708:
3442:Higher-dimensional algebra
2928:; Schanuel, Steve (1997),
2634:Scott-continuous functions
2588:. If all morphisms have a
1748:, and the prototype of an
1244:, forms a large category,
1156:Small and large categories
29:
3636:
3415:
3402:
3391:
3066:
3055:
2539:In many categories, e.g.
1533:of the topological space
791:, a binary operation hom(
65:to distinguish it from a
2988:What is category theory?
1684:generated by the graph.
1591:runs over all points of
1117:) refers) to denote the
972:left and right unit laws
817:composition of morphisms
779:for every three objects
3252:Cokernels and quotients
3175:Universal constructions
2910:(2nd ed.), Dover,
2716:Some authors write Mor(
2630:complete partial orders
1005:identity morphism for x
855:, and the composite of
166:as identity arrows and
3409:Higher category theory
3155:Natural transformation
2802:Awodey, Steve (2006),
2676:Higher category theory
1660:
1638:
1585:
1558:
1523:
1470:
1420:left and right inverse
1288:discrete category on I
1220:but without requiring
1190:locally small category
1121:of all morphisms from
191:Group-like structures
50:
18:Locally small category
2888:, Heldermann Verlag,
2705:Barr & Wells 2005
2614:A category is called
2603:A category is called
1934:-module homomorphisms
1658:
1639:
1586:
1584:{\displaystyle x_{0}}
1559:
1557:{\displaystyle x_{0}}
1524:
1471:
1469:{\displaystyle x_{0}}
1449:and fix a base point
1443:equivalence relations
1343:partially ordered set
1208:) is a set, called a
1033:, and every morphism
835:) such that dom(f) =
81:and whose arrows are
61:(sometimes called an
40:
3692:Algebraic structures
3278:Algebraic categories
2528:commutative diagrams
2026:continuous functions
2007:measurable functions
1864:graph homomorphisms
1827:monoid homomorphisms
1736:, consisting of all
1714:function composition
1646:fundamental groupoid
1599:
1568:
1541:
1484:
1453:
1414:there is a morphism
1347:equivalence relation
1242:function composition
1200:, the hom-class hom(
994:(some authors write
879:is often written as
77:, whose objects are
30:For other uses, see
3447:Homotopy hypothesis
3125:Commutative diagram
2556:, the hom-sets hom(
2534:Types of categories
1901:field homomorphisms
1846:group homomorphisms
1706:group homomorphisms
1537:and the base point
1366:algebraic structure
1214:algebraic structure
1079:is a morphism from
976:: for every object
192:
3160:Universal property
2948:Mac Lane, Saunders
2663:Mathematics portal
2628:, the category of
2584:, it is called an
2514:is an isomorphism.
2484:automorphism group
2285:for all morphisms
2220:for all morphisms
2159:Types of morphisms
2112:Product categories
2021:topological spaces
1882:ring homomorphisms
1661:
1634:
1581:
1554:
1519:
1466:
1325:. Furthermore, if
1260:(as objects) with
630:Commutative monoid
190:
156:topological spaces
154:, the category of
146:ring homomorphisms
140:, the category of
126:, the category of
51:
3674:
3673:
3632:
3631:
3628:
3627:
3610:monoidal category
3565:
3564:
3437:Enriched category
3389:
3388:
3385:
3384:
3362:Quotient category
3357:Opposite category
3272:
3271:
2997:978-88-7699-031-1
2973:"Category Theory"
2917:978-0-486-47187-7
2895:978-3-88538-001-6
2813:978-0-19-856861-2
2671:Enriched category
2586:additive category
2101:opposite category
2055:
2054:
1755:The class of all
1695:concrete category
1659:A directed graph.
1531:fundamental group
694:
693:
181:Saunders Mac Lane
67:concrete category
63:abstract category
16:(Redirected from
3699:
3664:
3663:
3654:
3653:
3644:
3643:
3479:
3457:Simplex category
3432:Categorification
3423:
3404:
3397:
3367:Product category
3352:Kleisli category
3347:Functor category
3192:Terminal objects
3180:
3115:Adjoint functors
3068:
3057:
3042:
3035:
3028:
3019:
3000:
2981:
2966:
2942:
2926:Lawvere, William
2920:
2904:Jacobson, Nathan
2898:
2880:
2861:
2842:
2816:
2797:
2786:
2770:
2759:
2737:
2714:
2708:
2702:
2665:
2660:
2659:
2616:cartesian closed
2598:abelian category
2126:product category
2040:smooth manifolds
1806:preordered sets
1766:
1750:abelian category
1742:full subcategory
1643:
1641:
1640:
1635:
1630:
1629:
1611:
1610:
1590:
1588:
1587:
1582:
1580:
1579:
1563:
1561:
1560:
1555:
1553:
1552:
1528:
1526:
1525:
1520:
1515:
1514:
1496:
1495:
1475:
1473:
1472:
1467:
1465:
1464:
1377:identity element
1373:binary operation
1279:. For any given
1262:binary relations
1256:consists of all
1087:". We write hom(
193:
75:category of sets
44:
21:
3707:
3706:
3702:
3701:
3700:
3698:
3697:
3696:
3687:Category theory
3677:
3676:
3675:
3670:
3624:
3594:
3561:
3538:
3529:
3486:
3470:
3421:
3411:
3398:
3381:
3330:
3268:
3237:Initial objects
3223:
3169:
3062:
3051:
3049:Category theory
3046:
3016:
2998:
2985:
2970:
2964:
2946:
2940:
2924:
2918:
2902:
2896:
2886:Category Theory
2883:
2865:
2859:
2846:
2820:
2814:
2804:Category theory
2801:
2795:
2778:
2768:
2757:
2750:
2746:
2741:
2740:
2715:
2711:
2703:
2699:
2694:
2661:
2654:
2651:
2554:
2536:
2473:is denoted aut(
2440:is denoted end(
2418:
2408:
2377:
2346:
2297:
2290:
2283:
2276:
2268:
2260:
2232:
2226:
2218:
2211:
2204:
2197:
2161:
2114:
2104:and is denoted
2088:
2083:
1948:
1621:
1602:
1597:
1596:
1571:
1566:
1565:
1544:
1539:
1538:
1506:
1487:
1482:
1481:
1456:
1451:
1450:
1230:
1176:) are actually
1158:
1100:
1054:
1024:
1002:
985:
902:associative law
710:
160:continuous maps
90:Category theory
42:
35:
28:
23:
22:
15:
12:
11:
5:
3705:
3703:
3695:
3694:
3689:
3679:
3678:
3672:
3671:
3669:
3668:
3658:
3648:
3637:
3634:
3633:
3630:
3629:
3626:
3625:
3623:
3622:
3617:
3612:
3598:
3592:
3587:
3582:
3576:
3574:
3567:
3566:
3563:
3562:
3560:
3559:
3554:
3543:
3541:
3536:
3531:
3530:
3528:
3527:
3522:
3517:
3512:
3507:
3502:
3491:
3489:
3484:
3476:
3472:
3471:
3469:
3464:
3462:String diagram
3459:
3454:
3452:Model category
3449:
3444:
3439:
3434:
3429:
3427:
3420:
3419:
3416:
3413:
3412:
3407:
3400:
3399:
3392:
3390:
3387:
3386:
3383:
3382:
3380:
3379:
3374:
3372:Comma category
3369:
3364:
3359:
3354:
3349:
3344:
3338:
3336:
3332:
3331:
3329:
3328:
3318:
3308:
3306:Abelian groups
3303:
3298:
3293:
3288:
3282:
3280:
3274:
3273:
3270:
3269:
3267:
3266:
3261:
3256:
3255:
3254:
3244:
3239:
3233:
3231:
3225:
3224:
3222:
3221:
3216:
3211:
3210:
3209:
3199:
3194:
3188:
3186:
3177:
3171:
3170:
3168:
3167:
3162:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3122:
3117:
3112:
3111:
3110:
3105:
3100:
3095:
3090:
3085:
3074:
3072:
3064:
3063:
3060:
3053:
3052:
3047:
3045:
3044:
3037:
3030:
3022:
3015:
3014:
3002:
2996:
2983:
2968:
2962:
2944:
2938:
2922:
2916:
2900:
2894:
2881:
2863:
2857:
2844:
2826:Wells, Charles
2818:
2812:
2799:
2793:
2776:
2766:
2747:
2745:
2742:
2739:
2738:
2709:
2696:
2695:
2693:
2690:
2689:
2688:
2683:
2678:
2673:
2667:
2666:
2650:
2647:
2646:
2645:
2637:
2612:
2601:
2566:abelian groups
2550:
2535:
2532:
2516:
2515:
2509:
2503:
2493:
2492:
2457:
2420:
2414:
2404:
2379:
2373:
2348:
2342:
2317:
2308:
2295:
2288:
2281:
2274:
2266:
2258:
2243:
2230:
2224:
2216:
2209:
2202:
2195:
2160:
2157:
2113:
2110:
2087:
2084:
2082:
2079:
2053:
2052:
2042:
2037:
2029:
2028:
2023:
2018:
2010:
2009:
2004:
2002:measure spaces
1999:
1991:
1990:
1985:
1980:
1972:
1971:
1962:
1950:
1944:
1937:
1936:
1928:
1915:
1904:
1903:
1898:
1893:
1885:
1884:
1879:
1874:
1866:
1865:
1862:
1857:
1849:
1848:
1843:
1838:
1830:
1829:
1824:
1819:
1811:
1810:
1807:
1804:
1796:
1795:
1790:
1785:
1777:
1776:
1773:
1770:
1738:abelian groups
1665:directed graph
1633:
1628:
1624:
1620:
1617:
1614:
1609:
1605:
1578:
1574:
1551:
1547:
1518:
1513:
1509:
1505:
1502:
1499:
1494:
1490:
1463:
1459:
1402:Similarly any
1368:with a single
1351:ordinal number
1299:preordered set
1229:
1226:
1182:proper classes
1157:
1154:
1096:
1075:, and we say "
1061:
1060:
1050:
1020:
998:
981:
966:
893:
892:
777:
766:
755:
738:
709:
706:
692:
691:
688:
685:
682:
679:
676:
670:
669:
666:
663:
660:
657:
654:
648:
647:
644:
641:
638:
635:
632:
626:
625:
622:
619:
616:
613:
610:
604:
603:
600:
597:
594:
591:
588:
581:
580:
577:
574:
571:
568:
565:
558:
557:
554:
551:
548:
545:
542:
535:
534:
531:
528:
525:
522:
519:
513:
512:
509:
506:
503:
500:
497:
490:
489:
486:
483:
480:
477:
474:
468:
467:
464:
461:
458:
455:
452:
445:
444:
441:
438:
435:
432:
429:
423:
422:
419:
416:
413:
410:
407:
400:
399:
396:
393:
390:
387:
384:
378:
377:
374:
371:
368:
365:
362:
355:
354:
351:
348:
345:
342:
339:
333:
332:
329:
326:
323:
320:
317:
310:
309:
306:
303:
300:
297:
294:
288:
287:
284:
281:
278:
275:
272:
270:Small category
266:
265:
262:
259:
256:
253:
250:
244:
243:
240:
237:
234:
231:
228:
222:
221:
216:
211:
206:
201:
196:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3704:
3693:
3690:
3688:
3685:
3684:
3682:
3667:
3659:
3657:
3649:
3647:
3639:
3638:
3635:
3621:
3618:
3616:
3613:
3611:
3607:
3603:
3599:
3597:
3595:
3588:
3586:
3583:
3581:
3578:
3577:
3575:
3572:
3568:
3558:
3555:
3552:
3548:
3545:
3544:
3542:
3540:
3532:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3510:Tetracategory
3508:
3506:
3503:
3500:
3499:pseudofunctor
3496:
3493:
3492:
3490:
3488:
3480:
3477:
3473:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3424:
3418:
3417:
3414:
3410:
3405:
3401:
3396:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3342:Free category
3340:
3339:
3337:
3333:
3326:
3325:Vector spaces
3322:
3319:
3316:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3283:
3281:
3279:
3275:
3265:
3262:
3260:
3257:
3253:
3250:
3249:
3248:
3245:
3243:
3240:
3238:
3235:
3234:
3232:
3230:
3226:
3220:
3219:Inverse limit
3217:
3215:
3212:
3208:
3205:
3204:
3203:
3200:
3198:
3195:
3193:
3190:
3189:
3187:
3185:
3181:
3178:
3176:
3172:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3145:Kan extension
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3080:
3079:
3076:
3075:
3073:
3069:
3065:
3058:
3054:
3050:
3043:
3038:
3036:
3031:
3029:
3024:
3023:
3020:
3013:
3011:
3006:
3003:
2999:
2993:
2989:
2984:
2980:
2979:
2974:
2969:
2965:
2963:0-387-98403-8
2959:
2955:
2954:
2949:
2945:
2941:
2939:0-521-47249-0
2935:
2931:
2927:
2923:
2919:
2913:
2909:
2908:Basic algebra
2905:
2901:
2897:
2891:
2887:
2882:
2878:
2874:
2873:
2868:
2864:
2860:
2858:0-521-06119-9
2854:
2850:
2845:
2841:
2837:
2833:
2832:
2827:
2823:
2822:Barr, Michael
2819:
2815:
2809:
2805:
2800:
2796:
2794:0-262-01125-5
2790:
2787:, MIT Press,
2785:
2784:
2777:
2774:
2769:
2767:0-471-60922-6
2763:
2756:
2755:
2749:
2748:
2743:
2735:
2731:
2727:
2723:
2719:
2713:
2710:
2706:
2701:
2698:
2691:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2668:
2664:
2658:
2653:
2648:
2642:
2638:
2635:
2631:
2627:
2623:
2622:
2617:
2613:
2610:
2607:if all small
2606:
2602:
2599:
2595:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2559:
2555:
2553:
2549:
2544:
2543:
2538:
2537:
2533:
2531:
2529:
2525:
2521:
2513:
2510:
2507:
2504:
2501:
2498:
2497:
2496:
2490:
2486:
2485:
2480:
2476:
2472:
2468:
2464:
2463:
2458:
2455:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2426:
2421:
2417:
2412:
2407:
2402:
2398:
2394:
2390:
2386:
2385:
2380:
2376:
2371:
2367:
2363:
2359:
2355:
2354:
2349:
2345:
2340:
2336:
2332:
2328:
2324:
2323:
2318:
2315:
2314:
2309:
2306:
2302:
2298:
2291:
2284:
2277:
2270:
2262:
2254:
2250:
2249:
2244:
2241:
2237:
2233:
2223:
2219:
2212:
2205:
2198:
2191:
2187:
2186:
2181:
2180:
2179:
2177:
2173:
2169:
2166:
2158:
2156:
2154:
2153:componentwise
2150:
2146:
2142:
2139:and one from
2138:
2134:
2130:
2127:
2123:
2119:
2111:
2109:
2107:
2103:
2102:
2098:
2093:
2090:Any category
2086:Dual category
2085:
2080:
2078:
2076:
2072:
2071:
2067:The category
2065:
2063:
2059:
2058:Fiber bundles
2050:
2046:
2043:
2041:
2038:
2036:
2035:
2031:
2030:
2027:
2024:
2022:
2019:
2017:
2016:
2012:
2011:
2008:
2005:
2003:
2000:
1998:
1997:
1993:
1992:
1989:
1986:
1984:
1983:metric spaces
1981:
1979:
1978:
1974:
1973:
1970:
1966:
1963:
1961:
1958:
1954:
1953:vector spaces
1951:
1949:
1947:
1943:
1939:
1938:
1935:
1933:
1929:
1926:
1922:
1920:
1916:
1914:
1913:
1911:
1906:
1905:
1902:
1899:
1897:
1894:
1892:
1891:
1887:
1886:
1883:
1880:
1878:
1875:
1873:
1872:
1868:
1867:
1863:
1861:
1858:
1856:
1855:
1851:
1850:
1847:
1844:
1842:
1839:
1837:
1836:
1832:
1831:
1828:
1825:
1823:
1820:
1818:
1817:
1813:
1812:
1808:
1805:
1803:
1802:
1798:
1797:
1794:
1791:
1789:
1786:
1784:
1783:
1779:
1778:
1774:
1771:
1768:
1767:
1764:
1761:
1758:
1753:
1751:
1747:
1743:
1739:
1735:
1734:
1729:
1725:
1721:
1720:
1715:
1711:
1707:
1702:
1700:
1696:
1692:
1691:
1685:
1683:
1682:
1681:free category
1677:
1673:
1669:
1666:
1657:
1653:
1651:
1647:
1626:
1622:
1618:
1615:
1607:
1603:
1594:
1576:
1572:
1549:
1545:
1536:
1532:
1511:
1507:
1503:
1500:
1492:
1488:
1479:
1461:
1457:
1448:
1444:
1440:
1439:group actions
1436:
1431:
1429:
1425:
1421:
1418:that is both
1417:
1413:
1409:
1405:
1400:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1371:
1367:
1363:
1358:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1331:antisymmetric
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1295:
1293:
1289:
1285:
1282:
1278:
1273:
1271:
1267:
1263:
1259:
1255:
1254:
1249:
1248:
1243:
1239:
1235:
1227:
1225:
1223:
1219:
1216:similar to a
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1188:otherwise. A
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1155:
1153:
1149:
1147:
1143:
1139:
1135:
1130:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1099:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1058:
1053:
1048:
1044:
1040:
1036:
1032:
1028:
1023:
1018:
1014:
1010:
1006:
1003:) called the
1001:
997:
993:
989:
984:
979:
975:
973:
967:
964:
960:
956:
952:
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
903:
898:
897:
896:
890:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
846:
842:
839:and cod(f) =
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
775:
771:
767:
764:
760:
756:
753:
749:
748:
743:
739:
736:
735:
730:
726:
722:
721:
720:
718:
715:
707:
705:
703:
699:
689:
686:
683:
680:
677:
675:
674:Abelian group
672:
671:
667:
664:
661:
658:
655:
653:
650:
649:
645:
642:
639:
636:
633:
631:
628:
627:
623:
620:
617:
614:
611:
609:
606:
605:
601:
598:
595:
592:
589:
587:
583:
582:
578:
575:
572:
569:
566:
564:
560:
559:
555:
552:
549:
546:
543:
541:
537:
536:
532:
529:
526:
523:
520:
518:
515:
514:
510:
507:
504:
501:
498:
496:
492:
491:
487:
484:
481:
478:
475:
473:
470:
469:
465:
462:
459:
456:
453:
451:
447:
446:
442:
439:
436:
433:
430:
428:
425:
424:
420:
417:
414:
411:
408:
406:
402:
401:
397:
394:
391:
388:
385:
383:
380:
379:
375:
372:
369:
366:
363:
361:
357:
356:
352:
349:
346:
343:
340:
338:
335:
334:
330:
327:
324:
321:
318:
316:
312:
311:
307:
304:
301:
298:
295:
293:
290:
289:
285:
282:
279:
276:
273:
271:
268:
267:
263:
260:
257:
254:
251:
249:
246:
245:
241:
238:
235:
232:
229:
227:
226:Partial magma
224:
223:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
194:
188:
186:
182:
178:
177:
171:
169:
165:
161:
157:
153:
152:
147:
143:
139:
138:
133:
132:set functions
129:
125:
124:
118:
116:
112:
107:
105:
100:
97:
92:
91:
86:
84:
80:
76:
72:
71:associatively
68:
64:
60:
56:
48:
39:
33:
19:
3590:
3571:Categorified
3475:n-categories
3426:Key concepts
3264:Direct limit
3247:Coequalizers
3165:Yoneda lemma
3077:
3071:Key concepts
3061:Key concepts
3009:
2987:
2976:
2951:
2929:
2907:
2885:
2870:
2848:
2830:
2803:
2782:
2753:
2733:
2729:
2725:
2724:) or simply
2721:
2717:
2712:
2700:
2625:
2619:
2561:
2557:
2551:
2547:
2540:
2523:
2519:
2517:
2511:
2505:
2499:
2494:
2488:
2482:
2474:
2470:
2466:
2462:automorphism
2460:
2452:and forms a
2449:
2445:
2441:
2437:
2433:
2429:
2425:endomorphism
2423:
2415:
2410:
2405:
2400:
2396:
2392:
2388:
2382:
2374:
2369:
2365:
2361:
2357:
2351:
2343:
2338:
2334:
2330:
2326:
2320:
2311:
2304:
2300:
2293:
2286:
2279:
2272:
2264:
2256:
2252:
2246:
2239:
2235:
2228:
2221:
2214:
2207:
2200:
2193:
2189:
2185:monomorphism
2183:
2175:
2171:
2167:
2162:
2148:
2144:
2140:
2136:
2132:
2128:
2125:
2121:
2117:
2115:
2105:
2100:
2096:
2091:
2089:
2068:
2066:
2056:
2044:
2033:
2013:
1994:
1975:
1964:
1959:
1945:
1941:
1931:
1924:
1918:
1909:
1907:
1888:
1869:
1852:
1833:
1814:
1799:
1780:
1762:
1754:
1745:
1731:
1727:
1723:
1717:
1703:
1698:
1688:
1686:
1679:
1662:
1649:
1592:
1534:
1477:
1446:
1432:
1423:
1415:
1411:
1407:
1401:
1396:
1392:
1388:
1384:
1380:
1359:
1339:transitivity
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1296:
1291:
1287:
1283:
1274:
1251:
1245:
1231:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1173:
1169:
1165:
1161:
1159:
1150:
1145:
1141:
1137:
1133:
1131:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1097:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1062:
1056:
1051:
1046:
1042:
1038:
1034:
1030:
1026:
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740:a class mor(
732:
728:
719:consists of
716:
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711:
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561:Associative
538:Commutative
493:Commutative
450:unital magma
448:Commutative
427:Unital magma
403:Commutative
358:Commutative
313:Commutative
269:
248:Semigroupoid
214:Cancellation
174:
172:
164:identity map
149:
135:
121:
119:
110:
108:
101:
88:
87:
62:
58:
52:
46:
3539:-categories
3515:Kan complex
3505:Tricategory
3487:-categories
3377:Subcategory
3135:Exponential
3103:Preadditive
3098:Pre-abelian
2707:, Chapter 1
2574:preadditive
2384:isomorphism
2248:epimorphism
2147:and one in
2062:bundle maps
1969:linear maps
1428:isomorphism
1370:associative
1355:ordered set
1335:reflexivity
1168:if both ob(
1160:A category
1140:instead of
1019:satisfies 1
819:. Here hom(
219:Commutative
204:Associative
168:composition
55:mathematics
3681:Categories
3557:3-category
3547:2-category
3520:β-groupoid
3495:Bicategory
3242:Coproducts
3202:Equalizers
3108:Bicategory
2867:"Category"
2744:References
2736:) instead.
2681:Quantaloid
2582:coproducts
2322:retraction
2313:bimorphism
2178:is called
1988:short maps
1927:is a ring
1775:Morphisms
1693:. It is a
1408:invertible
1270:allegories
1172:) and hom(
1164:is called
1045:satisfies
708:Definition
586:quasigroup
563:quasigroup
405:quasigroup
382:Quasigroup
185:References
115:equivalent
96:set theory
3606:Symmetric
3551:2-functor
3291:Relations
3214:Pullbacks
2877:EMS Press
2760:, Wiley,
1955:over the
1793:functions
1769:Category
1710:morphisms
1668:generates
1604:π
1489:π
1383:. (Here,
1266:relations
1238:functions
1119:hom-class
1095:) (or hom
1063:We write
815:) called
747:morphisms
690:Required
668:Unneeded
646:Required
624:Unneeded
602:Required
579:Unneeded
556:Required
540:semigroup
533:Unneeded
517:Semigroup
511:Required
488:Unneeded
466:Required
443:Unneeded
421:Required
398:Unneeded
376:Required
353:Unneeded
331:Required
308:Unneeded
286:Unneeded
264:Unneeded
242:Unneeded
111:different
83:functions
3666:Glossary
3646:Category
3620:n-monoid
3573:concepts
3229:Colimits
3197:Products
3150:Morphism
3093:Concrete
3088:Additive
3078:Category
3005:category
2950:(1998),
2906:(2009),
2828:(2005),
2649:See also
2605:complete
2594:cokernel
2578:products
2570:bilinear
2391: :
2360: :
2329: :
2299: :
2271:implies
2234: :
2206:implies
2170: :
2165:morphism
2075:functors
1923:, where
1921:-modules
1772:Objects
1672:vertices
1435:groupoid
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1337:and the
1277:discrete
1228:Examples
1180:and not
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770:codomain
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395:Required
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386:Required
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370:Unneeded
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341:Required
328:Required
325:Required
322:Required
319:Unneeded
315:Groupoid
305:Required
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299:Required
296:Unneeded
292:Groupoid
283:Unneeded
280:Required
277:Required
274:Unneeded
261:Unneeded
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252:Unneeded
239:Unneeded
236:Unneeded
233:Unneeded
230:Unneeded
209:Identity
59:category
3656:Outline
3615:n-group
3580:2-group
3535:Strict
3525:β-topos
3321:Modules
3259:Pushout
3207:Kernels
3140:Functor
3083:Abelian
3007:at the
2879:, 2001
2840:2178101
2773:GNU FDL
2644:theory.
2448:) is a
2353:section
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1822:monoids
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1529:is the
1480:, then
1375:and an
1222:closure
734:objects
199:Closure
3602:Traced
3585:2-ring
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3296:Magmas
3184:Limits
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2592:and a
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1860:graphs
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1362:monoid
1286:, the
1218:monoid
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1184:, and
774:target
763:source
759:domain
752:arrows
698:monoid
608:Monoid
148:; and
3596:-ring
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3467:Topos
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2692:Notes
2641:topos
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2190:monic
2060:with
2051:maps
1957:field
1890:Field
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1676:loops
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