5411:
4548:
595:
5658:
1513:
5678:
5668:
2867:
3005:
4286:
3890:
3989:
2087:
1940:
2316:
2745:
4064:
4733:
2872:
3742:
4124:
3811:
2109:
required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set
4500:
3206:
749:
4652:
1716:
4332:
3646:
2701:
2593:
2269:
3648:
How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a
1661:
1782:
904:
855:
806:
1196:
3129:
2233:
1280:
4372:
2017:
3399:
2560:
584:
1568:
1353:
362:
4542:
4431:
407:
3540:
1615:
313:
3074:
2466:
1446:
532:
484:
4590:
4208:
1865:
3421:
2202:
1824:
2160:
1478:
961:
934:
2510:
1507:
3451:
2426:
1070:
1041:
681:
2740:
2371:
1147:
1120:
651:
247:
220:
170:
143:
991:
4200:
3258:
1230:
270:
193:
116:
4177:
4157:
3603:
3583:
3560:
3509:
3477:
3025:
2633:
2613:
2486:
2391:
2336:
2127:
1962:
1747:
1397:
1377:
1304:
1090:
1011:
624:
435:
3816:
3899:
2021:
4661:
1869:
2277:
4290:
5055:
4951:
4918:
4874:
3994:
4827:
5340:
5011:
4336:
4658:
is one in which this morphism is actually an isomorphism. Thus in a distributive category, there is the canonical isomorphism
3655:
4069:
3747:
4440:
3134:
4595:
1666:
3611:
2862:{\displaystyle \prod _{i\in I}X_{i}:=\left\{\left(x_{i}\right)_{i\in I}:x_{i}\in X_{i}{\text{ for all }}i\in I\right\}}
2638:
2565:
2241:
5702:
4995:
4967:
4907:
4854:
3000:{\displaystyle \pi _{j}:\prod _{i\in I}X_{i}\to X_{j},\quad \pi _{j}\left(\left(x_{i}\right)_{i\in I}\right):=x_{j}.}
1620:
2179:
1752:
4434:
3275:
2129:
considered as a discrete category. The definition of the product then coincides with the definition of the limit,
1152:
594:
3079:
2207:
1237:
4986:. Encyclopedia of mathematics and its applications 50–51, 53 . Vol. 1. Cambridge University Press. p.
3227:
3215:
1967:
860:
811:
754:
3369:
2515:
537:
5048:
4956:
4544:
guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):
697:
5252:
5207:
1527:
1309:
318:
5681:
5621:
4509:
4399:
4379:
3260:
the product is the
Cartesian product with addition defined componentwise and distributive multiplication.
366:
5330:
3514:
3345:
can be treated as a category, using the order relation as the morphisms. In this case the products and
1573:
275:
5671:
5457:
5321:
5229:
3030:
2431:
1512:
1402:
488:
440:
1725:
Alternatively, the product may be defined through equations. So, for example, for the binary product:
5630:
5274:
5135:
5015:
4781:
2106:
5661:
5617:
5222:
5041:
4557:
3279:
3268:
1829:
35:
3404:
2185:
5217:
5199:
4769:
4763:
2715:, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets
2098:
1789:
5424:
5190:
5170:
5093:
4987:
4980:
4747:
3218:, the product is the space whose underlying set is the Cartesian product and which carries the
2175:
2163:
2132:
39:
4790: – Most general completion of a commutative square given two morphisms with same codomain
1451:
939:
912:
5306:
5145:
4655:
4391:
3342:
3286:
2495:
1483:
687:
3426:
2396:
1045:
1016:
656:
5118:
5113:
4787:
2718:
2344:
1125:
1098:
965:
629:
225:
198:
148:
121:
976:
8:
5462:
5410:
5336:
5140:
4203:
3234:
587:
59:
4927:
4880:
4182:
3240:
2105:(a family of objects without any morphisms, other than their identity morphisms) as the
1212:
252:
175:
98:
5316:
5311:
5293:
5175:
5150:
4547:
4162:
4142:
3588:
3568:
3545:
3494:
3462:
3366:
An example in which the product does not exist: In the category of fields, the product
3354:
3350:
3264:
3010:
2618:
2598:
2489:
2471:
2376:
2321:
2112:
1947:
1732:
1382:
1362:
1289:
1075:
996:
609:
420:
410:
63:
4281:{\displaystyle X\times (Y\times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z,}
5625:
5562:
5550:
5452:
5377:
5372:
5326:
5108:
5103:
4991:
4963:
4946:
4903:
4850:
4132:(although some authors use this phrase to mean "a category with all finite limits").
3301:
3223:
2102:
71:
51:
47:
5586:
5472:
5447:
5382:
5367:
5362:
5301:
5130:
5098:
4753:
3885:{\displaystyle \left\langle f_{1}\circ \pi _{1},f_{2}\circ \pi _{2}\right\rangle .}
3219:
2712:
2272:
2236:
67:
42:
is a notion designed to capture the essence behind constructions in other areas of
20:
5498:
5064:
4128:
A category where every finite set of objects has a product is sometimes called a
3484:
3305:
27:
5014:
which generates examples of products in the category of finite sets. Written by
5535:
5530:
5514:
5477:
5467:
5387:
4894:
3488:
3316:
3290:
75:
55:
3984:{\displaystyle \left\{X\right\}_{i},\left\{Y\right\}_{i},f_{i}:X_{i}\to Y_{i}}
5696:
5525:
5357:
5234:
5160:
4775:
3456:
3323:
3293:. (This may come as a bit of a surprise given that the category of sets is a
2082:{\displaystyle \left\langle \pi _{1}\circ g,\pi _{2}\circ g\right\rangle =g.}
5279:
5180:
4757:
3312:
2339:
1786:
Commutativity of the diagrams above is guaranteed by the equality: for all
1935:{\displaystyle \pi _{i}\circ \left\langle f_{1},f_{2}\right\rangle =f_{i}}
5540:
5520:
5392:
5262:
4159:
is a
Cartesian category, product functors have been chosen as above, and
4136:
3294:
3271:
given by the
Cartesian product with multiplication defined componentwise.
1205:
Instead of two objects, we can start with an arbitrary family of objects
43:
5021:
4772: – Set of arguments where two or more functions have the same value
5572:
5510:
5123:
3331:
2311:{\displaystyle \Delta :\mathbf {C} \to \mathbf {C} \times \mathbf {C} }
4892:
5566:
5257:
4743:
4503:
3649:
3480:
3346:
3327:
1206:
5635:
5267:
5165:
4926:. Les Publications CRM Montreal (publication PM023). Archived from
4378:; a Cartesian category with its finite products is an example of a
3401:
does not exist, since there is no field with homomorphisms to both
79:
5033:
4506:. To see this, note that the universal property of the coproduct
5605:
5595:
5244:
5155:
3606:
4374:
These properties are formally similar to those of a commutative
2178:, so is the product. Starting with the definition given for the
5600:
4375:
4433:
of a category with finite products and coproducts, there is a
5482:
4059:{\displaystyle \prod _{i\in I}X_{i}\to \prod _{i\in I}Y_{i}.}
3896:. Second, consider the general product functor. For families
684:
4828:"Banach spaces (and Lawvere metrics, and closed categories)"
4728:{\displaystyle X\times (Y+Z)\simeq (X\times Y)+(X\times Z).}
5025:
4872:
3585:
is a set such that all products for families indexed with
3491:, do not have a terminal object: given any infinite group
4893:
Adámek, Jiří; Horst
Herrlich; George E. Strecker (1990).
3737:{\displaystyle f_{1}:X_{1}\to Y_{1},f_{2}:X_{2}\to Y_{2}}
690:, because of the universal property, so one may speak of
4849:(1st ed.). New York: Springer-Verlag. p. 37.
4792:
Pages displaying short descriptions of redirect targets
4119:{\displaystyle \left\{f_{i}\circ \pi _{i}\right\}_{i}.}
3806:{\displaystyle X_{1}\times X_{2}\to Y_{1}\times Y_{2}.}
78:
of objects is the "most general" object which admits a
4810:
4654:
induced by the dashed arrows in the above diagram. A
751:
is another product, there exists a unique isomorphism
4664:
4598:
4560:
4512:
4495:{\displaystyle X\times Y+X\times Z\to X\times (Y+Z),}
4443:
4402:
4339:
4293:
4211:
4185:
4165:
4145:
4072:
3997:
3902:
3819:
3750:
3658:
3614:
3591:
3571:
3548:
3517:
3497:
3465:
3429:
3407:
3372:
3243:
3201:{\displaystyle f(y):=\left(f_{i}(y)\right)_{i\in I}.}
3137:
3082:
3033:
3013:
2875:
2748:
2721:
2641:
2621:
2601:
2568:
2518:
2498:
2474:
2434:
2399:
2379:
2347:
2324:
2280:
2244:
2210:
2188:
2135:
2115:
2024:
1970:
1950:
1872:
1832:
1792:
1755:
1735:
1669:
1623:
1576:
1530:
1486:
1454:
1405:
1385:
1365:
1312:
1292:
1240:
1215:
1155:
1128:
1101:
1078:
1048:
1019:
999:
979:
942:
915:
863:
814:
757:
700:
659:
632:
612:
540:
491:
443:
423:
369:
321:
278:
255:
228:
201:
178:
151:
124:
101:
4647:{\displaystyle X\times Y+X\times Z\to X\times (Y+Z)}
1711:{\displaystyle \langle f_{1},\ldots ,f_{n}\rangle .}
2204:as the discrete category with two objects, so that
4979:
4727:
4646:
4584:
4536:
4494:
4425:
4366:
4327:{\displaystyle X\times 1\simeq 1\times X\simeq X,}
4326:
4280:
4194:
4171:
4151:
4118:
4058:
3983:
3884:
3805:
3736:
3640:
3597:
3577:
3554:
3534:
3503:
3471:
3445:
3415:
3393:
3252:
3200:
3123:
3068:
3019:
2999:
2861:
2734:
2695:
2627:
2607:
2587:
2554:
2504:
2480:
2460:
2420:
2385:
2365:
2330:
2310:
2263:
2227:
2196:
2154:
2121:
2081:
2011:
1956:
1934:
1859:
1818:
1776:
1741:
1710:
1655:
1609:
1562:
1501:
1472:
1440:
1391:
1371:
1347:
1298:
1274:
1224:
1190:
1141:
1114:
1084:
1064:
1035:
1005:
985:
955:
928:
898:
849:
800:
743:
675:
645:
618:
578:
526:
478:
429:
401:
356:
307:
264:
241:
214:
187:
164:
137:
110:
4916:
3641:{\displaystyle \mathbf {C} ^{I}\to \mathbf {C} .}
2696:{\displaystyle (X,X)\to \left(X_{1},X_{2}\right)}
2166:and projections being the limit (limiting cone).
1480:such that the following diagrams commute for all
1200:
5694:
2588:{\displaystyle \mathbf {C} \times \mathbf {C} .}
2264:{\displaystyle \mathbf {C} \times \mathbf {C} .}
3487:, and some categories, such as the category of
1656:{\displaystyle X_{1}\times \cdots \times X_{n}}
4813:Introduction to Higher-Order Categorical Logic
2595:This universal morphism consists of an object
1777:{\displaystyle \langle \cdot ,\cdot \rangle .}
1355:satisfying the following universal property:
5049:
4825:
3605:exist, then one can treat each product as a
2143:
2136:
1851:
1839:
1768:
1756:
1749:is guaranteed by existence of the operation
1702:
1670:
1601:
1583:
1191:{\displaystyle \langle f_{1},f_{2}\rangle .}
1182:
1156:
694:product. This has the following meaning: if
4784: – Type of category in category theory
3124:{\displaystyle f:Y\to \prod _{i\in I}X_{i}}
2228:{\displaystyle \mathbf {C} ^{\mathbf {J} }}
2174:Just as the limit is a special case of the
1275:{\displaystyle \left(X_{i}\right)_{i\in I}}
5677:
5667:
5423:
5056:
5042:
4876:Category Theory – Lecture Notes for ESSLLI
4815:. Cambridge University Press. p. 304.
4367:{\displaystyle X\times Y\simeq Y\times X.}
3519:
3409:
3374:
2012:{\displaystyle g:Y\to X_{1}\times X_{2},}
899:{\displaystyle \pi _{2}'=\pi _{2}\circ h}
850:{\displaystyle \pi _{1}'=\pi _{1}\circ h}
801:{\displaystyle h:X'\to X_{1}\times X_{2}}
90:
4952:Categories for the Working Mathematician
4945:
4868:
4866:
4847:Categories for the working mathematician
3394:{\displaystyle \mathbb {Q} \times F_{p}}
2555:{\displaystyle \left(X_{1},X_{2}\right)}
1720:
1663:and the product of morphisms is denoted
579:{\displaystyle f:Y\to X_{1}\times X_{2}}
4977:
4778: – Construction in category theory
1964:is guaranteed by the equality: for all
606:Whether a product exists may depend on
5695:
4819:
4554:The universal property of the product
3892:This operation on morphisms is called
744:{\displaystyle X',\pi _{1}',\pi _{2}'}
5422:
5075:
5037:
4920:Category Theory for Computing Science
4917:Barr, Michael; Charles Wells (1999).
4863:
4502:where the plus sign here denotes the
3349:correspond to greatest lower bounds (
2169:
1563:{\displaystyle \prod _{i\in I}X_{i}.}
16:Generalized object in category theory
4873:Michael Barr, Charles Wells (1999).
4844:
3511:there are infinitely many morphisms
1348:{\displaystyle \pi _{i}:X\to X_{i},}
993:alliterates with projection. Given
357:{\displaystyle \pi _{1}:X\to X_{1},}
5063:
4537:{\displaystyle X\times Y+X\times Z}
4426:{\displaystyle X,Y,{\text{ and }}Z}
4066:We choose the product of morphisms
2097:The product is a special case of a
402:{\displaystyle \pi _{2}:X\to X_{2}}
13:
4592:then guarantees a unique morphism
4546:
3535:{\displaystyle \mathbb {Z} \to G,}
3226:for which all the projections are
2499:
2281:
1610:{\displaystyle I=\{1,\ldots ,n\},}
1511:
593:
315:equipped with a pair of morphisms
308:{\displaystyle X_{1}\times X_{2},}
14:
5714:
5005:
4385:
3069:{\displaystyle f_{i}:Y\to X_{i},}
2461:{\displaystyle X_{1}\times X_{2}}
1441:{\displaystyle f_{i}:Y\to X_{i},}
598:Universal property of the product
527:{\displaystyle f_{2}:Y\to X_{2},}
479:{\displaystyle f_{1}:Y\to X_{1},}
5676:
5666:
5657:
5656:
5409:
5076:
4896:Abstract and Concrete Categories
3631:
3617:
2578:
2570:
2304:
2296:
2288:
2254:
2246:
2219:
2213:
2190:
1516:Universal product of the product
586:such that the following diagram
74:. Essentially, the product of a
4982:Handbook of categorical algebra
4811:Lambek J., Scott P. J. (1988).
2931:
2869:with the canonical projections
1448:there exists a unique morphism
683:If it does exist, it is unique
534:there exists a unique morphism
4838:
4826:Qiaochu Yuan (June 23, 2012).
4804:
4719:
4707:
4701:
4689:
4683:
4671:
4641:
4629:
4620:
4579:
4567:
4486:
4474:
4465:
4248:
4236:
4230:
4218:
4024:
3968:
3894:Cartesian product of morphisms
3774:
3721:
3682:
3627:
3523:
3315:, the product is given by the
3304:, the product is given by the
3297:of the category of relations.)
3289:, the product is given by the
3222:. The product topology is the
3216:category of topological spaces
3175:
3169:
3147:
3141:
3092:
3050:
2915:
2657:
2654:
2642:
2412:
2400:
2360:
2348:
2292:
2101:. This may be seen by using a
1980:
1464:
1422:
1329:
1201:Product of an arbitrary family
772:
550:
508:
460:
386:
338:
82:to each of the given objects.
1:
4957:Graduate Texts in Mathematics
4797:
4585:{\displaystyle X\times (Y+Z)}
4179:denotes a terminal object of
3361:
2092:
1860:{\displaystyle i\in \{1,2\},}
1399:-indexed family of morphisms
85:
4879:. p. 62. Archived from
4766: – Mathematical concept
3416:{\displaystyle \mathbb {Q} }
2703:which contains projections.
2197:{\displaystyle \mathbf {J} }
2180:universal property of limits
437:and every pair of morphisms
7:
5351:Constructions on categories
4737:
4380:symmetric monoidal category
3027:with a family of functions
2706:
1819:{\displaystyle f_{1},f_{2}}
1286:of the family is an object
10:
5719:
5458:Higher-dimensional algebra
4962:(2nd ed.). Springer.
4389:
3991:we should find a morphism
3744:we should find a morphism
3353:) and least upper bounds (
3330:, the product carries the
2742:the product is defined as
18:
5652:
5585:
5549:
5497:
5490:
5441:
5431:
5418:
5407:
5350:
5292:
5243:
5198:
5189:
5086:
5082:
5071:
4978:Borceux, Francis (1994).
4902:. John Wiley & Sons.
4782:Cartesian closed category
2155:{\displaystyle \{f\}_{i}}
409:satisfying the following
5703:Limits (category theory)
3280:tensor product of graphs
3269:direct product of groups
1473:{\displaystyle f:Y\to X}
1306:equipped with morphisms
956:{\displaystyle \pi _{2}}
929:{\displaystyle \pi _{1}}
19:Not to be confused with
5268:Cokernels and quotients
5191:Universal constructions
4760:of the product functor.
2505:{\displaystyle \Delta }
2318:assigns to each object
1524:The product is denoted
1502:{\displaystyle i\in I:}
5425:Higher category theory
5171:Natural transformation
4729:
4648:
4586:
4551:
4538:
4496:
4427:
4368:
4328:
4282:
4196:
4173:
4153:
4120:
4060:
3985:
3886:
3807:
3738:
3642:
3599:
3579:
3556:
3536:
3505:
3473:
3447:
3446:{\displaystyle F_{p}.}
3417:
3395:
3254:
3202:
3125:
3070:
3021:
3001:
2863:
2736:
2697:
2629:
2609:
2589:
2556:
2506:
2482:
2462:
2422:
2421:{\displaystyle (f,f).}
2387:
2367:
2332:
2312:
2265:
2229:
2198:
2176:universal construction
2156:
2123:
2083:
2013:
1958:
1936:
1861:
1820:
1778:
1743:
1712:
1657:
1611:
1564:
1517:
1503:
1474:
1442:
1393:
1373:
1349:
1300:
1276:
1226:
1192:
1143:
1116:
1086:
1066:
1065:{\displaystyle f_{2},}
1037:
1036:{\displaystyle f_{1},}
1007:
987:
957:
930:
900:
851:
802:
745:
677:
676:{\displaystyle X_{2}.}
647:
620:
599:
580:
528:
480:
431:
403:
358:
309:
266:
243:
216:
189:
166:
139:
112:
91:Product of two objects
5012:Interactive Web page
4845:Lane, S. Mac (1988).
4730:
4656:distributive category
4649:
4587:
4550:
4539:
4497:
4428:
4392:Distributive category
4369:
4329:
4283:
4197:
4174:
4154:
4121:
4061:
3986:
3887:
3808:
3739:
3643:
3600:
3580:
3557:
3537:
3506:
3474:
3448:
3418:
3396:
3343:partially ordered set
3287:category of relations
3278:, the product is the
3267:, the product is the
3255:
3203:
3126:
3071:
3022:
3002:
2864:
2737:
2735:{\displaystyle X_{i}}
2698:
2630:
2610:
2590:
2557:
2507:
2483:
2463:
2423:
2388:
2373:and to each morphism
2368:
2366:{\displaystyle (X,X)}
2333:
2313:
2266:
2230:
2199:
2157:
2124:
2084:
2014:
1959:
1937:
1862:
1821:
1779:
1744:
1721:Equational definition
1713:
1658:
1612:
1565:
1515:
1504:
1475:
1443:
1394:
1374:
1350:
1301:
1277:
1227:
1193:
1144:
1142:{\displaystyle f_{2}}
1117:
1115:{\displaystyle f_{1}}
1087:
1067:
1038:
1008:
988:
966:canonical projections
958:
931:
901:
852:
803:
746:
688:canonical isomorphism
678:
648:
646:{\displaystyle X_{1}}
621:
597:
581:
529:
481:
432:
404:
359:
310:
267:
244:
242:{\displaystyle X_{2}}
217:
215:{\displaystyle X_{1}}
190:
167:
165:{\displaystyle X_{2}}
140:
138:{\displaystyle X_{1}}
113:
5294:Algebraic categories
4976:Definition 2.1.1 in
4788:Categorical pullback
4662:
4596:
4558:
4510:
4441:
4400:
4337:
4291:
4209:
4204:natural isomorphisms
4183:
4163:
4143:
4070:
3995:
3900:
3817:
3748:
3656:
3612:
3589:
3569:
3562:cannot be terminal.
3546:
3515:
3495:
3463:
3455:Another example: An
3427:
3405:
3370:
3313:semi-abelian monoids
3241:
3135:
3080:
3076:the universal arrow
3031:
3011:
2873:
2746:
2719:
2639:
2619:
2599:
2566:
2516:
2496:
2472:
2432:
2397:
2377:
2345:
2322:
2278:
2242:
2208:
2186:
2133:
2113:
2022:
1968:
1948:
1870:
1830:
1790:
1753:
1733:
1667:
1621:
1574:
1528:
1484:
1452:
1403:
1383:
1363:
1310:
1290:
1238:
1213:
1153:
1126:
1099:
1094:product of morphisms
1076:
1072:the unique morphism
1046:
1017:
997:
986:{\displaystyle \pi }
977:
971:projection morphisms
940:
913:
861:
812:
755:
698:
657:
630:
610:
538:
489:
441:
421:
367:
319:
276:
253:
226:
199:
176:
149:
122:
99:
5463:Homotopy hypothesis
5141:Commutative diagram
3483:) is the same as a
3322:In the category of
3311:In the category of
3302:algebraic varieties
3300:In the category of
3235:category of modules
2842: for all
1617:then it is denoted
876:
827:
740:
724:
5176:Universal property
4947:Mac Lane, Saunders
4832:Annoying Precision
4764:Limit and colimits
4725:
4644:
4582:
4552:
4534:
4492:
4423:
4364:
4324:
4278:
4195:{\displaystyle C.}
4192:
4169:
4149:
4130:Cartesian category
4116:
4056:
4042:
4013:
3981:
3882:
3803:
3734:
3638:
3595:
3575:
3552:
3532:
3501:
3469:
3443:
3413:
3391:
3276:category of graphs
3265:category of groups
3253:{\displaystyle R,}
3250:
3198:
3121:
3110:
3066:
3017:
2997:
2904:
2859:
2764:
2732:
2693:
2625:
2605:
2585:
2552:
2502:
2490:universal morphism
2478:
2458:
2418:
2383:
2363:
2328:
2308:
2261:
2225:
2194:
2170:Universal property
2152:
2119:
2079:
2009:
1954:
1932:
1857:
1816:
1774:
1739:
1708:
1653:
1607:
1560:
1546:
1518:
1499:
1470:
1438:
1389:
1369:
1345:
1296:
1272:
1225:{\displaystyle I.}
1222:
1188:
1139:
1112:
1082:
1062:
1033:
1003:
983:
953:
926:
896:
864:
847:
815:
798:
741:
728:
712:
673:
643:
616:
600:
576:
524:
476:
427:
411:universal property
399:
354:
305:
272:typically denoted
265:{\displaystyle X,}
262:
239:
212:
188:{\displaystyle C.}
185:
162:
135:
111:{\displaystyle C.}
108:
72:topological spaces
5690:
5689:
5648:
5647:
5644:
5643:
5626:monoidal category
5581:
5580:
5453:Enriched category
5405:
5404:
5401:
5400:
5378:Quotient category
5373:Opposite category
5288:
5287:
4418:
4172:{\displaystyle 1}
4152:{\displaystyle C}
4027:
3998:
3598:{\displaystyle I}
3578:{\displaystyle I}
3555:{\displaystyle G}
3504:{\displaystyle G}
3472:{\displaystyle I}
3224:coarsest topology
3095:
3020:{\displaystyle Y}
2889:
2843:
2749:
2628:{\displaystyle C}
2608:{\displaystyle X}
2492:from the functor
2481:{\displaystyle C}
2386:{\displaystyle f}
2331:{\displaystyle X}
2122:{\displaystyle I}
2103:discrete category
1957:{\displaystyle f}
1742:{\displaystyle f}
1531:
1392:{\displaystyle I}
1372:{\displaystyle Y}
1359:For every object
1299:{\displaystyle X}
1085:{\displaystyle f}
1006:{\displaystyle Y}
619:{\displaystyle C}
430:{\displaystyle Y}
417:For every object
48:Cartesian product
34:of two (or more)
5710:
5680:
5679:
5670:
5669:
5660:
5659:
5495:
5494:
5473:Simplex category
5448:Categorification
5439:
5438:
5420:
5419:
5413:
5383:Product category
5368:Kleisli category
5363:Functor category
5208:Terminal objects
5196:
5195:
5131:Adjoint functors
5084:
5083:
5073:
5072:
5058:
5051:
5044:
5035:
5034:
5001:
4985:
4973:
4941:
4939:
4938:
4932:
4925:
4913:
4901:
4885:
4884:
4870:
4861:
4860:
4842:
4836:
4835:
4823:
4817:
4816:
4808:
4793:
4754:Diagonal functor
4734:
4732:
4731:
4726:
4653:
4651:
4650:
4645:
4591:
4589:
4588:
4583:
4543:
4541:
4540:
4535:
4501:
4499:
4498:
4493:
4432:
4430:
4429:
4424:
4419:
4416:
4396:For any objects
4373:
4371:
4370:
4365:
4333:
4331:
4330:
4325:
4287:
4285:
4284:
4279:
4201:
4199:
4198:
4193:
4178:
4176:
4175:
4170:
4158:
4156:
4155:
4150:
4125:
4123:
4122:
4117:
4112:
4111:
4106:
4102:
4101:
4100:
4088:
4087:
4065:
4063:
4062:
4057:
4052:
4051:
4041:
4023:
4022:
4012:
3990:
3988:
3987:
3982:
3980:
3979:
3967:
3966:
3954:
3953:
3941:
3940:
3935:
3920:
3919:
3914:
3891:
3889:
3888:
3883:
3878:
3874:
3873:
3872:
3860:
3859:
3847:
3846:
3834:
3833:
3812:
3810:
3809:
3804:
3799:
3798:
3786:
3785:
3773:
3772:
3760:
3759:
3743:
3741:
3740:
3735:
3733:
3732:
3720:
3719:
3707:
3706:
3694:
3693:
3681:
3680:
3668:
3667:
3647:
3645:
3644:
3639:
3634:
3626:
3625:
3620:
3604:
3602:
3601:
3596:
3584:
3582:
3581:
3576:
3561:
3559:
3558:
3553:
3541:
3539:
3538:
3533:
3522:
3510:
3508:
3507:
3502:
3478:
3476:
3475:
3470:
3452:
3450:
3449:
3444:
3439:
3438:
3422:
3420:
3419:
3414:
3412:
3400:
3398:
3397:
3392:
3390:
3389:
3377:
3336:
3259:
3257:
3256:
3251:
3220:product topology
3210:Other examples:
3207:
3205:
3204:
3199:
3194:
3193:
3182:
3178:
3168:
3167:
3130:
3128:
3127:
3122:
3120:
3119:
3109:
3075:
3073:
3072:
3067:
3062:
3061:
3043:
3042:
3026:
3024:
3023:
3018:
3006:
3004:
3003:
2998:
2993:
2992:
2980:
2976:
2975:
2964:
2960:
2959:
2941:
2940:
2927:
2926:
2914:
2913:
2903:
2885:
2884:
2868:
2866:
2865:
2860:
2858:
2854:
2844:
2841:
2839:
2838:
2826:
2825:
2813:
2812:
2801:
2797:
2796:
2774:
2773:
2763:
2741:
2739:
2738:
2733:
2731:
2730:
2713:category of sets
2702:
2700:
2699:
2694:
2692:
2688:
2687:
2686:
2674:
2673:
2634:
2632:
2631:
2626:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2581:
2573:
2561:
2559:
2558:
2553:
2551:
2547:
2546:
2545:
2533:
2532:
2511:
2509:
2508:
2503:
2487:
2485:
2484:
2479:
2467:
2465:
2464:
2459:
2457:
2456:
2444:
2443:
2427:
2425:
2424:
2419:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2337:
2335:
2334:
2329:
2317:
2315:
2314:
2309:
2307:
2299:
2291:
2273:diagonal functor
2270:
2268:
2267:
2262:
2257:
2249:
2237:product category
2234:
2232:
2231:
2226:
2224:
2223:
2222:
2216:
2203:
2201:
2200:
2195:
2193:
2161:
2159:
2158:
2153:
2151:
2150:
2128:
2126:
2125:
2120:
2088:
2086:
2085:
2080:
2069:
2065:
2058:
2057:
2039:
2038:
2018:
2016:
2015:
2010:
2005:
2004:
1992:
1991:
1963:
1961:
1960:
1955:
1941:
1939:
1938:
1933:
1931:
1930:
1918:
1914:
1913:
1912:
1900:
1899:
1882:
1881:
1866:
1864:
1863:
1858:
1825:
1823:
1822:
1817:
1815:
1814:
1802:
1801:
1783:
1781:
1780:
1775:
1748:
1746:
1745:
1740:
1717:
1715:
1714:
1709:
1701:
1700:
1682:
1681:
1662:
1660:
1659:
1654:
1652:
1651:
1633:
1632:
1616:
1614:
1613:
1608:
1569:
1567:
1566:
1561:
1556:
1555:
1545:
1508:
1506:
1505:
1500:
1479:
1477:
1476:
1471:
1447:
1445:
1444:
1439:
1434:
1433:
1415:
1414:
1398:
1396:
1395:
1390:
1378:
1376:
1375:
1370:
1354:
1352:
1351:
1346:
1341:
1340:
1322:
1321:
1305:
1303:
1302:
1297:
1281:
1279:
1278:
1273:
1271:
1270:
1259:
1255:
1254:
1231:
1229:
1228:
1223:
1197:
1195:
1194:
1189:
1181:
1180:
1168:
1167:
1148:
1146:
1145:
1140:
1138:
1137:
1121:
1119:
1118:
1113:
1111:
1110:
1091:
1089:
1088:
1083:
1071:
1069:
1068:
1063:
1058:
1057:
1042:
1040:
1039:
1034:
1029:
1028:
1012:
1010:
1009:
1004:
992:
990:
989:
984:
962:
960:
959:
954:
952:
951:
935:
933:
932:
927:
925:
924:
905:
903:
902:
897:
889:
888:
872:
856:
854:
853:
848:
840:
839:
823:
807:
805:
804:
799:
797:
796:
784:
783:
771:
750:
748:
747:
742:
736:
720:
708:
682:
680:
679:
674:
669:
668:
652:
650:
649:
644:
642:
641:
625:
623:
622:
617:
585:
583:
582:
577:
575:
574:
562:
561:
533:
531:
530:
525:
520:
519:
501:
500:
485:
483:
482:
477:
472:
471:
453:
452:
436:
434:
433:
428:
408:
406:
405:
400:
398:
397:
379:
378:
363:
361:
360:
355:
350:
349:
331:
330:
314:
312:
311:
306:
301:
300:
288:
287:
271:
269:
268:
263:
248:
246:
245:
240:
238:
237:
221:
219:
218:
213:
211:
210:
194:
192:
191:
186:
171:
169:
168:
163:
161:
160:
144:
142:
141:
136:
134:
133:
117:
115:
114:
109:
21:Product category
5718:
5717:
5713:
5712:
5711:
5709:
5708:
5707:
5693:
5692:
5691:
5686:
5640:
5610:
5577:
5554:
5545:
5502:
5486:
5437:
5427:
5414:
5397:
5346:
5284:
5253:Initial objects
5239:
5185:
5078:
5067:
5065:Category theory
5062:
5008:
4998:
4970:
4936:
4934:
4930:
4923:
4910:
4899:
4889:
4888:
4871:
4864:
4857:
4843:
4839:
4824:
4820:
4809:
4805:
4800:
4791:
4740:
4663:
4660:
4659:
4597:
4594:
4593:
4559:
4556:
4555:
4511:
4508:
4507:
4442:
4439:
4438:
4417: and
4415:
4401:
4398:
4397:
4394:
4388:
4338:
4335:
4334:
4292:
4289:
4288:
4210:
4207:
4206:
4184:
4181:
4180:
4164:
4161:
4160:
4144:
4141:
4140:
4135:The product is
4107:
4096:
4092:
4083:
4079:
4078:
4074:
4073:
4071:
4068:
4067:
4047:
4043:
4031:
4018:
4014:
4002:
3996:
3993:
3992:
3975:
3971:
3962:
3958:
3949:
3945:
3936:
3925:
3924:
3915:
3904:
3903:
3901:
3898:
3897:
3868:
3864:
3855:
3851:
3842:
3838:
3829:
3825:
3824:
3820:
3818:
3815:
3814:
3794:
3790:
3781:
3777:
3768:
3764:
3755:
3751:
3749:
3746:
3745:
3728:
3724:
3715:
3711:
3702:
3698:
3689:
3685:
3676:
3672:
3663:
3659:
3657:
3654:
3653:
3630:
3621:
3616:
3615:
3613:
3610:
3609:
3590:
3587:
3586:
3570:
3567:
3566:
3547:
3544:
3543:
3518:
3516:
3513:
3512:
3496:
3493:
3492:
3489:infinite groups
3485:terminal object
3464:
3461:
3460:
3434:
3430:
3428:
3425:
3424:
3408:
3406:
3403:
3402:
3385:
3381:
3373:
3371:
3368:
3367:
3364:
3332:
3306:Segre embedding
3242:
3239:
3238:
3237:over some ring
3183:
3163:
3159:
3158:
3154:
3153:
3136:
3133:
3132:
3115:
3111:
3099:
3081:
3078:
3077:
3057:
3053:
3038:
3034:
3032:
3029:
3028:
3012:
3009:
3008:
2988:
2984:
2965:
2955:
2951:
2947:
2946:
2942:
2936:
2932:
2922:
2918:
2909:
2905:
2893:
2880:
2876:
2874:
2871:
2870:
2840:
2834:
2830:
2821:
2817:
2802:
2792:
2788:
2784:
2783:
2782:
2778:
2769:
2765:
2753:
2747:
2744:
2743:
2726:
2722:
2720:
2717:
2716:
2709:
2682:
2678:
2669:
2665:
2664:
2660:
2640:
2637:
2636:
2635:and a morphism
2620:
2617:
2616:
2600:
2597:
2596:
2577:
2569:
2567:
2564:
2563:
2541:
2537:
2528:
2524:
2523:
2519:
2517:
2514:
2513:
2497:
2494:
2493:
2473:
2470:
2469:
2452:
2448:
2439:
2435:
2433:
2430:
2429:
2398:
2395:
2394:
2378:
2375:
2374:
2346:
2343:
2342:
2323:
2320:
2319:
2303:
2295:
2287:
2279:
2276:
2275:
2253:
2245:
2243:
2240:
2239:
2218:
2217:
2212:
2211:
2209:
2206:
2205:
2189:
2187:
2184:
2183:
2172:
2146:
2142:
2134:
2131:
2130:
2114:
2111:
2110:
2095:
2053:
2049:
2034:
2030:
2029:
2025:
2023:
2020:
2019:
2000:
1996:
1987:
1983:
1969:
1966:
1965:
1949:
1946:
1945:
1926:
1922:
1908:
1904:
1895:
1891:
1890:
1886:
1877:
1873:
1871:
1868:
1867:
1831:
1828:
1827:
1810:
1806:
1797:
1793:
1791:
1788:
1787:
1754:
1751:
1750:
1734:
1731:
1730:
1723:
1696:
1692:
1677:
1673:
1668:
1665:
1664:
1647:
1643:
1628:
1624:
1622:
1619:
1618:
1575:
1572:
1571:
1551:
1547:
1535:
1529:
1526:
1525:
1485:
1482:
1481:
1453:
1450:
1449:
1429:
1425:
1410:
1406:
1404:
1401:
1400:
1384:
1381:
1380:
1364:
1361:
1360:
1336:
1332:
1317:
1313:
1311:
1308:
1307:
1291:
1288:
1287:
1260:
1250:
1246:
1242:
1241:
1239:
1236:
1235:
1234:Given a family
1214:
1211:
1210:
1203:
1176:
1172:
1163:
1159:
1154:
1151:
1150:
1149:and is denoted
1133:
1129:
1127:
1124:
1123:
1106:
1102:
1100:
1097:
1096:
1077:
1074:
1073:
1053:
1049:
1047:
1044:
1043:
1024:
1020:
1018:
1015:
1014:
998:
995:
994:
978:
975:
974:
963:are called the
947:
943:
941:
938:
937:
920:
916:
914:
911:
910:
884:
880:
868:
862:
859:
858:
835:
831:
819:
813:
810:
809:
792:
788:
779:
775:
764:
756:
753:
752:
732:
716:
701:
699:
696:
695:
664:
660:
658:
655:
654:
637:
633:
631:
628:
627:
611:
608:
607:
570:
566:
557:
553:
539:
536:
535:
515:
511:
496:
492:
490:
487:
486:
467:
463:
448:
444:
442:
439:
438:
422:
419:
418:
393:
389:
374:
370:
368:
365:
364:
345:
341:
326:
322:
320:
317:
316:
296:
292:
283:
279:
277:
274:
273:
254:
251:
250:
233:
229:
227:
224:
223:
206:
202:
200:
197:
196:
177:
174:
173:
156:
152:
150:
147:
146:
129:
125:
123:
120:
119:
100:
97:
96:
95:Fix a category
93:
88:
28:category theory
24:
17:
12:
11:
5:
5716:
5706:
5705:
5688:
5687:
5685:
5684:
5674:
5664:
5653:
5650:
5649:
5646:
5645:
5642:
5641:
5639:
5638:
5633:
5628:
5614:
5608:
5603:
5598:
5592:
5590:
5583:
5582:
5579:
5578:
5576:
5575:
5570:
5559:
5557:
5552:
5547:
5546:
5544:
5543:
5538:
5533:
5528:
5523:
5518:
5507:
5505:
5500:
5492:
5488:
5487:
5485:
5480:
5478:String diagram
5475:
5470:
5468:Model category
5465:
5460:
5455:
5450:
5445:
5443:
5436:
5435:
5432:
5429:
5428:
5416:
5415:
5408:
5406:
5403:
5402:
5399:
5398:
5396:
5395:
5390:
5388:Comma category
5385:
5380:
5375:
5370:
5365:
5360:
5354:
5352:
5348:
5347:
5345:
5344:
5334:
5324:
5322:Abelian groups
5319:
5314:
5309:
5304:
5298:
5296:
5290:
5289:
5286:
5285:
5283:
5282:
5277:
5272:
5271:
5270:
5260:
5255:
5249:
5247:
5241:
5240:
5238:
5237:
5232:
5227:
5226:
5225:
5215:
5210:
5204:
5202:
5193:
5187:
5186:
5184:
5183:
5178:
5173:
5168:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5127:
5126:
5121:
5116:
5111:
5106:
5101:
5090:
5088:
5080:
5079:
5069:
5068:
5061:
5060:
5053:
5046:
5038:
5032:
5031:
5019:
5007:
5006:External links
5004:
5003:
5002:
4996:
4974:
4968:
4943:
4914:
4908:
4887:
4886:
4883:on 2011-04-13.
4862:
4855:
4837:
4818:
4802:
4801:
4799:
4796:
4795:
4794:
4785:
4779:
4773:
4767:
4761:
4751:
4750:of the product
4739:
4736:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4422:
4414:
4411:
4408:
4405:
4390:Main article:
4387:
4386:Distributivity
4384:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4191:
4188:
4168:
4148:
4115:
4110:
4105:
4099:
4095:
4091:
4086:
4082:
4077:
4055:
4050:
4046:
4040:
4037:
4034:
4030:
4026:
4021:
4017:
4011:
4008:
4005:
4001:
3978:
3974:
3970:
3965:
3961:
3957:
3952:
3948:
3944:
3939:
3934:
3931:
3928:
3923:
3918:
3913:
3910:
3907:
3881:
3877:
3871:
3867:
3863:
3858:
3854:
3850:
3845:
3841:
3837:
3832:
3828:
3823:
3802:
3797:
3793:
3789:
3784:
3780:
3776:
3771:
3767:
3763:
3758:
3754:
3731:
3727:
3723:
3718:
3714:
3710:
3705:
3701:
3697:
3692:
3688:
3684:
3679:
3675:
3671:
3666:
3662:
3637:
3633:
3629:
3624:
3619:
3594:
3574:
3551:
3531:
3528:
3525:
3521:
3500:
3468:
3442:
3437:
3433:
3411:
3388:
3384:
3380:
3376:
3363:
3360:
3359:
3358:
3339:
3320:
3317:history monoid
3309:
3298:
3291:disjoint union
3283:
3272:
3261:
3249:
3246:
3231:
3197:
3192:
3189:
3186:
3181:
3177:
3174:
3171:
3166:
3162:
3157:
3152:
3149:
3146:
3143:
3140:
3131:is defined by
3118:
3114:
3108:
3105:
3102:
3098:
3094:
3091:
3088:
3085:
3065:
3060:
3056:
3052:
3049:
3046:
3041:
3037:
3016:
3007:Given any set
2996:
2991:
2987:
2983:
2979:
2974:
2971:
2968:
2963:
2958:
2954:
2950:
2945:
2939:
2935:
2930:
2925:
2921:
2917:
2912:
2908:
2902:
2899:
2896:
2892:
2888:
2883:
2879:
2857:
2853:
2850:
2847:
2837:
2833:
2829:
2824:
2820:
2816:
2811:
2808:
2805:
2800:
2795:
2791:
2787:
2781:
2777:
2772:
2768:
2762:
2759:
2756:
2752:
2729:
2725:
2708:
2705:
2691:
2685:
2681:
2677:
2672:
2668:
2663:
2659:
2656:
2653:
2650:
2647:
2644:
2624:
2604:
2584:
2580:
2576:
2572:
2550:
2544:
2540:
2536:
2531:
2527:
2522:
2512:to the object
2501:
2488:is given by a
2477:
2455:
2451:
2447:
2442:
2438:
2417:
2414:
2411:
2408:
2405:
2402:
2382:
2362:
2359:
2356:
2353:
2350:
2327:
2306:
2302:
2298:
2294:
2290:
2286:
2283:
2260:
2256:
2252:
2248:
2235:is simply the
2221:
2215:
2192:
2171:
2168:
2149:
2145:
2141:
2138:
2118:
2094:
2091:
2090:
2089:
2078:
2075:
2072:
2068:
2064:
2061:
2056:
2052:
2048:
2045:
2042:
2037:
2033:
2028:
2008:
2003:
1999:
1995:
1990:
1986:
1982:
1979:
1976:
1973:
1953:
1944:Uniqueness of
1942:
1929:
1925:
1921:
1917:
1911:
1907:
1903:
1898:
1894:
1889:
1885:
1880:
1876:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1813:
1809:
1805:
1800:
1796:
1784:
1773:
1770:
1767:
1764:
1761:
1758:
1738:
1722:
1719:
1707:
1704:
1699:
1695:
1691:
1688:
1685:
1680:
1676:
1672:
1650:
1646:
1642:
1639:
1636:
1631:
1627:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1559:
1554:
1550:
1544:
1541:
1538:
1534:
1522:
1521:
1520:
1519:
1498:
1495:
1492:
1489:
1469:
1466:
1463:
1460:
1457:
1437:
1432:
1428:
1424:
1421:
1418:
1413:
1409:
1388:
1368:
1344:
1339:
1335:
1331:
1328:
1325:
1320:
1316:
1295:
1282:of objects, a
1269:
1266:
1263:
1258:
1253:
1249:
1245:
1221:
1218:
1202:
1199:
1187:
1184:
1179:
1175:
1171:
1166:
1162:
1158:
1136:
1132:
1109:
1105:
1092:is called the
1081:
1061:
1056:
1052:
1032:
1027:
1023:
1002:
982:
950:
946:
923:
919:
909:The morphisms
895:
892:
887:
883:
879:
875:
871:
867:
846:
843:
838:
834:
830:
826:
822:
818:
795:
791:
787:
782:
778:
774:
770:
767:
763:
760:
739:
735:
731:
727:
723:
719:
715:
711:
707:
704:
672:
667:
663:
640:
636:
615:
604:
603:
602:
601:
573:
569:
565:
560:
556:
552:
549:
546:
543:
523:
518:
514:
510:
507:
504:
499:
495:
475:
470:
466:
462:
459:
456:
451:
447:
426:
396:
392:
388:
385:
382:
377:
373:
353:
348:
344:
340:
337:
334:
329:
325:
304:
299:
295:
291:
286:
282:
261:
258:
236:
232:
209:
205:
184:
181:
172:be objects of
159:
155:
132:
128:
107:
104:
92:
89:
87:
84:
56:direct product
15:
9:
6:
4:
3:
2:
5715:
5704:
5701:
5700:
5698:
5683:
5675:
5673:
5665:
5663:
5655:
5654:
5651:
5637:
5634:
5632:
5629:
5627:
5623:
5619:
5615:
5613:
5611:
5604:
5602:
5599:
5597:
5594:
5593:
5591:
5588:
5584:
5574:
5571:
5568:
5564:
5561:
5560:
5558:
5556:
5548:
5542:
5539:
5537:
5534:
5532:
5529:
5527:
5526:Tetracategory
5524:
5522:
5519:
5516:
5515:pseudofunctor
5512:
5509:
5508:
5506:
5504:
5496:
5493:
5489:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5449:
5446:
5444:
5440:
5434:
5433:
5430:
5426:
5421:
5417:
5412:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5358:Free category
5356:
5355:
5353:
5349:
5342:
5341:Vector spaces
5338:
5335:
5332:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5299:
5297:
5295:
5291:
5281:
5278:
5276:
5273:
5269:
5266:
5265:
5264:
5261:
5259:
5256:
5254:
5251:
5250:
5248:
5246:
5242:
5236:
5235:Inverse limit
5233:
5231:
5228:
5224:
5221:
5220:
5219:
5216:
5214:
5211:
5209:
5206:
5205:
5203:
5201:
5197:
5194:
5192:
5188:
5182:
5179:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5161:Kan extension
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5096:
5095:
5092:
5091:
5089:
5085:
5081:
5074:
5070:
5066:
5059:
5054:
5052:
5047:
5045:
5040:
5039:
5036:
5030:
5028:
5023:
5020:
5017:
5016:Jocelyn Paine
5013:
5010:
5009:
4999:
4997:0-521-44178-1
4993:
4989:
4984:
4983:
4975:
4971:
4969:0-387-98403-8
4965:
4961:
4958:
4954:
4953:
4948:
4944:
4933:on 2016-03-04
4929:
4922:
4921:
4915:
4911:
4909:0-471-60922-6
4905:
4898:
4897:
4891:
4890:
4882:
4878:
4877:
4869:
4867:
4858:
4856:0-387-90035-7
4852:
4848:
4841:
4833:
4829:
4822:
4814:
4807:
4803:
4789:
4786:
4783:
4780:
4777:
4776:Inverse limit
4774:
4771:
4768:
4765:
4762:
4759:
4755:
4752:
4749:
4745:
4742:
4741:
4735:
4722:
4716:
4713:
4710:
4704:
4698:
4695:
4692:
4686:
4680:
4677:
4674:
4668:
4665:
4657:
4638:
4635:
4632:
4626:
4623:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4576:
4573:
4570:
4564:
4561:
4549:
4545:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4505:
4489:
4483:
4480:
4477:
4471:
4468:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4436:
4420:
4412:
4409:
4406:
4403:
4393:
4383:
4381:
4377:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4245:
4242:
4239:
4233:
4227:
4224:
4221:
4215:
4212:
4205:
4202:We then have
4189:
4186:
4166:
4146:
4138:
4133:
4131:
4126:
4113:
4108:
4103:
4097:
4093:
4089:
4084:
4080:
4075:
4053:
4048:
4044:
4038:
4035:
4032:
4028:
4019:
4015:
4009:
4006:
4003:
3999:
3976:
3972:
3963:
3959:
3955:
3950:
3946:
3942:
3937:
3932:
3929:
3926:
3921:
3916:
3911:
3908:
3905:
3895:
3879:
3875:
3869:
3865:
3861:
3856:
3852:
3848:
3843:
3839:
3835:
3830:
3826:
3821:
3800:
3795:
3791:
3787:
3782:
3778:
3769:
3765:
3761:
3756:
3752:
3729:
3725:
3716:
3712:
3708:
3703:
3699:
3695:
3690:
3686:
3677:
3673:
3669:
3664:
3660:
3651:
3635:
3622:
3608:
3592:
3572:
3563:
3549:
3529:
3526:
3498:
3490:
3486:
3482:
3466:
3458:
3457:empty product
3453:
3440:
3435:
3431:
3386:
3382:
3378:
3356:
3352:
3348:
3344:
3340:
3337:
3335:
3329:
3325:
3324:Banach spaces
3321:
3318:
3314:
3310:
3307:
3303:
3299:
3296:
3292:
3288:
3284:
3281:
3277:
3273:
3270:
3266:
3262:
3247:
3244:
3236:
3232:
3229:
3225:
3221:
3217:
3213:
3212:
3211:
3208:
3195:
3190:
3187:
3184:
3179:
3172:
3164:
3160:
3155:
3150:
3144:
3138:
3116:
3112:
3106:
3103:
3100:
3096:
3089:
3086:
3083:
3063:
3058:
3054:
3047:
3044:
3039:
3035:
3014:
2994:
2989:
2985:
2981:
2977:
2972:
2969:
2966:
2961:
2956:
2952:
2948:
2943:
2937:
2933:
2928:
2923:
2919:
2910:
2906:
2900:
2897:
2894:
2890:
2886:
2881:
2877:
2855:
2851:
2848:
2845:
2835:
2831:
2827:
2822:
2818:
2814:
2809:
2806:
2803:
2798:
2793:
2789:
2785:
2779:
2775:
2770:
2766:
2760:
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2238:
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2167:
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2147:
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2100:
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2031:
2026:
2006:
2001:
1997:
1993:
1988:
1984:
1977:
1974:
1971:
1951:
1943:
1927:
1923:
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1762:
1759:
1736:
1729:Existence of
1728:
1727:
1726:
1718:
1705:
1697:
1693:
1689:
1686:
1683:
1678:
1674:
1648:
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1267:
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1256:
1251:
1247:
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1232:
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1216:
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1198:
1185:
1177:
1173:
1169:
1164:
1160:
1134:
1130:
1107:
1103:
1095:
1079:
1059:
1054:
1050:
1030:
1025:
1021:
1000:
980:
973:; the letter
972:
968:
967:
948:
944:
921:
917:
907:
893:
890:
885:
881:
877:
873:
869:
865:
844:
841:
836:
832:
828:
824:
820:
816:
793:
789:
785:
780:
776:
768:
765:
761:
758:
737:
733:
729:
725:
721:
717:
713:
709:
705:
702:
693:
689:
686:
670:
665:
661:
638:
634:
613:
596:
592:
591:
589:
571:
567:
563:
558:
554:
547:
544:
541:
521:
516:
512:
505:
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473:
468:
464:
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454:
449:
445:
424:
416:
415:
414:
412:
394:
390:
383:
380:
375:
371:
351:
346:
342:
335:
332:
327:
323:
302:
297:
293:
289:
284:
280:
259:
256:
249:is an object
234:
230:
207:
203:
195:A product of
182:
179:
157:
153:
130:
126:
105:
102:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
5606:
5587:Categorified
5491:n-categories
5442:Key concepts
5280:Direct limit
5263:Coequalizers
5212:
5181:Yoneda lemma
5087:Key concepts
5077:Key concepts
5026:
4981:
4959:
4950:
4935:. Retrieved
4928:the original
4919:
4895:
4881:the original
4875:
4846:
4840:
4831:
4821:
4812:
4806:
4758:left adjoint
4756: – the
4746: – the
4553:
4395:
4134:
4129:
4127:
3893:
3564:
3454:
3365:
3333:
3209:
2710:
2428:The product
2340:ordered pair
2173:
2096:
1724:
1523:
1283:
1233:
1204:
1093:
970:
964:
908:
691:
605:
94:
46:such as the
31:
25:
5555:-categories
5531:Kan complex
5521:Tricategory
5503:-categories
5393:Subcategory
5151:Exponential
5119:Preadditive
5114:Pre-abelian
4137:associative
3295:subcategory
44:mathematics
5573:3-category
5563:2-category
5536:∞-groupoid
5511:Bicategory
5258:Coproducts
5218:Equalizers
5124:Bicategory
4942:Chapter 5.
4937:2016-03-21
4798:References
4139:. Suppose
3813:We choose
3459:(that is,
3362:Discussion
3347:coproducts
3328:short maps
3228:continuous
2093:As a limit
1379:and every
808:such that
86:Definition
66:, and the
5622:Symmetric
5567:2-functor
5307:Relations
5230:Pullbacks
4770:Equalizer
4744:Coproduct
4714:×
4696:×
4687:≃
4669:×
4627:×
4621:→
4615:×
4603:×
4565:×
4529:×
4517:×
4504:coproduct
4472:×
4466:→
4460:×
4448:×
4437:morphism
4435:canonical
4356:×
4350:≃
4344:×
4316:≃
4310:×
4304:≃
4298:×
4270:×
4264:×
4258:≃
4252:×
4243:×
4234:≃
4225:×
4216:×
4094:π
4090:∘
4036:∈
4029:∏
4025:→
4007:∈
4000:∏
3969:→
3866:π
3862:∘
3840:π
3836:∘
3788:×
3775:→
3762:×
3722:→
3683:→
3650:bifunctor
3628:→
3524:→
3481:empty set
3379:×
3188:∈
3104:∈
3097:∏
3093:→
3051:→
2970:∈
2934:π
2916:→
2898:∈
2891:∏
2878:π
2849:∈
2828:∈
2807:∈
2758:∈
2751:∏
2658:→
2575:×
2500:Δ
2446:×
2393:the pair
2301:×
2293:→
2282:Δ
2251:×
2060:∘
2051:π
2041:∘
2032:π
1994:×
1981:→
1884:∘
1875:π
1837:∈
1769:⟩
1766:⋅
1760:⋅
1757:⟨
1703:⟩
1687:…
1671:⟨
1641:×
1638:⋯
1635:×
1593:…
1540:∈
1533:∏
1491:∈
1465:→
1423:→
1330:→
1315:π
1265:∈
1209:by a set
1183:⟩
1157:⟨
981:π
945:π
918:π
891:∘
882:π
866:π
842:∘
833:π
817:π
786:×
773:→
730:π
714:π
564:×
551:→
509:→
461:→
387:→
372:π
339:→
324:π
290:×
5697:Category
5682:Glossary
5662:Category
5636:n-monoid
5589:concepts
5245:Colimits
5213:Products
5166:Morphism
5109:Concrete
5104:Additive
5094:Category
4949:(1998).
4738:See also
3876:⟩
3822:⟨
2707:Examples
2162:being a
2067:⟩
2027:⟨
1916:⟩
1888:⟨
1826:and all
874:′
825:′
769:′
738:′
722:′
706:′
588:commutes
80:morphism
40:category
5672:Outline
5631:n-group
5596:2-group
5551:Strict
5541:∞-topos
5337:Modules
5275:Pushout
5223:Kernels
5156:Functor
5099:Abelian
5024:at the
5022:Product
3607:functor
3479:is the
3285:In the
3274:In the
3263:In the
3233:In the
3214:In the
2711:In the
2182:, take
2107:diagram
1284:product
1207:indexed
68:product
36:objects
32:product
5618:Traced
5601:2-ring
5331:Fields
5317:Groups
5312:Magmas
5200:Limits
4994:
4966:
4906:
4853:
4376:monoid
3652:. For
626:or on
76:family
60:groups
54:, the
30:, the
5612:-ring
5499:Weak
5483:Topos
5327:Rings
4931:(PDF)
4924:(PDF)
4900:(PDF)
3355:joins
3351:meets
3338:norm.
2099:limit
685:up to
64:rings
38:in a
5302:Sets
4992:ISBN
4964:ISBN
4904:ISBN
4851:ISBN
4748:dual
3423:and
3326:and
2338:the
2271:The
2164:cone
1122:and
1013:and
936:and
857:and
653:and
222:and
145:and
118:Let
52:sets
5146:End
5136:CCC
5029:Lab
3565:If
3542:so
2615:of
2562:in
2468:in
1570:If
969:or
692:the
70:of
62:or
58:of
50:of
26:In
5699::
5624:)
5620:)(
4990:.
4988:39
4955:.
4865:^
4830:.
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3341:A
3151::=
2982::=
2776::=
906:.
590::
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5609:n
5607:E
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2071:=
2063:g
2055:2
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2036:1
2007:,
2002:2
1998:X
1989:1
1985:X
1978:Y
1975::
1972:g
1952:f
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1924:f
1920:=
1910:2
1906:f
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1852:}
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1001:Y
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894:h
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878:=
870:2
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790:X
781:1
777:X
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762::
759:h
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718:1
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