4608:
4855:
4875:
4865:
1405:
802:
1692:, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the
2972:
1788:
The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space
3194:
1793:
by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute. This pattern holds for any
3629:
3079:
3284:
That it is an injection follows from the universal construction which stipulates the uniqueness of such maps. The naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant
2851:
2756:
526:
460:
3674:
3531:
3995:
3841:
2219:
3439:
1153:
3716:
1553:
3279:
2034:
1988:
697:
3368:
3317:
3957:
3925:
2668:
1630:
1507:
2622:
2067:
1942:
1464:
3485:
1369:
792:
4106:
2589:
2148:
911:
869:
743:
351:
308:
645:
3893:
2452:
2350:
2314:
2282:
2102:
1288:
1222:
596:
3997:. This may be extended by induction to a canonical morphism from any finite coproduct to the corresponding product. This morphism need not in general be an isomorphism; in
101:
reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category.
3779:
953:
394:
3102:
2478:
1587:
1320:
2416:
2174:
1399:
1006:
265:
2698:
1657:
1060:
1033:
235:
185:
158:
2376:
1104:
208:
4073:
4053:
3861:
3799:
3753:
3391:
3341:
3221:
2840:
2816:
2796:
2776:
2556:
2536:
2513:
2396:
2243:
1909:
1889:
1862:
1429:
1242:
1173:
826:
550:
127:
3554:
4252:
2991:
4179:
1751:
4899:
4196:
2967:{\displaystyle \operatorname {Hom} _{C}\left(\coprod _{j\in J}X_{j},Y\right)\cong \prod _{j\in J}\operatorname {Hom} _{C}(X_{j},Y)}
4157:
1731:
4220:
85:. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a
2712:
4245:
4184:
4449:
4404:
4878:
4818:
1795:
465:
399:
3635:
3490:
2558:
exist, then it is possible to choose the products in a compatible fashion so that the coproduct turns into a functor
3962:
3808:
2186:
4868:
4654:
4518:
4426:
3400:
1739:
1720:
many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)
1703:
4827:
4471:
4409:
4332:
4121:
1762:
1109:
94:
62:
4858:
4814:
4419:
4238:
3680:
1512:
3229:
1993:
1947:
650:
3346:
3295:
4414:
4396:
4126:
3930:
3898:
2630:
1592:
1469:
2594:
2039:
1914:
4621:
4387:
4367:
4290:
1434:
130:
90:
3444:
1325:
748:
4503:
4342:
4078:
2561:
2107:
873:
831:
702:
313:
270:
601:
3866:
2421:
2319:
2287:
2251:
2075:
1247:
1181:
555:
30:
This article is about coproducts in categories. For "coproduct" in the sense of comultiplication, see
4315:
4310:
4224:
4021:
3289:
changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the
1944:
will have a coproduct in general, but if it does, then the coproduct is unique in a strong sense: if
78:
3758:
3189:{\displaystyle \coprod _{j\in J}f_{j}\in \operatorname {Hom} \left(\coprod _{j\in J}X_{j},Y\right).}
4659:
4607:
4537:
4533:
4337:
3546:
2843:
1372:
1064:
795:
70:
915:
356:
4513:
4508:
4490:
4372:
4347:
2180:
1829:
1693:
1069:
529:
2457:
4822:
4759:
4747:
4649:
4574:
4569:
4527:
4523:
4305:
4300:
4192:
4174:
4029:
3726:
3290:
3093:
1869:
1758:
1743:
1560:
1293:
58:
2401:
4783:
4669:
4644:
4579:
4564:
4559:
4498:
4327:
4295:
4202:
3089:
2222:
2153:
1724:
1668:
1378:
958:
240:
2676:
1817:
sum, which cannot be so easily conceptualized as an "almost disjoint" sum, but does have a
1785:(which amounts to joining a collection of spaces with base points at a common base point).
1635:
1038:
1011:
213:
163:
136:
4695:
4261:
4206:
4188:
3802:
1778:
42:
4732:
2355:
1086:
190:
4727:
4711:
4674:
4664:
4584:
4058:
4038:
3846:
3784:
3738:
3538:
3376:
3326:
3206:
2825:
2801:
2781:
2761:
2541:
2521:
2498:
2492:
2381:
2228:
1894:
1874:
1847:
1825:
1689:
1673:
1414:
1227:
1158:
1080:
811:
535:
112:
54:
34:. For additional substances that result from the manufacturing of another product, see
4893:
4722:
4554:
4431:
4357:
2702:
1802:
1774:
1707:
1685:
4476:
4377:
4136:
4017:
3624:{\displaystyle X\oplus (Y\oplus Z)\cong (X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z}
2246:
1698:
1073:
82:
66:
4737:
4223:
which generates examples of coproducts in the category of finite sets. Written by
4717:
4589:
4459:
4131:
4013:
4002:
3733:
3286:
3074:{\displaystyle (f_{j})_{j\in J}\in \prod _{j\in J}\operatorname {Hom} (X_{j},Y)}
2070:
1790:
4024:, this morphism is an isomorphism and the corresponding object is known as the
4769:
4707:
4320:
3394:
3200:
1717:
1712:
74:
35:
4763:
4025:
2978:
2484:
1818:
1806:
1782:
31:
1404:
801:
17:
4832:
4464:
4362:
1810:
1766:
1176:
98:
86:
4802:
4792:
4441:
4352:
2819:
1865:
1841:
1840:
The coproduct construction given above is actually a special case of a
97:, which means the definition is the same as the product but with all
4797:
3722:
1765:. That is, it is a disjoint union of the underlying sets, and the
4679:
4230:
3537:, coproduct functors have been chosen as above, and 0 denotes the
2982:
3725:; a category with finite coproducts is an example of a symmetric
1716:, consists of the elements of the direct product which have only
3721:
These properties are formally similar to those of a commutative
4619:
4272:
4234:
2183:, the coproduct can be understood as a universal morphism. Let
2069:, then (by the definition of coproducts) there exists a unique
1821:
almost-disjointly generated by the unit ball is the cofactors.
1079:
The definition of a coproduct can be extended to an arbitrary
4158:"Banach spaces (and Lawvere metrics, and closed categories)"
3203:
follows from the commutativity of the diagram: any morphism
2538:
is a set such that all coproducts for families indexed with
1403:
800:
1750:, the coproduct is a quotient of the tensor algebra (see
2751:{\displaystyle \operatorname {Hom} _{C}\left(U,V\right)}
4028:. A category with all finite biproducts is known as a
3895:
as in the preceding paragraph. We thus have morphisms
4081:
4061:
4041:
3965:
3933:
3901:
3869:
3849:
3811:
3787:
3761:
3741:
3683:
3638:
3557:
3493:
3447:
3403:
3379:
3349:
3329:
3298:
3232:
3209:
3105:
2994:
2854:
2828:
2804:
2784:
2764:
2715:
2679:
2633:
2597:
2564:
2544:
2524:
2501:
2460:
2424:
2404:
2384:
2358:
2322:
2290:
2254:
2231:
2189:
2156:
2110:
2078:
2042:
1996:
1950:
1917:
1897:
1877:
1850:
1638:
1595:
1563:
1515:
1472:
1437:
1417:
1381:
1328:
1296:
1250:
1230:
1184:
1161:
1112:
1089:
1041:
1014:
961:
918:
876:
834:
814:
751:
705:
653:
604:
558:
538:
468:
402:
359:
316:
273:
243:
216:
193:
166:
139:
115:
4782:
4746:
4694:
4687:
4638:
4547:
4489:
4440:
4395:
4386:
4283:
3545:corresponding to the empty coproduct. We then have
3323:is continuous; it preserves limits (a coproduct in
53:, is a construction which includes as examples the
4100:
4067:
4047:
3989:
3951:
3919:
3887:
3855:
3835:
3793:
3773:
3747:
3710:
3668:
3623:
3525:
3479:
3433:
3385:
3362:
3335:
3311:
3273:
3215:
3188:
3073:
2966:
2834:
2810:
2790:
2770:
2750:
2692:
2662:
2616:
2583:
2550:
2530:
2507:
2472:
2446:
2410:
2390:
2370:
2344:
2308:
2276:
2237:
2213:
2168:
2142:
2096:
2061:
2028:
1982:
1936:
1903:
1883:
1856:
1702:, is quite complicated. On the other hand, in the
1651:
1624:
1581:
1547:
1501:
1458:
1423:
1393:
1363:
1314:
1282:
1236:
1216:
1167:
1147:
1098:
1054:
1027:
1000:
947:
905:
863:
820:
786:
737:
691:
639:
590:
544:
520:
454:
388:
345:
302:
259:
229:
202:
179:
152:
121:
3863:is also initial, we have a canonical isomorphism
3096:, so it is a tuple of morphisms) to the morphism
1773:, in a rather evident sense. In the category of
4108:. Note that, like the product, this functor is
2398:is given by a universal morphism to the functor
1844:in category theory. The coproduct in a category
521:{\displaystyle i_{2}:X_{2}\to X_{1}\sqcup X_{2}}
455:{\displaystyle i_{1}:X_{1}\to X_{1}\sqcup X_{2}}
3669:{\displaystyle X\oplus 0\cong 0\oplus X\cong X}
3526:{\displaystyle X_{1}\oplus \ldots \oplus X_{n}}
3990:{\displaystyle X\oplus Y\rightarrow X\times Y}
3836:{\displaystyle X\oplus Y\rightarrow Z\oplus Y}
2214:{\displaystyle \Delta :C\rightarrow C\times C}
4246:
3434:{\displaystyle J=\lbrace 1,\ldots ,n\rbrace }
1632:to indicate its dependence on the individual
8:
4187:. Vol. 5 (2nd ed.). New York, NY:
3428:
3410:
2611:
2598:
2056:
2043:
1931:
1918:
1761:, coproducts are disjoint unions with their
1148:{\displaystyle \left\{X_{j}:j\in J\right\}}
828:making this diagram commute may be denoted
4874:
4864:
4691:
4635:
4616:
4392:
4280:
4269:
4253:
4239:
4231:
4086:
4080:
4075:, then the coproduct comprises a functor
4060:
4040:
3964:
3959:, by which we infer a canonical morphism
3932:
3900:
3868:
3848:
3810:
3786:
3760:
3740:
3711:{\displaystyle X\oplus Y\cong Y\oplus X.}
3682:
3637:
3556:
3533:. Suppose all finite coproducts exist in
3517:
3498:
3492:
3471:
3452:
3446:
3402:
3378:
3354:
3348:
3328:
3303:
3297:
3256:
3246:
3231:
3208:
3166:
3150:
3126:
3110:
3104:
3056:
3031:
3012:
3002:
2993:
2949:
2933:
2917:
2893:
2877:
2859:
2853:
2827:
2803:
2783:
2763:
2720:
2714:
2684:
2678:
2654:
2638:
2632:
2605:
2596:
2569:
2563:
2543:
2523:
2500:
2459:
2423:
2403:
2383:
2357:
2321:
2289:
2253:
2230:
2188:
2155:
2134:
2121:
2109:
2077:
2050:
2041:
2014:
2001:
1995:
1968:
1955:
1949:
1925:
1916:
1896:
1876:
1849:
1796:variety in the sense of universal algebra
1643:
1637:
1616:
1600:
1594:
1562:
1548:{\displaystyle \bigoplus _{j\in J}X_{j}.}
1536:
1520:
1514:
1493:
1477:
1471:
1446:
1436:
1416:
1380:
1352:
1333:
1327:
1295:
1268:
1255:
1249:
1229:
1202:
1189:
1183:
1160:
1122:
1111:
1088:
1046:
1040:
1019:
1013:
984:
971:
960:
936:
923:
917:
894:
881:
875:
852:
839:
833:
813:
775:
756:
750:
729:
710:
704:
677:
664:
652:
622:
609:
603:
576:
563:
557:
537:
512:
499:
486:
473:
467:
446:
433:
420:
407:
401:
377:
364:
358:
334:
321:
315:
291:
278:
272:
248:
242:
221:
215:
192:
171:
165:
144:
138:
114:
4180:Categories for the Working Mathematician
3274:{\displaystyle (f\circ i_{j})_{j\in J}.}
2029:{\displaystyle k_{j}:X_{j}\rightarrow Y}
1983:{\displaystyle i_{j}:X_{j}\rightarrow X}
692:{\displaystyle f:X_{1}\sqcup X_{2}\to Y}
4148:
3363:{\displaystyle C^{\operatorname {op} }}
3312:{\displaystyle C^{\operatorname {op} }}
4035:If all families of objects indexed by
3952:{\displaystyle X\oplus Y\rightarrow Y}
3920:{\displaystyle X\oplus Y\rightarrow X}
2663:{\displaystyle \coprod _{j\in J}X_{j}}
1625:{\displaystyle \coprod _{j\in J}f_{j}}
1502:{\displaystyle \coprod _{j\in J}X_{j}}
2758:denote the set of all morphisms from
2617:{\displaystyle \lbrace X_{j}\rbrace }
2062:{\displaystyle \lbrace X_{j}\rbrace }
1937:{\displaystyle \lbrace X_{j}\rbrace }
1864:can be defined as the colimit of any
210:An object is called the coproduct of
7:
1752:free product of associative algebras
1459:{\displaystyle \left\{X_{j}\right\}}
3480:{\displaystyle X_{1},\ldots ,X_{n}}
1364:{\displaystyle f_{j}=f\circ i_{j}.}
787:{\displaystyle f_{2}=f\circ i_{2}.}
4101:{\displaystyle C^{J}\rightarrow C}
2584:{\displaystyle C^{J}\rightarrow C}
2405:
2190:
2143:{\displaystyle f\circ i_{j}=k_{j}}
906:{\displaystyle f_{1}\oplus f_{2},}
864:{\displaystyle f_{1}\sqcup f_{2},}
738:{\displaystyle f_{1}=f\circ i_{1}}
346:{\displaystyle X_{1}\oplus X_{2},}
303:{\displaystyle X_{1}\sqcup X_{2},}
25:
3755:, then we have a unique morphism
2036:are two coproducts of the family
1801:The coproduct in the category of
640:{\displaystyle f_{2}:X_{2}\to Y,}
4873:
4863:
4854:
4853:
4606:
3888:{\displaystyle Z\oplus Y\cong Y}
3441:, then the coproduct of objects
2447:{\displaystyle \left(X,Y\right)}
2345:{\displaystyle \left(f,f\right)}
2309:{\displaystyle f:X\rightarrow Y}
2277:{\displaystyle \left(X,X\right)}
2097:{\displaystyle f:X\rightarrow Y}
1283:{\displaystyle f_{j}:X_{j}\to Y}
1244:and any collection of morphisms
1217:{\displaystyle i_{j}:X_{j}\to X}
591:{\displaystyle f_{1}:X_{1}\to Y}
89:. It is the category-theoretic
3223:is the coproduct of the tuple
1371:That is, the following diagram
1290:there exists a unique morphism
794:That is, the following diagram
647:there exists a unique morphism
27:Category-theoretic construction
4156:Qiaochu Yuan (June 23, 2012).
4092:
3975:
3943:
3911:
3821:
3774:{\displaystyle X\rightarrow Z}
3765:
3594:
3582:
3576:
3564:
3253:
3233:
3068:
3049:
3009:
2995:
2961:
2942:
2591:. The coproduct of the family
2575:
2300:
2199:
2088:
2020:
1974:
1573:
1306:
1274:
1208:
1175:together with a collection of
683:
628:
582:
492:
426:
1:
4185:Graduate Texts in Mathematics
2483:The coproduct indexed by the
2225:which assigns to each object
1744:category of (noncommutative)
1710:), the coproduct, called the
1106:The coproduct of the family
1083:of objects indexed by a set
1068:, although they need not be
948:{\displaystyle f_{1}+f_{2},}
389:{\displaystyle X_{1}+X_{2},}
4548:Constructions on categories
4916:
4655:Higher-dimensional algebra
2624:is then often denoted by
1771:open in each of the spaces
1704:category of abelian groups
1224:such that, for any object
29:
4849:
4628:
4615:
4604:
4279:
4268:
2473:{\displaystyle C\times C}
1763:disjoint union topologies
528:satisfying the following
396:if there exist morphisms
4900:Limits (category theory)
1732:category of commutative
1582:{\displaystyle f:X\to Y}
1315:{\displaystyle f:X\to Y}
4465:Cokernels and quotients
4388:Universal constructions
2411:{\displaystyle \Delta }
1781:, the coproduct is the
1730:, the coproduct in the
1557:Sometimes the morphism
4622:Higher category theory
4368:Natural transformation
4102:
4069:
4049:
3991:
3953:
3921:
3889:
3857:
3837:
3805:) and thus a morphism
3795:
3775:
3749:
3732:If the category has a
3712:
3670:
3625:
3527:
3481:
3435:
3387:
3364:
3337:
3313:
3275:
3217:
3190:
3075:
2968:
2836:
2812:
2792:
2772:
2752:
2694:
2664:
2618:
2585:
2552:
2532:
2509:
2474:
2448:
2412:
2392:
2372:
2346:
2310:
2278:
2239:
2215:
2170:
2169:{\displaystyle j\in J}
2144:
2098:
2063:
2030:
1984:
1938:
1905:
1885:
1858:
1653:
1626:
1583:
1549:
1503:
1460:
1425:
1408:
1395:
1394:{\displaystyle j\in J}
1365:
1316:
1284:
1238:
1218:
1169:
1149:
1100:
1056:
1029:
1002:
1001:{\displaystyle \left.}
949:
907:
865:
822:
805:
788:
739:
693:
641:
592:
546:
522:
456:
390:
347:
304:
261:
260:{\displaystyle X_{2},}
231:
204:
181:
154:
123:
4221:Interactive Web page
4103:
4070:
4050:
4030:semiadditive category
3992:
3954:
3922:
3890:
3858:
3838:
3796:
3776:
3750:
3713:
3671:
3626:
3528:
3482:
3436:
3388:
3365:
3338:
3314:
3276:
3218:
3191:
3076:
2969:
2837:
2813:
2793:
2773:
2753:
2695:
2693:{\displaystyle i_{j}}
2665:
2619:
2586:
2553:
2533:
2510:
2475:
2449:
2413:
2393:
2373:
2352:. Then the coproduct
2347:
2311:
2284:and to each morphism
2279:
2240:
2216:
2171:
2145:
2099:
2064:
2031:
1985:
1939:
1906:
1886:
1859:
1667:The coproduct in the
1654:
1652:{\displaystyle f_{j}}
1627:
1584:
1550:
1504:
1461:
1426:
1407:
1396:
1366:
1317:
1285:
1239:
1219:
1170:
1150:
1101:
1057:
1055:{\displaystyle i_{2}}
1030:
1028:{\displaystyle i_{1}}
1003:
950:
908:
866:
823:
804:
789:
740:
694:
642:
593:
547:
523:
457:
391:
348:
305:
262:
232:
230:{\displaystyle X_{1}}
205:
182:
180:{\displaystyle X_{2}}
155:
153:{\displaystyle X_{1}}
124:
63:of topological spaces
4491:Algebraic categories
4079:
4059:
4039:
4022:preadditive category
3963:
3931:
3899:
3867:
3847:
3809:
3785:
3759:
3739:
3681:
3636:
3555:
3547:natural isomorphisms
3491:
3487:is often denoted by
3445:
3401:
3377:
3347:
3327:
3296:
3230:
3207:
3103:
2992:
2852:
2826:
2802:
2782:
2762:
2713:
2677:
2631:
2595:
2562:
2542:
2522:
2499:
2491:) is the same as an
2458:
2422:
2402:
2382:
2356:
2320:
2288:
2252:
2229:
2187:
2154:
2108:
2076:
2040:
1994:
1948:
1915:
1895:
1875:
1848:
1636:
1593:
1561:
1513:
1470:
1435:
1415:
1379:
1326:
1294:
1248:
1228:
1182:
1159:
1110:
1087:
1065:canonical injections
1039:
1012:
959:
916:
874:
832:
812:
749:
703:
651:
602:
556:
536:
466:
400:
357:
353:or sometimes simply
314:
271:
241:
214:
191:
164:
137:
113:
4660:Homotopy hypothesis
4338:Commutative diagram
4127:Limits and colimits
4055:have coproducts in
3199:That this map is a
2844:natural isomorphism
2371:{\displaystyle X+Y}
1911:. Not every family
1824:The coproduct of a
95:categorical product
4373:Universal property
4175:Mac Lane, Saunders
4162:Annoying Precision
4098:
4065:
4045:
3987:
3949:
3917:
3885:
3853:
3833:
3791:
3771:
3745:
3708:
3666:
3621:
3523:
3477:
3431:
3383:
3360:
3333:
3309:
3271:
3213:
3186:
3161:
3121:
3071:
3042:
2964:
2928:
2888:
2832:
2808:
2788:
2768:
2748:
2703:natural injections
2690:
2660:
2649:
2614:
2581:
2548:
2528:
2505:
2470:
2444:
2408:
2388:
2368:
2342:
2306:
2274:
2235:
2211:
2181:universal property
2166:
2140:
2094:
2059:
2026:
1980:
1934:
1901:
1881:
1854:
1759:topological spaces
1694:category of groups
1649:
1622:
1611:
1579:
1545:
1531:
1499:
1488:
1456:
1421:
1409:
1391:
1361:
1312:
1280:
1234:
1214:
1165:
1145:
1099:{\displaystyle J.}
1096:
1052:
1025:
998:
945:
903:
861:
818:
806:
784:
735:
689:
637:
588:
552:and any morphisms
542:
530:universal property
518:
452:
386:
343:
300:
257:
227:
203:{\displaystyle C.}
200:
177:
150:
119:
4887:
4886:
4845:
4844:
4841:
4840:
4823:monoidal category
4778:
4777:
4650:Enriched category
4602:
4601:
4598:
4597:
4575:Quotient category
4570:Opposite category
4485:
4484:
4068:{\displaystyle C}
4048:{\displaystyle J}
4016:) it is a proper
4012:(the category of
3856:{\displaystyle Z}
3794:{\displaystyle Z}
3748:{\displaystyle Z}
3727:monoidal category
3386:{\displaystyle J}
3336:{\displaystyle C}
3291:opposite category
3216:{\displaystyle f}
3146:
3106:
3094:Cartesian product
3027:
2981:which maps every
2913:
2873:
2835:{\displaystyle C}
2811:{\displaystyle C}
2791:{\displaystyle V}
2771:{\displaystyle U}
2700:are known as the
2634:
2551:{\displaystyle J}
2531:{\displaystyle J}
2508:{\displaystyle C}
2391:{\displaystyle C}
2238:{\displaystyle X}
1904:{\displaystyle C}
1884:{\displaystyle J}
1870:discrete category
1857:{\displaystyle C}
1777:, fundamental in
1706:(and equally for
1596:
1516:
1473:
1466:is often denoted
1424:{\displaystyle X}
1237:{\displaystyle Y}
1168:{\displaystyle X}
821:{\displaystyle f}
808:The unique arrow
545:{\displaystyle Y}
532:: for any object
122:{\displaystyle C}
16:(Redirected from
4907:
4877:
4876:
4867:
4866:
4857:
4856:
4692:
4670:Simplex category
4645:Categorification
4636:
4617:
4610:
4580:Product category
4565:Kleisli category
4560:Functor category
4405:Terminal objects
4393:
4328:Adjoint functors
4281:
4270:
4255:
4248:
4241:
4232:
4210:
4166:
4165:
4153:
4107:
4105:
4104:
4099:
4091:
4090:
4074:
4072:
4071:
4066:
4054:
4052:
4051:
4046:
3996:
3994:
3993:
3988:
3958:
3956:
3955:
3950:
3926:
3924:
3923:
3918:
3894:
3892:
3891:
3886:
3862:
3860:
3859:
3854:
3842:
3840:
3839:
3834:
3800:
3798:
3797:
3792:
3780:
3778:
3777:
3772:
3754:
3752:
3751:
3746:
3717:
3715:
3714:
3709:
3675:
3673:
3672:
3667:
3630:
3628:
3627:
3622:
3532:
3530:
3529:
3524:
3522:
3521:
3503:
3502:
3486:
3484:
3483:
3478:
3476:
3475:
3457:
3456:
3440:
3438:
3437:
3432:
3392:
3390:
3389:
3384:
3369:
3367:
3366:
3361:
3359:
3358:
3343:is a product in
3342:
3340:
3339:
3334:
3318:
3316:
3315:
3310:
3308:
3307:
3280:
3278:
3277:
3272:
3267:
3266:
3251:
3250:
3222:
3220:
3219:
3214:
3195:
3193:
3192:
3187:
3182:
3178:
3171:
3170:
3160:
3131:
3130:
3120:
3090:category of sets
3080:
3078:
3077:
3072:
3061:
3060:
3041:
3023:
3022:
3007:
3006:
2973:
2971:
2970:
2965:
2954:
2953:
2938:
2937:
2927:
2909:
2905:
2898:
2897:
2887:
2864:
2863:
2841:
2839:
2838:
2833:
2817:
2815:
2814:
2809:
2797:
2795:
2794:
2789:
2777:
2775:
2774:
2769:
2757:
2755:
2754:
2749:
2747:
2743:
2725:
2724:
2699:
2697:
2696:
2691:
2689:
2688:
2669:
2667:
2666:
2661:
2659:
2658:
2648:
2623:
2621:
2620:
2615:
2610:
2609:
2590:
2588:
2587:
2582:
2574:
2573:
2557:
2555:
2554:
2549:
2537:
2535:
2534:
2529:
2514:
2512:
2511:
2506:
2479:
2477:
2476:
2471:
2453:
2451:
2450:
2445:
2443:
2439:
2418:from the object
2417:
2415:
2414:
2409:
2397:
2395:
2394:
2389:
2377:
2375:
2374:
2369:
2351:
2349:
2348:
2343:
2341:
2337:
2315:
2313:
2312:
2307:
2283:
2281:
2280:
2275:
2273:
2269:
2244:
2242:
2241:
2236:
2223:diagonal functor
2220:
2218:
2217:
2212:
2175:
2173:
2172:
2167:
2149:
2147:
2146:
2141:
2139:
2138:
2126:
2125:
2103:
2101:
2100:
2095:
2068:
2066:
2065:
2060:
2055:
2054:
2035:
2033:
2032:
2027:
2019:
2018:
2006:
2005:
1989:
1987:
1986:
1981:
1973:
1972:
1960:
1959:
1943:
1941:
1940:
1935:
1930:
1929:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1863:
1861:
1860:
1855:
1815:
1725:commutative ring
1669:category of sets
1658:
1656:
1655:
1650:
1648:
1647:
1631:
1629:
1628:
1623:
1621:
1620:
1610:
1588:
1586:
1585:
1580:
1554:
1552:
1551:
1546:
1541:
1540:
1530:
1508:
1506:
1505:
1500:
1498:
1497:
1487:
1465:
1463:
1462:
1457:
1455:
1451:
1450:
1430:
1428:
1427:
1422:
1400:
1398:
1397:
1392:
1370:
1368:
1367:
1362:
1357:
1356:
1338:
1337:
1321:
1319:
1318:
1313:
1289:
1287:
1286:
1281:
1273:
1272:
1260:
1259:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1207:
1206:
1194:
1193:
1174:
1172:
1171:
1166:
1154:
1152:
1151:
1146:
1144:
1140:
1127:
1126:
1105:
1103:
1102:
1097:
1061:
1059:
1058:
1053:
1051:
1050:
1034:
1032:
1031:
1026:
1024:
1023:
1007:
1005:
1004:
999:
994:
990:
989:
988:
976:
975:
954:
952:
951:
946:
941:
940:
928:
927:
912:
910:
909:
904:
899:
898:
886:
885:
870:
868:
867:
862:
857:
856:
844:
843:
827:
825:
824:
819:
793:
791:
790:
785:
780:
779:
761:
760:
744:
742:
741:
736:
734:
733:
715:
714:
698:
696:
695:
690:
682:
681:
669:
668:
646:
644:
643:
638:
627:
626:
614:
613:
597:
595:
594:
589:
581:
580:
568:
567:
551:
549:
548:
543:
527:
525:
524:
519:
517:
516:
504:
503:
491:
490:
478:
477:
461:
459:
458:
453:
451:
450:
438:
437:
425:
424:
412:
411:
395:
393:
392:
387:
382:
381:
369:
368:
352:
350:
349:
344:
339:
338:
326:
325:
309:
307:
306:
301:
296:
295:
283:
282:
266:
264:
263:
258:
253:
252:
236:
234:
233:
228:
226:
225:
209:
207:
206:
201:
186:
184:
183:
178:
176:
175:
159:
157:
156:
151:
149:
148:
128:
126:
125:
120:
21:
4915:
4914:
4910:
4909:
4908:
4906:
4905:
4904:
4890:
4889:
4888:
4883:
4837:
4807:
4774:
4751:
4742:
4699:
4683:
4634:
4624:
4611:
4594:
4543:
4481:
4450:Initial objects
4436:
4382:
4275:
4264:
4262:Category theory
4259:
4217:
4199:
4189:Springer-Verlag
4173:
4170:
4169:
4155:
4154:
4150:
4145:
4118:
4082:
4077:
4076:
4057:
4056:
4037:
4036:
4011:
4001:it is a proper
3961:
3960:
3929:
3928:
3897:
3896:
3865:
3864:
3845:
3844:
3807:
3806:
3783:
3782:
3757:
3756:
3737:
3736:
3679:
3678:
3634:
3633:
3553:
3552:
3513:
3494:
3489:
3488:
3467:
3448:
3443:
3442:
3399:
3398:
3375:
3374:
3350:
3345:
3344:
3325:
3324:
3299:
3294:
3293:
3252:
3242:
3228:
3227:
3205:
3204:
3162:
3145:
3141:
3122:
3101:
3100:
3092:, which is the
3052:
3008:
2998:
2990:
2989:
2945:
2929:
2889:
2872:
2868:
2855:
2850:
2849:
2824:
2823:
2800:
2799:
2780:
2779:
2760:
2759:
2733:
2729:
2716:
2711:
2710:
2680:
2675:
2674:
2650:
2629:
2628:
2601:
2593:
2592:
2565:
2560:
2559:
2540:
2539:
2520:
2519:
2497:
2496:
2489:empty coproduct
2456:
2455:
2429:
2425:
2420:
2419:
2400:
2399:
2380:
2379:
2354:
2353:
2327:
2323:
2318:
2317:
2286:
2285:
2259:
2255:
2250:
2249:
2227:
2226:
2185:
2184:
2152:
2151:
2130:
2117:
2106:
2105:
2074:
2073:
2046:
2038:
2037:
2010:
1997:
1992:
1991:
1964:
1951:
1946:
1945:
1921:
1913:
1912:
1893:
1892:
1873:
1872:
1846:
1845:
1838:
1811:
1779:homotopy theory
1757:In the case of
1690:direct products
1682:
1665:
1639:
1634:
1633:
1612:
1591:
1590:
1589:may be denoted
1559:
1558:
1532:
1511:
1510:
1489:
1468:
1467:
1442:
1438:
1433:
1432:
1413:
1412:
1377:
1376:
1348:
1329:
1324:
1323:
1292:
1291:
1264:
1251:
1246:
1245:
1226:
1225:
1198:
1185:
1180:
1179:
1157:
1156:
1118:
1117:
1113:
1108:
1107:
1085:
1084:
1042:
1037:
1036:
1015:
1010:
1009:
980:
967:
966:
962:
957:
956:
932:
919:
914:
913:
890:
877:
872:
871:
848:
835:
830:
829:
810:
809:
771:
752:
747:
746:
725:
706:
701:
700:
673:
660:
649:
648:
618:
605:
600:
599:
572:
559:
554:
553:
534:
533:
508:
495:
482:
469:
464:
463:
442:
429:
416:
403:
398:
397:
373:
360:
355:
354:
330:
317:
312:
311:
287:
274:
269:
268:
244:
239:
238:
217:
212:
211:
189:
188:
167:
162:
161:
140:
135:
134:
111:
110:
107:
51:categorical sum
43:category theory
39:
28:
23:
22:
15:
12:
11:
5:
4913:
4911:
4903:
4902:
4892:
4891:
4885:
4884:
4882:
4881:
4871:
4861:
4850:
4847:
4846:
4843:
4842:
4839:
4838:
4836:
4835:
4830:
4825:
4811:
4805:
4800:
4795:
4789:
4787:
4780:
4779:
4776:
4775:
4773:
4772:
4767:
4756:
4754:
4749:
4744:
4743:
4741:
4740:
4735:
4730:
4725:
4720:
4715:
4704:
4702:
4697:
4689:
4685:
4684:
4682:
4677:
4675:String diagram
4672:
4667:
4665:Model category
4662:
4657:
4652:
4647:
4642:
4640:
4633:
4632:
4629:
4626:
4625:
4620:
4613:
4612:
4605:
4603:
4600:
4599:
4596:
4595:
4593:
4592:
4587:
4585:Comma category
4582:
4577:
4572:
4567:
4562:
4557:
4551:
4549:
4545:
4544:
4542:
4541:
4531:
4521:
4519:Abelian groups
4516:
4511:
4506:
4501:
4495:
4493:
4487:
4486:
4483:
4482:
4480:
4479:
4474:
4469:
4468:
4467:
4457:
4452:
4446:
4444:
4438:
4437:
4435:
4434:
4429:
4424:
4423:
4422:
4412:
4407:
4401:
4399:
4390:
4384:
4383:
4381:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4324:
4323:
4318:
4313:
4308:
4303:
4298:
4287:
4285:
4277:
4276:
4273:
4266:
4265:
4260:
4258:
4257:
4250:
4243:
4235:
4229:
4228:
4216:
4215:External links
4213:
4212:
4211:
4197:
4168:
4167:
4147:
4146:
4144:
4141:
4140:
4139:
4134:
4129:
4124:
4117:
4114:
4097:
4094:
4089:
4085:
4064:
4044:
4009:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3948:
3945:
3942:
3939:
3936:
3916:
3913:
3910:
3907:
3904:
3884:
3881:
3878:
3875:
3872:
3852:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3790:
3770:
3767:
3764:
3744:
3719:
3718:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3676:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3631:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3539:initial object
3520:
3516:
3512:
3509:
3506:
3501:
3497:
3474:
3470:
3466:
3463:
3460:
3455:
3451:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3382:
3357:
3353:
3332:
3306:
3302:
3282:
3281:
3270:
3265:
3262:
3259:
3255:
3249:
3245:
3241:
3238:
3235:
3212:
3197:
3196:
3185:
3181:
3177:
3174:
3169:
3165:
3159:
3156:
3153:
3149:
3144:
3140:
3137:
3134:
3129:
3125:
3119:
3116:
3113:
3109:
3084:(a product in
3082:
3081:
3070:
3067:
3064:
3059:
3055:
3051:
3048:
3045:
3040:
3037:
3034:
3030:
3026:
3021:
3018:
3015:
3011:
3005:
3001:
2997:
2975:
2974:
2963:
2960:
2957:
2952:
2948:
2944:
2941:
2936:
2932:
2926:
2923:
2920:
2916:
2912:
2908:
2904:
2901:
2896:
2892:
2886:
2883:
2880:
2876:
2871:
2867:
2862:
2858:
2831:
2807:
2787:
2767:
2746:
2742:
2739:
2736:
2732:
2728:
2723:
2719:
2687:
2683:
2671:
2670:
2657:
2653:
2647:
2644:
2641:
2637:
2613:
2608:
2604:
2600:
2580:
2577:
2572:
2568:
2547:
2527:
2504:
2493:initial object
2469:
2466:
2463:
2442:
2438:
2435:
2432:
2428:
2407:
2387:
2367:
2364:
2361:
2340:
2336:
2333:
2330:
2326:
2305:
2302:
2299:
2296:
2293:
2272:
2268:
2265:
2262:
2258:
2234:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2165:
2162:
2159:
2137:
2133:
2129:
2124:
2120:
2116:
2113:
2093:
2090:
2087:
2084:
2081:
2058:
2053:
2049:
2045:
2025:
2022:
2017:
2013:
2009:
2004:
2000:
1979:
1976:
1971:
1967:
1963:
1958:
1954:
1933:
1928:
1924:
1920:
1900:
1880:
1853:
1837:
1834:
1830:join operation
1826:poset category
1775:pointed spaces
1740:tensor product
1686:inclusion maps
1680:
1677:with the maps
1674:disjoint union
1671:is simply the
1664:
1661:
1646:
1642:
1619:
1615:
1609:
1606:
1603:
1599:
1578:
1575:
1572:
1569:
1566:
1544:
1539:
1535:
1529:
1526:
1523:
1519:
1496:
1492:
1486:
1483:
1480:
1476:
1454:
1449:
1445:
1441:
1431:of the family
1420:
1411:The coproduct
1390:
1387:
1384:
1360:
1355:
1351:
1347:
1344:
1341:
1336:
1332:
1311:
1308:
1305:
1302:
1299:
1279:
1276:
1271:
1267:
1263:
1258:
1254:
1233:
1213:
1210:
1205:
1201:
1197:
1192:
1188:
1164:
1143:
1139:
1136:
1133:
1130:
1125:
1121:
1116:
1095:
1092:
1067:
1049:
1045:
1022:
1018:
1008:The morphisms
997:
993:
987:
983:
979:
974:
970:
965:
944:
939:
935:
931:
926:
922:
902:
897:
893:
889:
884:
880:
860:
855:
851:
847:
842:
838:
817:
783:
778:
774:
770:
767:
764:
759:
755:
732:
728:
724:
721:
718:
713:
709:
688:
685:
680:
676:
672:
667:
663:
659:
656:
636:
633:
630:
625:
621:
617:
612:
608:
587:
584:
579:
575:
571:
566:
562:
541:
515:
511:
507:
502:
498:
494:
489:
485:
481:
476:
472:
449:
445:
441:
436:
432:
428:
423:
419:
415:
410:
406:
385:
380:
376:
372:
367:
363:
342:
337:
333:
329:
324:
320:
299:
294:
290:
286:
281:
277:
256:
251:
247:
224:
220:
199:
196:
187:be objects of
174:
170:
147:
143:
118:
106:
103:
55:disjoint union
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4912:
4901:
4898:
4897:
4895:
4880:
4872:
4870:
4862:
4860:
4852:
4851:
4848:
4834:
4831:
4829:
4826:
4824:
4820:
4816:
4812:
4810:
4808:
4801:
4799:
4796:
4794:
4791:
4790:
4788:
4785:
4781:
4771:
4768:
4765:
4761:
4758:
4757:
4755:
4753:
4745:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4723:Tetracategory
4721:
4719:
4716:
4713:
4712:pseudofunctor
4709:
4706:
4705:
4703:
4701:
4693:
4690:
4686:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4637:
4631:
4630:
4627:
4623:
4618:
4614:
4609:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4555:Free category
4553:
4552:
4550:
4546:
4539:
4538:Vector spaces
4535:
4532:
4529:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4496:
4494:
4492:
4488:
4478:
4475:
4473:
4470:
4466:
4463:
4462:
4461:
4458:
4456:
4453:
4451:
4448:
4447:
4445:
4443:
4439:
4433:
4432:Inverse limit
4430:
4428:
4425:
4421:
4418:
4417:
4416:
4413:
4411:
4408:
4406:
4403:
4402:
4400:
4398:
4394:
4391:
4389:
4385:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4358:Kan extension
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4293:
4292:
4289:
4288:
4286:
4282:
4278:
4271:
4267:
4263:
4256:
4251:
4249:
4244:
4242:
4237:
4236:
4233:
4226:
4225:Jocelyn Paine
4222:
4219:
4218:
4214:
4208:
4204:
4200:
4198:0-387-98403-8
4194:
4190:
4186:
4182:
4181:
4176:
4172:
4171:
4163:
4159:
4152:
4149:
4142:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4119:
4115:
4113:
4111:
4095:
4087:
4083:
4062:
4042:
4033:
4031:
4027:
4023:
4019:
4015:
4008:
4004:
4000:
3984:
3981:
3978:
3972:
3969:
3966:
3946:
3940:
3937:
3934:
3914:
3908:
3905:
3902:
3882:
3879:
3876:
3873:
3870:
3850:
3830:
3827:
3824:
3818:
3815:
3812:
3804:
3788:
3768:
3762:
3742:
3735:
3730:
3728:
3724:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3677:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3632:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3591:
3588:
3585:
3579:
3573:
3570:
3567:
3561:
3558:
3551:
3550:
3549:
3548:
3544:
3540:
3536:
3518:
3514:
3510:
3507:
3504:
3499:
3495:
3472:
3468:
3464:
3461:
3458:
3453:
3449:
3425:
3422:
3419:
3416:
3413:
3407:
3404:
3396:
3380:
3371:
3355:
3351:
3330:
3322:
3304:
3300:
3292:
3288:
3268:
3263:
3260:
3257:
3247:
3243:
3239:
3236:
3226:
3225:
3224:
3210:
3202:
3183:
3179:
3175:
3172:
3167:
3163:
3157:
3154:
3151:
3147:
3142:
3138:
3135:
3132:
3127:
3123:
3117:
3114:
3111:
3107:
3099:
3098:
3097:
3095:
3091:
3087:
3065:
3062:
3057:
3053:
3046:
3043:
3038:
3035:
3032:
3028:
3024:
3019:
3016:
3013:
3003:
2999:
2988:
2987:
2986:
2985:of morphisms
2984:
2980:
2977:given by the
2958:
2955:
2950:
2946:
2939:
2934:
2930:
2924:
2921:
2918:
2914:
2910:
2906:
2902:
2899:
2894:
2890:
2884:
2881:
2878:
2874:
2869:
2865:
2860:
2856:
2848:
2847:
2846:
2845:
2842:), we have a
2829:
2821:
2805:
2785:
2765:
2744:
2740:
2737:
2734:
2730:
2726:
2721:
2717:
2707:
2705:
2704:
2685:
2681:
2673:and the maps
2655:
2651:
2645:
2642:
2639:
2635:
2627:
2626:
2625:
2606:
2602:
2578:
2570:
2566:
2545:
2525:
2516:
2502:
2494:
2490:
2487:(that is, an
2486:
2481:
2467:
2464:
2461:
2440:
2436:
2433:
2430:
2426:
2385:
2365:
2362:
2359:
2338:
2334:
2331:
2328:
2324:
2303:
2297:
2294:
2291:
2270:
2266:
2263:
2260:
2256:
2248:
2232:
2224:
2208:
2205:
2202:
2196:
2193:
2182:
2177:
2163:
2160:
2157:
2135:
2131:
2127:
2122:
2118:
2114:
2111:
2091:
2085:
2082:
2079:
2072:
2051:
2047:
2023:
2015:
2011:
2007:
2002:
1998:
1977:
1969:
1965:
1961:
1956:
1952:
1926:
1922:
1898:
1878:
1871:
1867:
1851:
1843:
1835:
1833:
1831:
1827:
1822:
1820:
1816:
1814:
1808:
1804:
1803:Banach spaces
1799:
1797:
1792:
1786:
1784:
1780:
1776:
1772:
1768:
1764:
1760:
1755:
1753:
1749:
1747:
1741:
1737:
1735:
1729:
1726:
1721:
1719:
1715:
1714:
1709:
1708:vector spaces
1705:
1701:
1700:
1696:, called the
1695:
1691:
1687:
1683:
1676:
1675:
1670:
1662:
1660:
1644:
1640:
1617:
1613:
1607:
1604:
1601:
1597:
1576:
1570:
1567:
1564:
1555:
1542:
1537:
1533:
1527:
1524:
1521:
1517:
1494:
1490:
1484:
1481:
1478:
1474:
1452:
1447:
1443:
1439:
1418:
1406:
1402:
1388:
1385:
1382:
1374:
1358:
1353:
1349:
1345:
1342:
1339:
1334:
1330:
1309:
1303:
1300:
1297:
1277:
1269:
1265:
1261:
1256:
1252:
1231:
1211:
1203:
1199:
1195:
1190:
1186:
1178:
1162:
1155:is an object
1141:
1137:
1134:
1131:
1128:
1123:
1119:
1114:
1093:
1090:
1082:
1077:
1075:
1071:
1066:
1063:
1047:
1043:
1020:
1016:
995:
991:
985:
981:
977:
972:
968:
963:
942:
937:
933:
929:
924:
920:
900:
895:
891:
887:
882:
878:
858:
853:
849:
845:
840:
836:
815:
803:
799:
797:
781:
776:
772:
768:
765:
762:
757:
753:
730:
726:
722:
719:
716:
711:
707:
686:
678:
674:
670:
665:
661:
657:
654:
634:
631:
623:
619:
615:
610:
606:
585:
577:
573:
569:
564:
560:
539:
531:
513:
509:
505:
500:
496:
487:
483:
479:
474:
470:
447:
443:
439:
434:
430:
421:
417:
413:
408:
404:
383:
378:
374:
370:
365:
361:
340:
335:
331:
327:
322:
318:
297:
292:
288:
284:
279:
275:
254:
249:
245:
222:
218:
197:
194:
172:
168:
145:
141:
132:
116:
104:
102:
100:
96:
92:
88:
84:
83:vector spaces
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
37:
33:
19:
4803:
4784:Categorified
4688:n-categories
4639:Key concepts
4477:Direct limit
4460:Coequalizers
4454:
4378:Yoneda lemma
4284:Key concepts
4274:Key concepts
4178:
4161:
4151:
4137:Direct limit
4109:
4034:
4018:monomorphism
4014:pointed sets
4006:
3998:
3731:
3720:
3542:
3534:
3372:
3320:
3283:
3198:
3085:
3083:
2976:
2818:(that is, a
2708:
2701:
2672:
2517:
2488:
2482:
2247:ordered pair
2179:As with any
2178:
1839:
1823:
1812:
1800:
1787:
1770:
1756:
1745:
1733:
1727:
1722:
1711:
1699:free product
1697:
1678:
1672:
1666:
1556:
1410:
1078:
807:
108:
67:free product
50:
46:
40:
4752:-categories
4728:Kan complex
4718:Tricategory
4700:-categories
4590:Subcategory
4348:Exponential
4316:Preadditive
4311:Pre-abelian
4132:Coequalizer
4003:epimorphism
3734:zero object
3287:hom-functor
2071:isomorphism
1062:are called
91:dual notion
4770:3-category
4760:2-category
4733:∞-groupoid
4708:Bicategory
4455:Coproducts
4415:Equalizers
4321:Bicategory
4207:0906.18001
4143:References
4020:. In any
3395:finite set
3201:surjection
2104:such that
1836:Discussion
1807:short maps
1713:direct sum
1684:being the
1322:such that
1070:injections
699:such that
105:Definition
75:direct sum
73:, and the
36:by-product
18:Coproducts
4819:Symmetric
4764:2-functor
4504:Relations
4427:Pullbacks
4110:covariant
4093:→
4026:biproduct
4005:while in
3982:×
3976:→
3970:⊕
3944:→
3938:⊕
3912:→
3906:⊕
3880:≅
3874:⊕
3843:. Since
3828:⊕
3822:→
3816:⊕
3766:→
3700:⊕
3694:≅
3688:⊕
3661:≅
3655:⊕
3649:≅
3643:⊕
3616:⊕
3610:⊕
3604:≅
3598:⊕
3589:⊕
3580:≅
3571:⊕
3562:⊕
3511:⊕
3508:…
3505:⊕
3462:…
3420:…
3261:∈
3240:∘
3155:∈
3148:∐
3139:
3133:∈
3115:∈
3108:∐
3047:
3036:∈
3029:∏
3025:∈
3017:∈
2979:bijection
2940:
2922:∈
2915:∏
2911:≅
2882:∈
2875:∐
2866:
2727:
2643:∈
2636:∐
2576:→
2485:empty set
2465:×
2406:Δ
2316:the pair
2301:→
2206:×
2200:→
2191:Δ
2161:∈
2150:for each
2115:∘
2089:→
2021:→
1975:→
1819:unit ball
1783:wedge sum
1769:are sets
1767:open sets
1748:-algebras
1742:. In the
1736:-algebras
1688:. Unlike
1605:∈
1598:∐
1574:→
1525:∈
1518:⨁
1482:∈
1475:∐
1386:∈
1375:for each
1346:∘
1307:→
1275:→
1209:→
1177:morphisms
1135:∈
888:⊕
846:⊔
769:∘
723:∘
684:→
671:⊔
629:→
583:→
506:⊔
493:→
440:⊔
427:→
328:⊕
285:⊔
47:coproduct
32:Coalgebra
4894:Category
4879:Glossary
4859:Category
4833:n-monoid
4786:concepts
4442:Colimits
4410:Products
4363:Morphism
4306:Concrete
4301:Additive
4291:Category
4177:(1998).
4116:See also
3803:terminal
2709:Letting
1723:Given a
1718:finitely
1663:Examples
1373:commutes
1072:or even
796:commutes
267:written
133:and let
131:category
87:morphism
4869:Outline
4828:n-group
4793:2-group
4748:Strict
4738:∞-topos
4534:Modules
4472:Pushout
4420:Kernels
4353:Functor
4296:Abelian
4122:Product
3781:(since
2820:hom-set
2221:be the
1868:from a
1866:functor
1842:colimit
1828:is the
1809:is the
1791:spanned
1738:is the
93:to the
79:modules
4815:Traced
4798:2-ring
4528:Fields
4514:Groups
4509:Magmas
4397:Limits
4205:
4195:
3723:monoid
3397:, say
3088:, the
1081:family
99:arrows
71:groups
65:, the
45:, the
4809:-ring
4696:Weak
4680:Topos
4524:Rings
3393:is a
2983:tuple
1891:into
1805:with
1074:monic
129:be a
49:, or
4499:Sets
4193:ISBN
3927:and
2245:the
1990:and
1035:and
745:and
598:and
462:and
237:and
160:and
109:Let
81:and
61:and
59:sets
4343:End
4333:CCC
4203:Zbl
4007:Set
3999:Grp
3801:is
3541:of
3373:If
3370:).
3321:Set
3319:to
3136:Hom
3086:Set
3044:Hom
2931:Hom
2857:Hom
2822:in
2798:in
2778:to
2718:Hom
2518:If
2495:in
2454:in
2378:in
1754:).
1659:s.
1509:or
955:or
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