3080:
337:
270:
3327:
3347:
3337:
1700:
1450:
1847:
359:. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, though there may be other ways to construct it, they are all equivalent.
1695:{\displaystyle A\otimes _{C}B=\left\{\sum _{i\in I}(a_{i},b_{i})\;{\big |}\;a_{i}\in A,b_{i}\in B\right\}{\Bigg /}{\bigg \langle }(f(c)a,b)-(a,g(c)b)\;{\big |}\;a\in A,b\in B,c\in C{\bigg \rangle }}
1323:
1273:
2561:
2488:
1371:
771:
2640:
pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points to give a generalisation of the
Seifert-van Kampen Theorem.
724:
498:
613:
1762:
1749:
162:
1223:
648:
907:
797:
686:
201:
1077:
1045:
964:
2187:
Graphically this means that two pushout squares, placed side by side and sharing one morphism, form a larger pushout square when ignoring the inner shared morphism.
125:
of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are
2680:
2583:. Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when
2724:
2231:
Pushouts can be constructed from coproducts and coequalizers, as described below (the pushout is the coequalizer of the maps to the coproduct).
1709:
2591:, since then both homomorphisms above have trivial domain. Indeed, this is the case, since then the pushout (of groups) reduces to the
3371:
2627:
An introduction to categorical approaches to algebraic topology: the focus is on the algebra, and assumes a topological background.
2235:
All of the above examples may be regarded as special cases of the following very general construction, which works in any category
1162:
2596:
1158:
821:
2717:
2921:
2876:
2634:
982:
3350:
3290:
2228:
is the pushout of and , so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
2493:
2423:
1278:
1228:
3340:
3126:
2990:
2898:
2367:
1390:
1382:
978:
211:
729:
3299:
2881:
2804:
557:
52:
3330:
3286:
2891:
2710:
691:
87:
2886:
2868:
986:
456:
3093:
2859:
2839:
2762:
1842:{\displaystyle \left({\frac {\operatorname {lcm} (m,n)}{m}},{\frac {\operatorname {lcm} (m,n)}{n}}\right)}
1386:
1378:
356:
207:
1328:
2975:
2814:
1850:
586:
509:
1725:
2787:
2782:
505:
128:
83:
2595:, which is the coproduct in the category of groups. In a most general case we will be speaking of a
1195:
621:
3131:
3079:
3009:
3005:
2809:
1136:
998:
276:
110:
336:
2985:
2980:
2962:
2844:
2819:
1166:
1154:
1144:
879:
581:
298:
122:
1574:
776:
665:
167:
3294:
3231:
3219:
3121:
3046:
3041:
2999:
2995:
2777:
2772:
2401:
1434:
1185:
1177:
1050:
1018:
829:
376:
943:
803:. Cographs are dual to graphs of functions since the graph may be defined as the pullback of
3255:
3141:
3116:
3051:
3036:
3031:
2970:
2799:
2767:
2646:
2588:
1927:
1181:
817:
3167:
2733:
2373:
1104:
20:
3204:
343:
As with all universal constructions, the pushout, if it exists, is unique up to a unique
2681:"Does the concept of "cograph of a function" have natural generalisations / Extensions?"
3199:
3183:
3146:
3136:
3056:
2603:
2205:
1189:
1100:
412:
2392:
is also path-connected. (Assume also that the basepoint * lies in the intersection of
3365:
3194:
3026:
2903:
2829:
1002:
933:
917:. More generally, all identification spaces may be regarded as pushouts in this way.
849:
16:
Most general completion of a commutative square given two morphisms with same domain
2948:
2849:
2592:
1013:
269:
3209:
2372:
The
Seifert–van Kampen theorem answers the following question. Suppose we have a
1751:, considered as a category with one object, the pushout of two positive integers
3189:
3061:
2931:
2201:
344:
24:
773:. A function may be recovered by its cograph because each equivalence class in
3241:
3179:
2792:
990:
1393:(see the examples section) as dual notions to each other. In particular, let
3235:
2926:
2602:
There is a detailed exposition of this, in a slightly more general setting (
2197:
921:
3304:
2936:
2834:
2618:
2606:
2212:
Coproducts are a pushout from the initial object, and the coequalizer of
1931:
994:
424:
56:
3274:
3264:
2913:
2824:
2697:
2637:
1720:
1374:
48:
319:) for which the following diagram commutes, there must exist a unique
3269:
2420:? The answer is yes, provided we also know the induced homomorphisms
1716:
545:
3151:
2702:
301:
with respect to this diagram. That is, for any other such triple (
3091:
2744:
2706:
985:
with gluing" in the same way we think of adjunction spaces as "
2649:
Explains some uses of groupoids in group theory and topology.
2274:
with the same domain and the same target, the coequalizer of
618:
A specific case of this is the cograph of a function. If
580:, then the pushout can be canonically identified with the
1389:, we can think of the tensor product of rings and the
662:. In elementary terms, the cograph is the quotient of
2609:) in the book by J. P. May listed in the references.
2567:
is the pushout of these two induced maps. Of course,
2496:
2426:
2338:, then include into the coproduct, or we can go from
1987:) is an isomorphism, and so is the natural map coker(
1765:
1728:
1453:
1331:
1281:
1231:
1198:
1053:
1021:
946:
882:
779:
732:
694:
688:
by the equivalence relation generated by identifying
668:
624:
589:
459:
170:
131:
2563:
The theorem then says that the fundamental group of
3254:
3218:
3166:
3159:
3110:
3019:
2961:
2912:
2867:
2858:
2755:
2555:
2482:
2350:, then include into the coproduct. The pushout of
2290:In this setup, we obtain the pushout of morphisms
1897:exists as well and there is a natural isomorphism
1841:
1743:
1694:
1365:
1317:
1267:
1217:
1071:
1039:
958:
901:
791:
765:
718:
680:
642:
607:
492:
195:
156:
1687:
1581:
1095:are arbitrary homomorphisms from a common domain
1079:. The pushout of these maps is the direct sum of
486:
1405:be objects (commutative rings with identity) in
966:, the space obtained by gluing the basepoint of
2556:{\displaystyle \pi _{1}(D,*)\to \pi _{1}(B,*).}
1861:. Note that the same pair is also the pullback.
1135:. A similar approach yields the pushout in the
355:Here are some examples of pushouts in familiar
2483:{\displaystyle \pi _{1}(D,*)\to \pi _{1}(A,*)}
2314:by first forming the coproduct of the targets
1318:{\displaystyle f':B\rightarrow A\otimes _{C}B}
1268:{\displaystyle g':A\rightarrow A\otimes _{C}B}
431:) are identified, together with the morphisms
2718:
1646:
1526:
1123:)). Thus we have "glued" along the images of
8:
2571:is the pushout of the two inclusion maps of
2192:Construction via coproducts and coequalizers
766:{\displaystyle f(x)\in Y\subseteq X\sqcup Y}
2380:, covered by path-connected open subspaces
2362:Application: the Seifert–van Kampen theorem
3346:
3336:
3163:
3107:
3088:
2864:
2752:
2741:
2725:
2711:
2703:
2416:, can we recover the fundamental group of
1651:
1643:
1531:
1523:
2647:"Categories and Groupoids" free download
2529:
2501:
2495:
2459:
2431:
2425:
2326:to this coproduct. We can either go from
1804:
1771:
1764:
1735:
1730:
1727:
1686:
1685:
1645:
1644:
1580:
1579:
1573:
1572:
1555:
1536:
1525:
1524:
1514:
1501:
1482:
1461:
1452:
1330:
1306:
1280:
1256:
1230:
1206:
1197:
1052:
1020:
945:
940:the one-point space. Then the pushout is
890:
881:
778:
731:
719:{\displaystyle x\in X\subseteq X\sqcup Y}
693:
667:
623:
588:
481:
458:
222:Explicitly, the pushout of the morphisms
184:
169:
145:
130:
2659:
2622:A concise course in algebraic topology.
2358:is the coequalizer of these new maps.
1930:all pushouts exist, and they preserve
1712:for the case of non-commutative rings.
1107:by the subgroup consisting of pairs (
876:. The result is the adjunction space
493:{\displaystyle P=(X\sqcup Y)/\!\sim }
7:
1849:, where the numerators are both the
1710:Free product of associative algebras
1373:. In fact, since the pushout is the
2624:University of Chicago Press, 1999.
1366:{\displaystyle f'\circ g=g'\circ f}
920:A special case of the above is the
2322:. We then have two morphisms from
799:contains precisely one element of
608:{\displaystyle X\cup Y\subseteq W}
423:, where elements sharing a common
403:are set functions. The pushout of
14:
1087:. Generalizing to the case where
981:, pushouts can be thought of as "
924:or one-point union; here we take
820:is an example of pushouts in the
331:also making the diagram commute:
3345:
3335:
3326:
3325:
3078:
2001:There is a natural isomorphism (
1744:{\displaystyle \mathbf {Z} _{+}}
1731:
1099:, one obtains for the pushout a
654:of a function is the pushout of
335:
268:
1188:), the pushout is given by the
909:, which is just the pushout of
658:along the identity function of
157:{\displaystyle P=X\sqcup _{Z}Y}
2597:free product with amalgamation
2547:
2535:
2522:
2519:
2507:
2477:
2465:
2452:
2449:
2437:
1825:
1813:
1792:
1780:
1640:
1634:
1628:
1616:
1610:
1598:
1592:
1586:
1520:
1494:
1296:
1246:
1218:{\displaystyle A\otimes _{C}B}
1159:free product with amalgamation
1103:of the direct sum; namely, we
1063:
1031:
822:category of topological spaces
742:
736:
643:{\displaystyle f\colon X\to Y}
634:
478:
466:
1:
1976:, then the natural map coker(
1441:. Then the tensor product is:
113:with the two given morphisms
86:. The pushout consists of an
2255:, their coproduct exists in
1934:in the following sense: if (
1157:, the pushout is called the
3020:Constructions on categories
2196:Pushouts are equivalent to
902:{\displaystyle X\cup _{f}Y}
506:finest equivalence relation
3388:
3127:Higher-dimensional algebra
2668:Category Theory in Context
2368:Seifert-van Kampen theorem
2365:
2036:. Explicitly, this means:
1163:Seifert–van Kampen theorem
979:category of abelian groups
3321:
3100:
3087:
3076:
2751:
2740:
2412:, and their intersection
864:using an "attaching map"
792:{\displaystyle X\sqcup Y}
681:{\displaystyle X\sqcup Y}
196:{\displaystyle P=X+_{Z}Y}
93:along with two morphisms
3372:Limits (category theory)
2638:"Topology and Groupoids"
1391:fibered product of rings
1072:{\displaystyle g:0\to B}
1040:{\displaystyle f:0\to A}
650:is a function, then the
121:. In fact, the defining
2937:Cokernels and quotients
2860:Universal constructions
959:{\displaystyle X\vee Y}
3094:Higher category theory
2840:Natural transformation
2557:
2484:
1843:
1745:
1715:In the multiplicative
1696:
1367:
1319:
1269:
1219:
1073:
1041:
960:
903:
807:along the identity of
793:
767:
720:
682:
644:
609:
568:the inclusion maps of
494:
264:such that the diagram
230:consists of an object
197:
158:
2558:
2485:
2208:) in the sense that:
1871:Whenever the pushout
1851:least common multiple
1844:
1746:
1697:
1368:
1320:
1270:
1220:
1161:. It shows up in the
1074:
1042:
961:
904:
824:. More precisely, if
794:
768:
721:
683:
645:
610:
495:
198:
159:
2963:Algebraic categories
2494:
2424:
2148:then the pushout of
1952:) is the pushout of
1763:
1726:
1451:
1329:
1279:
1229:
1196:
1051:
1019:
970:to the basepoint of
944:
880:
816:The construction of
777:
730:
692:
666:
622:
587:
536:. In particular, if
457:
351:Examples of pushouts
168:
129:
3132:Homotopy hypothesis
2810:Commutative diagram
2645:Philip J. Higgins,
2388:whose intersection
1225:with the morphisms
548:of some larger set
399: →
387: →
206:The pushout is the
2845:Universal property
2553:
2480:
2402:fundamental groups
2400:.) If we know the
2262:For any morphisms
1839:
1741:
1692:
1493:
1435:ring homomorphisms
1385:is the limit of a
1363:
1315:
1265:
1215:
1176:, the category of
1167:algebraic topology
1155:category of groups
1069:
1037:
989:with gluing". The
956:
899:
789:
763:
716:
678:
640:
605:
490:
234:and two morphisms
218:Universal property
193:
154:
123:universal property
111:commutative square
55:consisting of two
41:cocartesian square
3359:
3358:
3317:
3316:
3313:
3312:
3295:monoidal category
3250:
3249:
3122:Enriched category
3074:
3073:
3070:
3069:
3047:Quotient category
3042:Opposite category
2957:
2956:
1832:
1799:
1759:is just the pair
1478:
1186:category of rings
1178:commutative rings
856:to another space
818:adjunction spaces
33:fibered coproduct
3379:
3349:
3348:
3339:
3338:
3329:
3328:
3164:
3142:Simplex category
3117:Categorification
3108:
3089:
3082:
3052:Product category
3037:Kleisli category
3032:Functor category
2877:Terminal objects
2865:
2800:Adjoint functors
2753:
2742:
2727:
2720:
2713:
2704:
2698:pushout in nLab
2685:
2684:
2677:
2671:
2664:
2589:simply connected
2562:
2560:
2559:
2554:
2534:
2533:
2506:
2505:
2489:
2487:
2486:
2481:
2464:
2463:
2436:
2435:
2243:For any objects
2204:(if there is an
1991:) → coker(
1980:) → coker(
1928:abelian category
1848:
1846:
1845:
1840:
1838:
1834:
1833:
1828:
1805:
1800:
1795:
1772:
1750:
1748:
1747:
1742:
1740:
1739:
1734:
1701:
1699:
1698:
1693:
1691:
1690:
1650:
1649:
1585:
1584:
1578:
1577:
1571:
1567:
1560:
1559:
1541:
1540:
1530:
1529:
1519:
1518:
1506:
1505:
1492:
1466:
1465:
1372:
1370:
1369:
1364:
1356:
1339:
1324:
1322:
1321:
1316:
1311:
1310:
1289:
1274:
1272:
1271:
1266:
1261:
1260:
1239:
1224:
1222:
1221:
1216:
1211:
1210:
1182:full subcategory
1078:
1076:
1075:
1070:
1046:
1044:
1043:
1038:
965:
963:
962:
957:
908:
906:
905:
900:
895:
894:
810:
806:
802:
798:
796:
795:
790:
772:
770:
769:
764:
725:
723:
722:
717:
687:
685:
684:
679:
661:
657:
649:
647:
646:
641:
614:
612:
611:
606:
499:
497:
496:
491:
485:
339:
272:
208:categorical dual
202:
200:
199:
194:
189:
188:
163:
161:
160:
155:
150:
149:
109:that complete a
3387:
3386:
3382:
3381:
3380:
3378:
3377:
3376:
3362:
3361:
3360:
3355:
3309:
3279:
3246:
3223:
3214:
3171:
3155:
3106:
3096:
3083:
3066:
3015:
2953:
2922:Initial objects
2908:
2854:
2747:
2736:
2734:Category theory
2731:
2694:
2689:
2688:
2679:
2678:
2674:
2665:
2661:
2656:
2615:
2525:
2497:
2492:
2491:
2455:
2427:
2422:
2421:
2370:
2364:
2194:
2113:the pushout of
2078:the pushout of
2032:
2019:
2010:
1997:
1986:
1951:
1944:
1919:
1906:
1893:
1880:
1868:
1806:
1773:
1770:
1766:
1761:
1760:
1729:
1724:
1723:
1551:
1532:
1510:
1497:
1477:
1473:
1457:
1449:
1448:
1349:
1332:
1327:
1326:
1302:
1282:
1277:
1276:
1252:
1232:
1227:
1226:
1202:
1194:
1193:
1049:
1048:
1017:
1016:
942:
941:
886:
878:
877:
808:
804:
800:
775:
774:
728:
727:
690:
689:
664:
663:
659:
655:
620:
619:
585:
584:
455:
454:
444:
437:
353:
318:
311:
296:
289:
279:and such that (
255:
240:
220:
180:
166:
165:
141:
127:
126:
45:amalgamated sum
31:(also called a
21:category theory
17:
12:
11:
5:
3385:
3383:
3375:
3374:
3364:
3363:
3357:
3356:
3354:
3353:
3343:
3333:
3322:
3319:
3318:
3315:
3314:
3311:
3310:
3308:
3307:
3302:
3297:
3283:
3277:
3272:
3267:
3261:
3259:
3252:
3251:
3248:
3247:
3245:
3244:
3239:
3228:
3226:
3221:
3216:
3215:
3213:
3212:
3207:
3202:
3197:
3192:
3187:
3176:
3174:
3169:
3161:
3157:
3156:
3154:
3149:
3147:String diagram
3144:
3139:
3137:Model category
3134:
3129:
3124:
3119:
3114:
3112:
3105:
3104:
3101:
3098:
3097:
3092:
3085:
3084:
3077:
3075:
3072:
3071:
3068:
3067:
3065:
3064:
3059:
3057:Comma category
3054:
3049:
3044:
3039:
3034:
3029:
3023:
3021:
3017:
3016:
3014:
3013:
3003:
2993:
2991:Abelian groups
2988:
2983:
2978:
2973:
2967:
2965:
2959:
2958:
2955:
2954:
2952:
2951:
2946:
2941:
2940:
2939:
2929:
2924:
2918:
2916:
2910:
2909:
2907:
2906:
2901:
2896:
2895:
2894:
2884:
2879:
2873:
2871:
2862:
2856:
2855:
2853:
2852:
2847:
2842:
2837:
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2796:
2795:
2790:
2785:
2780:
2775:
2770:
2759:
2757:
2749:
2748:
2745:
2738:
2737:
2732:
2730:
2729:
2722:
2715:
2707:
2701:
2700:
2693:
2692:External links
2690:
2687:
2686:
2672:
2658:
2657:
2655:
2652:
2651:
2650:
2642:
2641:
2631:
2630:
2629:
2628:
2614:
2611:
2552:
2549:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2521:
2518:
2515:
2512:
2509:
2504:
2500:
2479:
2476:
2473:
2470:
2467:
2462:
2458:
2454:
2451:
2448:
2445:
2442:
2439:
2434:
2430:
2374:path-connected
2366:Main article:
2363:
2360:
2288:
2287:
2260:
2233:
2232:
2229:
2206:initial object
2193:
2190:
2189:
2188:
2184:
2183:
2182:
2181:
2146:
2111:
2076:
2028:
2015:
2006:
1999:
1995:
1984:
1949:
1942:
1924:
1915:
1902:
1889:
1876:
1867:
1864:
1863:
1862:
1837:
1831:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1803:
1798:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1769:
1738:
1733:
1713:
1705:
1704:
1703:
1702:
1689:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1648:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1583:
1576:
1570:
1566:
1563:
1558:
1554:
1550:
1547:
1544:
1539:
1535:
1528:
1522:
1517:
1513:
1509:
1504:
1500:
1496:
1491:
1488:
1485:
1481:
1476:
1472:
1469:
1464:
1460:
1456:
1443:
1442:
1433:be morphisms (
1362:
1359:
1355:
1352:
1348:
1345:
1342:
1338:
1335:
1314:
1309:
1305:
1301:
1298:
1295:
1292:
1288:
1285:
1264:
1259:
1255:
1251:
1248:
1245:
1242:
1238:
1235:
1214:
1209:
1205:
1201:
1190:tensor product
1170:
1151:
1101:quotient group
1068:
1065:
1062:
1059:
1056:
1036:
1033:
1030:
1027:
1024:
1003:abelian groups
987:disjoint union
975:
955:
952:
949:
934:pointed spaces
918:
898:
893:
889:
885:
852:we can "glue"
814:
813:
812:
788:
785:
782:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
715:
712:
709:
706:
703:
700:
697:
677:
674:
671:
639:
636:
633:
630:
627:
604:
601:
598:
595:
592:
520:) ~
489:
484:
480:
477:
474:
471:
468:
465:
462:
442:
435:
413:disjoint union
352:
349:
341:
340:
316:
309:
294:
287:
274:
273:
253:
238:
219:
216:
192:
187:
183:
179:
176:
173:
153:
148:
144:
140:
137:
134:
82:with a common
23:, a branch of
15:
13:
10:
9:
6:
4:
3:
2:
3384:
3373:
3370:
3369:
3367:
3352:
3344:
3342:
3334:
3332:
3324:
3323:
3320:
3306:
3303:
3301:
3298:
3296:
3292:
3288:
3284:
3282:
3280:
3273:
3271:
3268:
3266:
3263:
3262:
3260:
3257:
3253:
3243:
3240:
3237:
3233:
3230:
3229:
3227:
3225:
3217:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3195:Tetracategory
3193:
3191:
3188:
3185:
3184:pseudofunctor
3181:
3178:
3177:
3175:
3173:
3165:
3162:
3158:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3109:
3103:
3102:
3099:
3095:
3090:
3086:
3081:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3027:Free category
3025:
3024:
3022:
3018:
3011:
3010:Vector spaces
3007:
3004:
3001:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2968:
2966:
2964:
2960:
2950:
2947:
2945:
2942:
2938:
2935:
2934:
2933:
2930:
2928:
2925:
2923:
2920:
2919:
2917:
2915:
2911:
2905:
2904:Inverse limit
2902:
2900:
2897:
2893:
2890:
2889:
2888:
2885:
2883:
2880:
2878:
2875:
2874:
2872:
2870:
2866:
2863:
2861:
2857:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2830:Kan extension
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2764:
2761:
2760:
2758:
2754:
2750:
2743:
2739:
2735:
2728:
2723:
2721:
2716:
2714:
2709:
2708:
2705:
2699:
2696:
2695:
2691:
2682:
2676:
2673:
2669:
2663:
2660:
2653:
2648:
2644:
2643:
2639:
2636:
2633:
2632:
2626:
2625:
2623:
2620:
2617:
2616:
2612:
2610:
2608:
2605:
2600:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2566:
2550:
2544:
2541:
2538:
2530:
2526:
2516:
2513:
2510:
2502:
2498:
2474:
2471:
2468:
2460:
2456:
2446:
2443:
2440:
2432:
2428:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2369:
2361:
2359:
2357:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2285:
2281:
2277:
2273:
2269:
2265:
2261:
2258:
2254:
2250:
2246:
2242:
2241:
2240:
2238:
2230:
2227:
2223:
2219:
2215:
2211:
2210:
2209:
2207:
2203:
2199:
2191:
2186:
2185:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2144:
2140:
2136:
2132:
2128:
2124:
2121:is given by
2120:
2116:
2112:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2075:are given and
2074:
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2037:
2035:
2031:
2026:
2022:
2018:
2013:
2009:
2004:
2000:
1994:
1990:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1948:
1941:
1937:
1933:
1929:
1925:
1922:
1918:
1913:
1909:
1905:
1900:
1896:
1892:
1887:
1884:exists, then
1883:
1879:
1874:
1870:
1869:
1865:
1860:
1856:
1852:
1835:
1829:
1822:
1819:
1816:
1810:
1807:
1801:
1796:
1789:
1786:
1783:
1777:
1774:
1767:
1758:
1754:
1736:
1722:
1718:
1714:
1711:
1707:
1706:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1637:
1631:
1625:
1622:
1619:
1613:
1607:
1604:
1601:
1595:
1589:
1568:
1564:
1561:
1556:
1552:
1548:
1545:
1542:
1537:
1533:
1515:
1511:
1507:
1502:
1498:
1489:
1486:
1483:
1479:
1474:
1470:
1467:
1462:
1458:
1454:
1447:
1446:
1445:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1360:
1357:
1353:
1350:
1346:
1343:
1340:
1336:
1333:
1325:that satisfy
1312:
1307:
1303:
1299:
1293:
1290:
1286:
1283:
1262:
1257:
1253:
1249:
1243:
1240:
1236:
1233:
1212:
1207:
1203:
1199:
1191:
1187:
1183:
1179:
1175:
1171:
1168:
1164:
1160:
1156:
1152:
1149:
1146:
1142:
1140:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1066:
1060:
1057:
1054:
1034:
1028:
1025:
1022:
1015:
1014:homomorphisms
1011:
1007:
1004:
1001:, so for any
1000:
996:
992:
988:
984:
980:
976:
973:
969:
953:
950:
947:
939:
935:
931:
927:
923:
919:
916:
912:
896:
891:
887:
883:
875:
871:
867:
863:
859:
855:
851:
850:inclusion map
847:
843:
839:
835:
831:
827:
823:
819:
815:
786:
783:
780:
760:
757:
754:
751:
748:
745:
739:
733:
713:
710:
707:
704:
701:
698:
695:
675:
672:
669:
653:
637:
631:
628:
625:
617:
616:
602:
599:
596:
593:
590:
583:
579:
575:
571:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
487:
482:
475:
472:
469:
463:
460:
452:
448:
441:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
395: :
394:
390:
386:
383: :
382:
378:
375:as above are
374:
370:
366:
363:Suppose that
362:
361:
360:
358:
350:
348:
346:
338:
334:
333:
332:
330:
326:
322:
315:
308:
304:
300:
293:
286:
282:
278:
271:
267:
266:
265:
263:
259:
252:
248:
244:
237:
233:
229:
225:
217:
215:
213:
209:
204:
190:
185:
181:
177:
174:
171:
151:
146:
142:
138:
135:
132:
124:
120:
116:
112:
108:
104:
100:
96:
92:
89:
85:
81:
77:
73:
69:
65:
61:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3275:
3256:Categorified
3160:n-categories
3111:Key concepts
2949:Direct limit
2943:
2932:Coequalizers
2850:Yoneda lemma
2756:Key concepts
2746:Key concepts
2675:
2667:
2662:
2635:Ronald Brown
2621:
2601:
2593:free product
2584:
2580:
2576:
2572:
2568:
2564:
2417:
2413:
2409:
2405:
2397:
2393:
2389:
2385:
2381:
2377:
2371:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2289:
2283:
2279:
2275:
2271:
2267:
2263:
2256:
2252:
2248:
2244:
2239:satisfying:
2236:
2234:
2225:
2221:
2217:
2213:
2202:coequalizers
2195:
2177:
2173:
2169:
2165:
2161:
2157:
2156:is given by
2153:
2149:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2107:
2103:
2099:
2095:
2091:
2087:
2086:is given by
2083:
2079:
2072:
2068:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2033:
2029:
2024:
2020:
2016:
2011:
2007:
2002:
1992:
1988:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1946:
1939:
1935:
1920:
1916:
1911:
1907:
1903:
1898:
1894:
1890:
1885:
1881:
1877:
1872:
1858:
1854:
1756:
1752:
1719:of positive
1438:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1173:
1169:(see below).
1147:
1138:
1137:category of
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1096:
1092:
1088:
1084:
1080:
1009:
1005:
971:
967:
937:
929:
925:
914:
910:
873:
869:
865:
861:
857:
853:
845:
841:
837:
833:
825:
651:
577:
573:
569:
565:
561:
558:intersection
553:
549:
541:
537:
533:
529:
525:
521:
517:
513:
512:) such that
501:
450:
446:
439:
432:
428:
420:
416:
408:
404:
400:
396:
392:
388:
384:
380:
372:
368:
364:
354:
342:
328:
324:
320:
313:
306:
302:
291:
284:
280:
275:
261:
257:
250:
246:
242:
235:
231:
227:
223:
221:
205:
118:
114:
106:
102:
98:
94:
90:
79:
75:
71:
67:
63:
59:
44:
40:
36:
32:
28:
18:
3224:-categories
3200:Kan complex
3190:Tricategory
3172:-categories
3062:Subcategory
2820:Exponential
2788:Preadditive
2783:Pre-abelian
379:, and that
345:isomorphism
37:fibered sum
25:mathematics
3242:3-category
3232:2-category
3205:∞-groupoid
3180:Bicategory
2927:Coproducts
2887:Equalizers
2793:Bicategory
2654:References
2619:May, J. P.
2613:References
2282:exists in
2198:coproducts
1866:Properties
1012:, we have
991:zero group
983:direct sum
528:) for all
508:(cf. also
357:categories
3291:Symmetric
3236:2-functor
2976:Relations
2899:Pullbacks
2607:groupoids
2545:∗
2527:π
2523:→
2517:∗
2499:π
2475:∗
2457:π
2453:→
2447:∗
2429:π
2014:) ⊔
1932:cokernels
1811:
1778:
1680:∈
1668:∈
1656:∈
1614:−
1562:∈
1543:∈
1487:∈
1480:∑
1459:⊗
1358:∘
1341:∘
1304:⊗
1297:→
1254:⊗
1247:→
1204:⊗
1192:of rings
1115:), −
1064:→
1032:→
997:of every
951:∨
922:wedge sum
888:∪
784:⊔
758:⊔
752:⊆
746:∈
711:⊔
705:⊆
699:∈
673:⊔
635:→
629::
600:⊆
594:∪
556:is their
488:∼
473:⊔
299:universal
143:⊔
57:morphisms
47:) is the
3366:Category
3351:Glossary
3331:Category
3305:n-monoid
3258:concepts
2914:Colimits
2882:Products
2835:Morphism
2778:Concrete
2773:Additive
2763:Category
2670:, p. xii
2604:covering
2310:→
2306: :
2298:→
2294: :
2224:→
2220: :
2176:→
2172: :
2164:→
2160: :
2141:→
2137: :
2129:→
2125: :
2106:→
2102: :
2094:→
2090: :
2071:→
2067: :
2059:→
2055: :
2047:→
2043: :
2039:if maps
2023:≅
1972:→
1968: :
1960:→
1956: :
1914: ⊔
1910:≅
1901: ⊔
1888: ⊔
1875: ⊔
1721:integers
1688:⟩
1582:⟨
1429:→
1425: :
1417:→
1413: :
1409:and let
1383:pullback
1381:and the
1354:′
1337:′
1287:′
1237:′
1143:for any
1141:-modules
995:subgroup
872:→
868: :
844:→
840: :
830:subspace
425:preimage
327:→
323: :
277:commutes
260:→
256: :
245:→
241: :
212:pullback
105:→
97:→
78:→
74: :
66:→
62: :
3341:Outline
3300:n-group
3265:2-group
3220:Strict
3210:∞-topos
3006:Modules
2944:Pushout
2892:Kernels
2825:Functor
2768:Abelian
2666:Riehl,
1375:colimit
1184:of the
1153:In the
1105:mod out
977:In the
848:is the
652:cograph
560:, with
546:subsets
504:is the
453:, i.e.
411:is the
210:of the
53:diagram
49:colimit
29:pushout
3287:Traced
3270:2-ring
3000:Fields
2986:Groups
2981:Magmas
2869:Limits
2376:space
1926:In an
1717:monoid
1401:, and
1387:cospan
1127:under
932:to be
860:along
500:where
371:, and
88:object
84:domain
3281:-ring
3168:Weak
3152:Topos
2996:Rings
2575:into
2168:and
2110:, and
1439:CRing
1437:) in
1407:CRing
1377:of a
1174:CRing
999:group
993:is a
828:is a
726:with
582:union
572:into
445:from
297:) is
51:of a
2971:Sets
2579:and
2490:and
2396:and
2384:and
2354:and
2346:via
2334:via
2318:and
2302:and
2278:and
2266:and
2247:and
2200:and
2152:and
2133:and
2117:and
2098:and
2082:and
2063:and
1964:and
1857:and
1755:and
1708:See
1421:and
1379:span
1275:and
1145:ring
1131:and
1091:and
1083:and
1047:and
1008:and
936:and
928:and
913:and
836:and
576:and
564:and
552:and
544:are
540:and
510:this
449:and
427:(in
419:and
407:and
391:and
377:sets
249:and
226:and
164:and
117:and
101:and
70:and
27:, a
2815:End
2805:CCC
2587:is
2404:of
2342:to
2330:to
2270:of
2251:of
1853:of
1808:lcm
1775:lcm
1180:(a
1172:In
1165:of
832:of
532:in
415:of
43:or
39:or
35:or
19:In
3368::
3293:)
3289:)(
2599:.
2408:,
2216:,
2158:ki
2154:hg
2051:,
1998:).
1945:,
1938:,
1397:,
615:.
438:,
367:,
347:.
312:,
305:,
290:,
283:,
214:.
203:.
3285:(
3278:n
3276:E
3238:)
3234:(
3222:n
3186:)
3182:(
3170:n
3012:)
3008:(
3002:)
2998:(
2726:e
2719:t
2712:v
2683:.
2585:D
2581:B
2577:A
2573:D
2569:X
2565:X
2551:.
2548:)
2542:,
2539:B
2536:(
2531:1
2520:)
2514:,
2511:D
2508:(
2503:1
2478:)
2472:,
2469:A
2466:(
2461:1
2450:)
2444:,
2441:D
2438:(
2433:1
2418:X
2414:D
2410:B
2406:A
2398:B
2394:A
2390:D
2386:B
2382:A
2378:X
2356:g
2352:f
2348:g
2344:Y
2340:Z
2336:f
2332:X
2328:Z
2324:Z
2320:Y
2316:X
2312:Y
2308:Z
2304:g
2300:X
2296:Z
2292:f
2286:.
2284:C
2280:k
2276:j
2272:C
2268:k
2264:j
2259:;
2257:C
2253:C
2249:B
2245:A
2237:C
2226:Y
2222:X
2218:g
2214:f
2180:.
2178:Q
2174:D
2170:l
2166:Q
2162:A
2150:f
2145:,
2143:Q
2139:D
2135:l
2131:Q
2127:P
2123:k
2119:h
2115:j
2108:P
2104:B
2100:j
2096:P
2092:A
2088:i
2084:g
2080:f
2073:D
2069:B
2065:h
2061:B
2057:C
2053:g
2049:A
2045:C
2041:f
2034:D
2030:C
2027:⊔
2025:A
2021:D
2017:B
2012:B
2008:C
2005:⊔
2003:A
1996:1
1993:i
1989:g
1985:2
1982:i
1978:f
1974:Y
1970:Z
1966:g
1962:X
1958:Z
1954:f
1950:2
1947:i
1943:1
1940:i
1936:P
1923:.
1921:A
1917:C
1912:B
1908:B
1904:C
1899:A
1895:A
1891:C
1886:B
1882:B
1878:C
1873:A
1859:n
1855:m
1836:)
1830:n
1826:)
1823:n
1820:,
1817:m
1814:(
1802:,
1797:m
1793:)
1790:n
1787:,
1784:m
1781:(
1768:(
1757:n
1753:m
1737:+
1732:Z
1683:C
1677:c
1674:,
1671:B
1665:b
1662:,
1659:A
1653:a
1647:|
1641:)
1638:b
1635:)
1632:c
1629:(
1626:g
1623:,
1620:a
1617:(
1611:)
1608:b
1605:,
1602:a
1599:)
1596:c
1593:(
1590:f
1587:(
1575:/
1569:}
1565:B
1557:i
1553:b
1549:,
1546:A
1538:i
1534:a
1527:|
1521:)
1516:i
1512:b
1508:,
1503:i
1499:a
1495:(
1490:I
1484:i
1475:{
1471:=
1468:B
1463:C
1455:A
1431:B
1427:C
1423:g
1419:A
1415:C
1411:f
1403:C
1399:B
1395:A
1361:f
1351:g
1347:=
1344:g
1334:f
1313:B
1308:C
1300:A
1294:B
1291::
1284:f
1263:B
1258:C
1250:A
1244:A
1241::
1234:g
1213:B
1208:C
1200:A
1150:.
1148:R
1139:R
1133:g
1129:f
1125:Z
1121:z
1119:(
1117:g
1113:z
1111:(
1109:f
1097:Z
1093:g
1089:f
1085:B
1081:A
1067:B
1061:0
1058::
1055:g
1035:A
1029:0
1026::
1023:f
1010:B
1006:A
974:.
972:Y
968:X
954:Y
948:X
938:Z
930:Y
926:X
915:g
911:f
897:Y
892:f
884:X
874:X
870:Z
866:f
862:Z
858:X
854:Y
846:Y
842:Z
838:g
834:Y
826:Z
811:.
809:Y
805:f
801:Y
787:Y
781:X
761:Y
755:X
749:Y
743:)
740:x
737:(
734:f
714:Y
708:X
702:X
696:x
676:Y
670:X
660:X
656:f
638:Y
632:X
626:f
603:W
597:Y
591:X
578:Y
574:X
570:Z
566:g
562:f
554:Z
550:W
542:Y
538:X
534:Z
530:z
526:z
524:(
522:g
518:z
516:(
514:f
502:~
483:/
479:)
476:Y
470:X
467:(
464:=
461:P
451:Y
447:X
443:2
440:i
436:1
433:i
429:Z
421:Y
417:X
409:g
405:f
401:Y
397:Z
393:g
389:X
385:Z
381:f
373:Z
369:Y
365:X
329:Q
325:P
321:u
317:2
314:j
310:1
307:j
303:Q
295:2
292:i
288:1
285:i
281:P
262:P
258:Y
254:2
251:i
247:P
243:X
239:1
236:i
232:P
228:g
224:f
191:Y
186:Z
182:+
178:X
175:=
172:P
152:Y
147:Z
139:X
136:=
133:P
119:g
115:f
107:P
103:Y
99:P
95:X
91:P
80:Y
76:Z
72:g
68:X
64:Z
60:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
