2287:
2534:
2554:
2544:
1132:
723:
942:
580:
1489:. This concept makes sense, however: such matrices have no entries and so are completely determined by their size. While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object.
1127:{\displaystyle {\begin{pmatrix}f_{11}&f_{12}&\cdots &f_{1n}\\f_{21}&f_{22}&\cdots &f_{2n}\\\vdots &\vdots &\cdots &\vdots \\f_{m1}&f_{m2}&\cdots &f_{mn}\end{pmatrix}}}
1223:
718:{\displaystyle A\ {\xrightarrow {\quad \Delta \quad }}\ A\oplus A\oplus A\ {\xrightarrow {\alpha _{1}\,\oplus \,\alpha _{2}\,\oplus \,\alpha _{3}}}\ B\oplus B\oplus B\ {\xrightarrow {\quad \nabla \quad }}\ B}
139:
Thus an additive category is equivalently described as a preadditive category admitting all finitary products, or a preadditive category admitting all finitary coproducts.
285:
be a semiadditive category, so a category having all finitary biproducts. Then every hom-set has an addition, endowing it with the structure of an
1806:
1137:
1931:
2216:
1434:
This construction should be compared with the result that a ring is a preadditive category with just one object, shown
1886:
574:
This addition is both commutative and associative. The associativity can be seen by considering the composition
1722:
1266:
Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense.
296:
is additive, then the two additions on hom-sets must agree. In particular, a semiadditive category is additive
193:
136:
every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it).
1492:
Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects
1924:
2578:
2128:
2083:
238:
125:
117:
2557:
2497:
1435:
2206:
1697:
Almost all functors studied between additive categories are additive. In fact, it is a theorem that all
2547:
2333:
2197:
2105:
1753:
223:
2506:
2150:
2088:
2011:
256:
113:
1534:
is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.
2537:
2493:
2098:
1917:
1749:
176:
An additive category may then be defined as a semiadditive category in which every morphism has an
286:
2093:
2075:
1854:
2300:
2066:
2046:
1969:
62:
1876:
1702:
2182:
2021:
1575:. If the categories are additive, then a functor is additive if and only if it preserves all
1523:
1260:
1994:
1989:
1832:
1744:
1453:
270:
211:
39:
20:
1849:, Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191,
8:
2338:
2286:
2212:
2016:
1774:
1730:
260:
2192:
2187:
2169:
2051:
2026:
1786:
1781:
Many commonly studied additive categories are in fact abelian categories; for example,
1328:
249:
170:
2501:
2438:
2426:
2328:
2253:
2248:
2202:
1984:
1882:
1850:
1379:
1274:
106:
102:
1823:
2462:
2348:
2323:
2258:
2243:
2238:
2177:
2006:
1974:
1858:
1818:
1761:
1698:
200:
177:
133:
2374:
1940:
1899:
1828:
1256:
245:
31:
1862:
1255:, it follows that addition and composition of matrices obey the usual rules for
2411:
2406:
2390:
2353:
2343:
2263:
1353:
297:
166:
19:
For the weaker sense of the term "additive category" (without biproducts), see
2572:
2401:
2233:
2110:
2036:
1705:). Most of the interesting functors studied in category theory are adjoints.
310:
To define the addition law, we will use the convention that for a biproduct,
230:
181:
90:
273:
over a ring, thought of as a category as described below, is also additive.
65:
equipped with additional structure, and another as a category equipped with
2155:
2056:
1766:
1564:
1486:
94:
2416:
2396:
2268:
2138:
1836:
Sections 18 and 19 deal with the addition law in semiadditive categories.
1770:
1467:
158:
27:
2448:
2386:
1999:
1906:; Academic Press, Inc. (out of print) goes over all of this very slowly
165:. It is then a remarkable theorem that the Hom sets naturally admit an
61:
There are two equivalent definitions of an additive category: One as a
2442:
2133:
1576:
1218:{\displaystyle f_{kl}:=p_{k}\circ f\circ i_{l}\colon A_{l}\to B_{k}.}
234:
162:
129:
121:
50:
1846:
700:
630:
595:
2511:
2143:
2041:
1789:
provide an example of a category that is additive but not abelian.
157:
to be a category (note: not a preadditive category) which admits a
70:
47:
1909:
2481:
2471:
2120:
2031:
1568:
1543:
1504:
to 0 (just as there is exactly one 0-by-1 matrix with entries in
86:
1394:, and where composition is given by matrix multiplication. Then
276:
2476:
206:. These are categories enriched over the monoidal category of
2358:
303:
This shows that the addition law for an additive category is
1515:(just as there is exactly one 1-by-0 matrix with entries in
1500:
in an additive category, there is exactly one morphism from
1701:
between additive categories must be additive functors (see
289:, and such that the composition of morphisms is bilinear.
1904:
Abelian
Categories with Applications to Rings and Modules
184:
structure instead of merely an abelian monoid structure.
1522:) – this is just what it means to say that 0 is a
1748:
is an additive category in which every morphism has a
1717:-linear additive categories, one usually restricts to
951:
16:
Preadditive category that admits all finitary products
1357:
1140:
945:
857:
583:
1847:
Typing linear algebra: A biproduct-oriented approach
1269:
Recall that the morphisms from a single object
894:
in an additive category, we can represent morphisms
222:
The original example of an additive category is the
1217:
1126:
717:
2570:
1477:This may be confusing in the special case where
142:
128:in the case of an empty diagram), and hence a
1925:
1811:Bulletin of the American Mathematical Society
277:Internal characterisation of the addition law
76:
1360:) and letting the hom-set of morphisms from
836:. Using this, we can represent any morphism
752:It is also bilinear, using for example that
192:More generally, one also considers additive
1485:is zero, because we usually don't think of
2553:
2543:
2299:
1932:
1918:
1765:is a pre-abelian category such that every
1470:of the category of left modules over
317:will denote the projection morphisms, and
112:In a preadditive category, every finitary
1822:
660:
656:
645:
641:
300:every morphism has an additive inverse.
1874:
1804:
255:is additive, and so in particular, the
214:and admitting all finitary biproducts.
2571:
1526:. Furthermore, the zero morphism from
116:(including the empty product, i.e., a
2298:
1951:
1913:
1511:) and exactly one morphism from 0 to
324:will denote the injection morphisms.
233:, the addition of morphisms is given
1537:
1317:matrices with entries from the ring
1939:
1708:
147:We give an alternative definition.
13:
1713:When considering functors between
1559:between preadditive categories is
858:Matrix representation of morphisms
702:
597:
180:. This then gives the Hom sets an
14:
2590:
1487:matrices with 0 rows or 0 columns
474:, there exists a unique morphism
187:
2552:
2542:
2533:
2532:
2285:
1952:
1736:
93:and composition of morphisms is
1824:10.1090/S0002-9904-1950-09427-0
795:We remark that for a biproduct
705:
701:
600:
596:
1868:
1839:
1798:
1725:, so those functors giving an
1635:should be a biproduct of
1334:, we can form a category
1199:
237:, and biproducts are given by
1:
1792:
1405:is an additive category, and
173:of this fact is given below.
56:
1845:H.D. Macedo, J.N. Oliveira,
1785:is an abelian category. The
7:
2227:Constructions on categories
1863:10.1016/j.scico.2012.07.012
1441:If we interpret the object
217:
143:Via semiadditive categories
73:satisfy certain equations.
10:
2595:
2334:Higher-dimensional algebra
1881:, CRC Press, p. 466,
1875:Shastri, Anant R. (2013),
1805:MacLane, Saunders (1950),
1664:with projection morphisms
1606:with projection morphisms
1563:if it is an abelian group
457:Next, given two morphisms
224:category of abelian groups
85:is preadditive if all its
77:Via preadditive categories
18:
2528:
2461:
2425:
2373:
2366:
2317:
2307:
2294:
2283:
2226:
2168:
2119:
2074:
2065:
1962:
1958:
1947:
257:category of vector spaces
229:. The zero object is the
1878:Basic Algebraic Topology
1680:and injection morphisms
1615:and injection morphisms
551:We can therefore define
2144:Cokernels and quotients
2067:Universal constructions
1586:is a biproduct of
2301:Higher category theory
2047:Natural transformation
1352:indexed by the set of
1327:Conversely, given any
1297:, then morphisms from
1287:-fold product of
1219:
1128:
719:
244:More generally, every
69:but whose objects and
1220:
1129:
720:
2170:Algebraic categories
1807:"Duality for groups"
1745:pre-abelian category
1138:
943:
581:
40:preadditive category
21:preadditive category
2579:Additive categories
2339:Homotopy hypothesis
2017:Commutative diagram
1787:free abelian groups
1731:module homomorphism
1390:matrices over
1283:. If we denote the
1273:to itself form the
754:∆ ∘ β = (β ⊕ β) ∘ ∆
706:
671:
601:
548:, and 0 otherwise.
375:codiagonal morphism
120:) is necessarily a
109:of abelian groups.
2052:Universal property
1345:by taking objects
1215:
1124:
1118:
715:
307:to that category.
197:-linear categories
97:; in other words,
67:no extra structure
30:, specifically in
2566:
2565:
2524:
2523:
2520:
2519:
2502:monoidal category
2457:
2456:
2329:Enriched category
2281:
2280:
2277:
2276:
2254:Quotient category
2249:Opposite category
2164:
2163:
1733:on each hom-set.
1538:Additive functors
1275:endomorphism ring
711:
707:
694:
676:
672:
624:
606:
602:
589:
337:diagonal morphism
107:monoidal category
36:additive category
2586:
2556:
2555:
2546:
2545:
2536:
2535:
2371:
2370:
2349:Simplex category
2324:Categorification
2315:
2314:
2296:
2295:
2289:
2259:Product category
2244:Kleisli category
2239:Functor category
2084:Terminal objects
2072:
2071:
2007:Adjoint functors
1960:
1959:
1949:
1948:
1934:
1927:
1920:
1911:
1910:
1893:
1891:
1872:
1866:
1843:
1837:
1835:
1826:
1802:
1762:abelian category
1728:
1720:
1716:
1699:adjoint functors
1693:
1679:
1659:
1634:
1623:
1614:
1601:
1585:
1558:
1533:
1529:
1521:
1514:
1510:
1503:
1499:
1495:
1484:
1480:
1473:
1461:
1451:
1430:
1419:
1415:
1404:
1393:
1389:
1385:
1377:
1368:
1344:
1333:
1323:
1308:
1302:
1296:
1290:
1286:
1282:
1272:
1254:
1224:
1222:
1221:
1216:
1211:
1210:
1198:
1197:
1185:
1184:
1166:
1165:
1153:
1152:
1133:
1131:
1130:
1125:
1123:
1122:
1115:
1114:
1095:
1094:
1080:
1079:
1041:
1040:
1021:
1020:
1009:
1008:
995:
994:
975:
974:
963:
962:
935:
931:
927:
893:
877:
853:
835:
804:
791:
755:
748:
731:
724:
722:
721:
716:
709:
708:
696:
692:
674:
673:
670:
669:
655:
654:
640:
639:
626:
622:
604:
603:
591:
587:
570:
547:
537:
528:
500:
473:
453:
435:
418:
407:
390:
369:
352:
330:
327:For each object
265:
254:
209:
205:
201:commutative ring
196:
178:additive inverse
2594:
2593:
2589:
2588:
2587:
2585:
2584:
2583:
2569:
2568:
2567:
2562:
2516:
2486:
2453:
2430:
2421:
2378:
2362:
2313:
2303:
2290:
2273:
2222:
2160:
2129:Initial objects
2115:
2061:
1954:
1943:
1941:Category theory
1938:
1900:Nicolae Popescu
1896:
1889:
1873:
1869:
1844:
1840:
1803:
1799:
1795:
1739:
1726:
1723:linear functors
1718:
1714:
1711:
1690:
1681:
1677:
1665:
1656:
1646:
1636:
1625:
1621:
1616:
1612:
1607:
1599:
1593:
1587:
1583:
1546:
1540:
1531:
1527:
1516:
1512:
1505:
1501:
1497:
1493:
1482:
1478:
1471:
1464:matrix category
1457:
1450:
1442:
1428:
1421:
1417:
1414:
1406:
1395:
1391:
1387:
1383:
1375:
1370:
1366:
1361:
1354:natural numbers
1350:
1335:
1331:
1318:
1304:
1298:
1292:
1291:with itself by
1288:
1284:
1277:
1270:
1257:matrix addition
1252:
1243:
1235:
1229:
1206:
1202:
1193:
1189:
1180:
1176:
1161:
1157:
1145:
1141:
1139:
1136:
1135:
1117:
1116:
1107:
1103:
1101:
1096:
1087:
1083:
1081:
1072:
1068:
1065:
1064:
1059:
1054:
1049:
1043:
1042:
1033:
1029:
1027:
1022:
1016:
1012:
1010:
1004:
1000:
997:
996:
987:
983:
981:
976:
970:
966:
964:
958:
954:
947:
946:
944:
941:
940:
933:
929:
925:
919:
911:
905:
895:
891:
885:
879:
875:
869:
863:
860:
837:
833:
826:
819:
812:
806:
796:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
747:
740:
733:
729:
695:
665:
661:
650:
646:
635:
631:
625:
590:
582:
579:
578:
568:
564:
561: := ∇ ∘ (α
560:
556:
552:
539:
536:
530:
527:
518:
514:
510:
502:
483:
479:
475:
464:
458:
452:
446:
437:
434:
428:
420:
413:
406:
399:
392:
377:
368:
361:
354:
339:
328:
322:
315:
279:
269:The algebra of
263:
252:
246:module category
220:
207:
203:
194:
190:
161:and all binary
145:
79:
59:
32:category theory
24:
17:
12:
11:
5:
2592:
2582:
2581:
2564:
2563:
2561:
2560:
2550:
2540:
2529:
2526:
2525:
2522:
2521:
2518:
2517:
2515:
2514:
2509:
2504:
2490:
2484:
2479:
2474:
2468:
2466:
2459:
2458:
2455:
2454:
2452:
2451:
2446:
2435:
2433:
2428:
2423:
2422:
2420:
2419:
2414:
2409:
2404:
2399:
2394:
2383:
2381:
2376:
2368:
2364:
2363:
2361:
2356:
2354:String diagram
2351:
2346:
2344:Model category
2341:
2336:
2331:
2326:
2321:
2319:
2312:
2311:
2308:
2305:
2304:
2292:
2291:
2284:
2282:
2279:
2278:
2275:
2274:
2272:
2271:
2266:
2264:Comma category
2261:
2256:
2251:
2246:
2241:
2236:
2230:
2228:
2224:
2223:
2221:
2220:
2210:
2200:
2198:Abelian groups
2195:
2190:
2185:
2180:
2174:
2172:
2166:
2165:
2162:
2161:
2159:
2158:
2153:
2148:
2147:
2146:
2136:
2131:
2125:
2123:
2117:
2116:
2114:
2113:
2108:
2103:
2102:
2101:
2091:
2086:
2080:
2078:
2069:
2063:
2062:
2060:
2059:
2054:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2004:
2003:
2002:
1997:
1992:
1987:
1982:
1977:
1966:
1964:
1956:
1955:
1945:
1944:
1937:
1936:
1929:
1922:
1914:
1908:
1907:
1895:
1894:
1887:
1867:
1838:
1817:(6): 485–516,
1796:
1794:
1791:
1779:
1778:
1757:
1738:
1735:
1710:
1709:Generalization
1707:
1688:
1673:
1654:
1644:
1619:
1610:
1597:
1591:
1539:
1536:
1446:
1426:
1410:
1373:
1364:
1348:
1261:multiplication
1248:
1239:
1231:
1226:
1225:
1214:
1209:
1205:
1201:
1196:
1192:
1188:
1183:
1179:
1175:
1172:
1169:
1164:
1160:
1156:
1151:
1148:
1144:
1121:
1113:
1110:
1106:
1102:
1100:
1097:
1093:
1090:
1086:
1082:
1078:
1075:
1071:
1067:
1066:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1044:
1039:
1036:
1032:
1028:
1026:
1023:
1019:
1015:
1011:
1007:
1003:
999:
998:
993:
990:
986:
982:
980:
977:
973:
969:
965:
961:
957:
953:
952:
950:
923:
917:
909:
903:
889:
883:
873:
867:
862:Given objects
859:
856:
831:
824:
817:
810:
787:
783:
779:
775:
771:
767:
763:
759:
745:
738:
726:
725:
714:
704:
699:
691:
688:
685:
682:
679:
668:
664:
659:
653:
649:
644:
638:
634:
629:
621:
618:
615:
612:
609:
599:
594:
586:
566:
562:
558:
554:
532:
523:
516:
512:
506:
481:
477:
460:
448:
442:
430:
424:
410:
409:
404:
397:
371:
366:
359:
320:
313:
298:if and only if
287:abelian monoid
278:
275:
219:
216:
189:
188:Generalization
186:
167:abelian monoid
144:
141:
126:initial object
91:abelian groups
78:
75:
58:
55:
46:admitting all
15:
9:
6:
4:
3:
2:
2591:
2580:
2577:
2576:
2574:
2559:
2551:
2549:
2541:
2539:
2531:
2530:
2527:
2513:
2510:
2508:
2505:
2503:
2499:
2495:
2491:
2489:
2487:
2480:
2478:
2475:
2473:
2470:
2469:
2467:
2464:
2460:
2450:
2447:
2444:
2440:
2437:
2436:
2434:
2432:
2424:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2402:Tetracategory
2400:
2398:
2395:
2392:
2391:pseudofunctor
2388:
2385:
2384:
2382:
2380:
2372:
2369:
2365:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2316:
2310:
2309:
2306:
2302:
2297:
2293:
2288:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2234:Free category
2232:
2231:
2229:
2225:
2218:
2217:Vector spaces
2214:
2211:
2208:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2175:
2173:
2171:
2167:
2157:
2154:
2152:
2149:
2145:
2142:
2141:
2140:
2137:
2135:
2132:
2130:
2127:
2126:
2124:
2122:
2118:
2112:
2111:Inverse limit
2109:
2107:
2104:
2100:
2097:
2096:
2095:
2092:
2090:
2087:
2085:
2082:
2081:
2079:
2077:
2073:
2070:
2068:
2064:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2037:Kan extension
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1972:
1971:
1968:
1967:
1965:
1961:
1957:
1950:
1946:
1942:
1935:
1930:
1928:
1923:
1921:
1916:
1915:
1912:
1905:
1901:
1898:
1897:
1890:
1888:9781466562431
1884:
1880:
1879:
1871:
1864:
1860:
1856:
1852:
1848:
1842:
1834:
1830:
1825:
1820:
1816:
1812:
1808:
1801:
1797:
1790:
1788:
1784:
1776:
1772:
1768:
1764:
1763:
1758:
1755:
1751:
1747:
1746:
1741:
1740:
1737:Special cases
1734:
1732:
1724:
1706:
1704:
1700:
1695:
1691:
1684:
1676:
1672:
1668:
1663:
1657:
1650:
1643:
1639:
1632:
1628:
1622:
1613:
1605:
1600:
1590:
1580:
1578:
1574:
1570:
1566:
1562:
1557:
1553:
1549:
1545:
1535:
1525:
1520:
1509:
1490:
1488:
1475:
1469:
1465:
1460:
1455:
1449:
1445:
1439:
1437:
1432:
1425:
1413:
1409:
1402:
1398:
1381:
1376:
1367:
1359:
1355:
1351:
1342:
1338:
1330:
1325:
1322:
1316:
1312:
1307:
1301:
1295:
1281:
1276:
1267:
1264:
1262:
1258:
1251:
1247:
1242:
1238:
1234:
1212:
1207:
1203:
1194:
1190:
1186:
1181:
1177:
1173:
1170:
1167:
1162:
1158:
1154:
1149:
1146:
1142:
1119:
1111:
1108:
1104:
1098:
1091:
1088:
1084:
1076:
1073:
1069:
1061:
1056:
1051:
1046:
1037:
1034:
1030:
1024:
1017:
1013:
1005:
1001:
991:
988:
984:
978:
971:
967:
959:
955:
948:
939:
938:
937:
926:
916:
912:
902:
898:
892:
882:
876:
866:
855:
854:as a matrix.
852:
848:
844:
840:
830:
823:
816:
809:
803:
799:
793:
750:
744:
737:
732:, using that
712:
697:
689:
686:
683:
680:
677:
666:
662:
657:
651:
647:
642:
636:
632:
627:
619:
616:
613:
610:
607:
592:
584:
577:
576:
575:
572:
549:
546:
542:
535:
526:
522:
509:
505:
499:
495:
491:
487:
472:
468:
463:
455:
451:
445:
441:
433:
427:
423:
416:
403:
396:
389:
385:
381:
376:
372:
365:
358:
351:
347:
343:
338:
334:
333:
332:
331:, we define:
325:
323:
316:
308:
306:
301:
299:
295:
292:Moreover, if
290:
288:
284:
274:
272:
267:
266:is additive.
262:
258:
251:
247:
242:
240:
236:
232:
231:trivial group
228:
225:
215:
213:
202:
198:
185:
183:
182:abelian group
179:
174:
172:
169:structure. A
168:
164:
160:
156:
153:
148:
140:
137:
135:
131:
127:
123:
119:
115:
110:
108:
104:
100:
96:
92:
88:
84:
74:
72:
68:
64:
54:
52:
49:
45:
41:
37:
33:
29:
22:
2482:
2463:Categorified
2367:n-categories
2318:Key concepts
2156:Direct limit
2139:Coequalizers
2057:Yoneda lemma
1979:
1963:Key concepts
1953:Key concepts
1903:
1877:
1870:
1841:
1814:
1810:
1800:
1782:
1780:
1767:monomorphism
1760:
1743:
1712:
1696:
1686:
1682:
1674:
1670:
1666:
1661:
1652:
1648:
1641:
1637:
1630:
1626:
1617:
1608:
1603:
1595:
1588:
1582:That is, if
1581:
1572:
1565:homomorphism
1560:
1555:
1551:
1547:
1541:
1518:
1507:
1491:
1476:
1463:
1462:, then this
1458:
1452:as the left
1447:
1443:
1440:
1433:
1423:
1420:-fold power
1411:
1407:
1400:
1396:
1371:
1362:
1346:
1340:
1336:
1326:
1320:
1314:
1310:
1305:
1299:
1293:
1279:
1268:
1265:
1249:
1245:
1240:
1236:
1232:
1227:
921:
914:
907:
900:
896:
887:
880:
871:
864:
861:
850:
846:
842:
838:
828:
821:
814:
807:
801:
797:
794:
751:
742:
735:
727:
573:
550:
544:
540:
533:
524:
520:
507:
503:
497:
493:
489:
485:
470:
466:
461:
456:
449:
443:
439:
431:
425:
421:
414:
411:
401:
394:
387:
383:
379:
374:
363:
356:
349:
345:
341:
336:
326:
318:
311:
309:
304:
302:
293:
291:
282:
280:
268:
243:
226:
221:
191:
175:
154:
152:semiadditive
151:
149:
146:
138:
118:final object
111:
98:
82:
80:
66:
60:
43:
35:
25:
2431:-categories
2407:Kan complex
2397:Tricategory
2379:-categories
2269:Subcategory
2027:Exponential
1995:Preadditive
1990:Pre-abelian
1771:epimorphism
1524:zero object
1468:subcategory
1416:equals the
1356:(including
1228:Using that
239:direct sums
159:zero object
81:A category
28:mathematics
2449:3-category
2439:2-category
2412:∞-groupoid
2387:Bicategory
2134:Coproducts
2094:Equalizers
2000:Bicategory
1793:References
1579:diagrams.
1517:End
1506:End
1466:becomes a
1319:End
1278:End
501:such that
419:, we have
412:Then, for
163:biproducts
134:conversely
57:Definition
51:biproducts
2498:Symmetric
2443:2-functor
2183:Relations
2106:Pullbacks
1855:0167-6423
1647:), ... ,
1577:biproduct
1200:→
1187::
1174:∘
1168:∘
1099:⋯
1062:⋮
1057:⋯
1052:⋮
1047:⋮
1025:⋯
979:⋯
936:matrices
756:and that
730:α + 0 = α
703:∇
687:⊕
681:⊕
663:α
658:⊕
648:α
643:⊕
633:α
617:⊕
611:⊕
598:Δ
235:pointwise
150:Define a
130:biproduct
122:coproduct
105:over the
71:morphisms
2573:Category
2558:Glossary
2538:Category
2512:n-monoid
2465:concepts
2121:Colimits
2089:Products
2042:Morphism
1985:Concrete
1980:Additive
1970:Category
1902:; 1973;
1754:cokernel
1660:in
1602:in
1594:, ... ,
1567:on each
1561:additive
920:⊕ ⋅⋅⋅ ⊕
906:⊕ ⋅⋅⋅ ⊕
805:we have
734:α ⊕ 0 =
728:We have
698:→
628:→
593:→
305:internal
271:matrices
218:Examples
155:category
103:enriched
95:bilinear
87:hom-sets
63:category
48:finitary
2548:Outline
2507:n-group
2472:2-group
2427:Strict
2417:∞-topos
2213:Modules
2151:Pushout
2099:Kernels
2032:Functor
1975:Abelian
1833:0049192
1624:, then
1569:hom-set
1544:functor
1378:be the
886:, ...,
870:, ...,
529:equals
429:∘ ∆ = 1
259:over a
248:over a
212:modules
114:product
2494:Traced
2477:2-ring
2207:Fields
2193:Groups
2188:Magmas
2076:Limits
1885:
1853:
1831:
1775:normal
1752:and a
1750:kernel
1456:
1454:module
1134:where
782:) ⊕ (α
774:) = (α
766:) ∘ (β
741:∘ α ∘
710:
693:
675:
623:
605:
588:
417:= 1, 2
199:for a
132:, and
42:
2488:-ring
2375:Weak
2359:Topos
2203:Rings
569:) ∘ ∆
261:field
171:proof
38:is a
34:, an
2178:Sets
1883:ISBN
1851:ISSN
1769:and
1703:here
1496:and
1436:here
1386:-by-
1329:ring
1313:-by-
1309:are
1259:and
932:-by-
878:and
519:) ∘
511:∘ (α
438:∇ ∘
436:and
393:∇ =
373:the
355:∆ =
335:the
281:Let
250:ring
124:(or
89:are
2022:End
2012:CCC
1859:doi
1819:doi
1773:is
1759:An
1571:in
1530:to
1481:or
1397:Mat
1382:of
1380:set
1369:to
1337:Mat
1303:to
1253:= 1
928:as
834:= 1
786:∘ β
778:∘ β
770:⊕ β
762:⊕ α
565:⊕ α
557:+ α
538:if
515:⊕ α
480:⊕ α
447:= 1
391:by
378:∇:
353:by
340:∆:
101:is
26:In
2575::
2500:)
2496:)(
1857:,
1829:MR
1827:,
1815:56
1813:,
1809:,
1783:Ab
1742:A
1694:.
1554:→
1550::
1542:A
1474:.
1438:.
1431:.
1324:.
1263:.
1244:∘
1155::=
1018:22
1006:21
972:12
960:11
913:→
899::
849:⊕
845:→
841:⊕
827:∘
820:+
813:∘
800:⊕
792:.
758:(α
749:.
571:.
543:=
496:⊕
492:→
488:⊕
484::
469:→
465::
454:.
400:+
386:→
382:⊕
362:+
348:⊕
344:→
241:.
227:Ab
53:.
2492:(
2485:n
2483:E
2445:)
2441:(
2429:n
2393:)
2389:(
2377:n
2219:)
2215:(
2209:)
2205:(
1933:e
1926:t
1919:v
1892:.
1865:.
1861::
1821::
1777:.
1756:.
1729:-
1727:R
1721:-
1719:R
1715:R
1692:)
1689:j
1687:i
1685:(
1683:F
1678:)
1675:j
1671:p
1669:(
1667:F
1662:D
1658:)
1655:n
1653:A
1651:(
1649:F
1645:1
1642:A
1640:(
1638:F
1633:)
1631:B
1629:(
1627:F
1620:j
1618:i
1611:k
1609:p
1604:C
1598:n
1596:A
1592:1
1589:A
1584:B
1573:C
1556:D
1552:C
1548:F
1532:B
1528:A
1519:B
1513:B
1508:A
1502:A
1498:B
1494:A
1483:n
1479:m
1472:R
1459:R
1448:n
1444:A
1429:)
1427:1
1424:A
1422:(
1418:n
1412:n
1408:A
1403:)
1401:R
1399:(
1392:R
1388:n
1384:m
1374:m
1372:A
1365:n
1363:A
1358:0
1349:n
1347:A
1343:)
1341:R
1339:(
1332:R
1321:A
1315:n
1311:m
1306:A
1300:A
1294:A
1289:A
1285:n
1280:A
1271:A
1250:k
1246:p
1241:k
1237:i
1233:k
1230:∑
1213:.
1208:k
1204:B
1195:l
1191:A
1182:l
1178:i
1171:f
1163:k
1159:p
1150:l
1147:k
1143:f
1120:)
1112:n
1109:m
1105:f
1092:2
1089:m
1085:f
1077:1
1074:m
1070:f
1038:n
1035:2
1031:f
1014:f
1002:f
992:n
989:1
985:f
968:f
956:f
949:(
934:n
930:m
924:m
922:B
918:1
915:B
910:n
908:A
904:1
901:A
897:f
890:m
888:B
884:1
881:B
874:n
872:A
868:1
865:A
851:D
847:C
843:B
839:A
832:2
829:p
825:2
822:i
818:1
815:p
811:1
808:i
802:B
798:A
790:)
788:2
784:2
780:1
776:1
772:2
768:1
764:2
760:1
746:1
743:p
739:1
736:i
713:B
690:B
684:B
678:B
667:3
652:2
637:1
620:A
614:A
608:A
585:A
567:2
563:1
559:2
555:1
553:α
545:l
541:k
534:k
531:α
525:k
521:i
517:2
513:1
508:l
504:p
498:B
494:B
490:A
486:A
482:2
478:1
476:α
471:B
467:A
462:k
459:α
450:A
444:k
440:i
432:A
426:k
422:p
415:k
408:.
405:2
402:p
398:1
395:p
388:A
384:A
380:A
370:;
367:2
364:i
360:1
357:i
350:A
346:A
342:A
329:A
321:k
319:i
314:k
312:p
294:C
283:C
264:K
253:R
210:-
208:R
204:R
195:R
99:C
83:C
44:C
23:.
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