1746:
1344:
2099:
234:
1080:
541:
1618:
1407:
732:
2383:
1932:
653:
385:
2487:
1829:
2221:
1158:, where each member of the relation is associated with a set of morphisms. A functor is a special case of a profunctor in the same way that a function is a special case of a relation.
1987:
2270:
2333:
1491:
1254:
1222:
1113:
869:
457:
166:
269:
1629:
829:
1440:
576:
321:
1187:
685:
2301:
2439:
2182:
88:
1769:
600:
411:
2407:
1869:
1849:
1551:
1531:
1511:
1156:
1136:
1029:
1009:
989:
969:
949:
929:
909:
889:
803:
783:
756:
293:
131:
111:
1265:
2002:
17:
61:
181:
1038:
1556:
1352:
462:
690:
2109:
Composition of profunctors is associative only up to isomorphism (because the product is not strictly associative in
2338:
608:
2444:
2187:
2229:
2306:
1448:
1227:
1195:
1085:
842:
579:
139:
1877:
1937:
1741:{\displaystyle (\psi \phi )(e,c)=\left(\coprod _{d\in D}\psi (e,d)\times \phi (d,c)\right){\Bigg /}\sim }
330:
1774:
735:
242:
416:
808:
547:
173:
1416:
553:
298:
1118:
This also makes it clear that a profunctor can be thought of as a relation between the objects of
2516:
1169:
661:
2279:
2135:
1730:
91:
2412:
2155:
1994:
72:
1754:
585:
8:
2539:
390:
2392:
1854:
1834:
1536:
1516:
1496:
1141:
1121:
1014:
994:
974:
954:
934:
914:
894:
874:
788:
768:
741:
278:
116:
96:
2582:
1339:{\displaystyle \psi \phi =\mathrm {Lan} _{Y_{D}}({\hat {\psi }})\circ {\hat {\phi }}}
1031:. (These are also known as het-sets, since the corresponding morphisms can be called
272:
2386:
324:
2131:
1-cells between two small categories are the profunctors between those categories,
1035:.) The previous definition can be recovered by the restriction of the hom-functor
40:
28:
2512:
2559:
2125:
1443:
2094:{\displaystyle (\psi \phi )(e,c)=\int ^{d\colon D}\psi (e,d)\times \phi (d,c)}
991:. The sets in the formal definition above are the hom-sets between objects of
2576:
1410:
2547:
32:
2303:
has a right adjoint. Moreover, this is a characterization: a profunctor
2498:
2114:
871:
is a category whose objects are the disjoint union of the objects of
44:
229:{\displaystyle \phi :D^{\mathrm {op} }\times C\to \mathbf {Set} }
1075:{\displaystyle \phi ^{\text{op}}\times \phi \to \mathbf {Set} }
1993:
Equivalently, profunctor composition can be written using a
2563:
2551:
1613:{\displaystyle D(-,d):D^{\mathrm {op} }\to \mathrm {Set} }
951:, plus zero or more additional morphisms from objects of
1402:{\displaystyle \mathrm {Lan} _{Y_{D}}({\hat {\psi }})}
2447:
2415:
2395:
2341:
2309:
2282:
2232:
2190:
2158:
2005:
1940:
1880:
1857:
1837:
1777:
1757:
1632:
1559:
1539:
1519:
1499:
1451:
1419:
1355:
1268:
1230:
1198:
1172:
1144:
1124:
1088:
1041:
1017:
997:
977:
957:
937:
917:
897:
877:
845:
811:
791:
771:
744:
693:
664:
611:
588:
556:
465:
419:
393:
333:
301:
281:
245:
184:
142:
119:
99:
75:
536:{\displaystyle xf\in \phi (d,c),gx\in \phi (d',c')}
2481:
2433:
2401:
2377:
2327:
2295:
2264:
2215:
2176:
2147:
2093:
1981:
1926:
1863:
1843:
1823:
1763:
1740:
1612:
1545:
1525:
1505:
1485:
1434:
1401:
1338:
1248:
1216:
1181:
1150:
1130:
1107:
1074:
1023:
1003:
983:
963:
943:
923:
903:
883:
863:
823:
797:
777:
750:
727:{\displaystyle \mathrm {Set} ^{D^{\mathrm {op} }}}
726:
679:
647:
594:
570:
535:
451:
405:
379:
315:
287:
263:
228:
160:
125:
105:
82:
2113:). The best one can hope is therefore to build a
2574:
2378:{\displaystyle {\hat {\phi }}:C\to {\hat {D}}}
648:{\displaystyle {\hat {\phi }}:C\to {\hat {D}}}
1161:
2104:
1771:is the least equivalence relation such that
834:
2482:{\displaystyle {\hat {\phi }}=Y_{D}\circ F}
911:, and whose morphisms are the morphisms of
2223:by postcomposing with the Yoneda functor:
2216:{\displaystyle \phi _{F}:C\nrightarrow D}
839:An equivalent definition of a profunctor
76:
2134:2-cells between two profunctors are the
2527:
2511:
2276:It can be shown that such a profunctor
14:
2575:
2265:{\displaystyle \phi _{F}=Y_{D}\circ F}
2536:
2328:{\displaystyle \phi :C\nrightarrow D}
1486:{\displaystyle Y_{D}:D\to {\hat {D}}}
1249:{\displaystyle \psi :D\nrightarrow E}
1217:{\displaystyle \phi :C\nrightarrow D}
1108:{\displaystyle D^{\text{op}}\times C}
864:{\displaystyle \phi :C\nrightarrow D}
161:{\displaystyle \phi :C\nrightarrow D}
1927:{\displaystyle y'=vy\in \psi (e,d')}
2335:has a right adjoint if and only if
1982:{\displaystyle x'v=x\in \phi (d,c)}
380:{\displaystyle f:d\to d',g:c\to c'}
24:
1606:
1603:
1600:
1590:
1587:
1364:
1361:
1358:
1286:
1283:
1280:
716:
713:
702:
699:
696:
255:
252:
200:
197:
25:
2594:
1831:whenever there exists a morphism
1824:{\displaystyle (y',x')\sim (y,x)}
264:{\displaystyle D^{\mathrm {op} }}
1068:
1065:
1062:
564:
561:
558:
452:{\displaystyle x\in \phi (d',c)}
309:
306:
303:
222:
219:
216:
18:Correspondence (category theory)
2530:Handbook of Categorical Algebra
2148:Lifting functors to profunctors
824:{\displaystyle D\nrightarrow C}
2454:
2425:
2409:, i.e. there exists a functor
2369:
2360:
2348:
2168:
2088:
2076:
2067:
2055:
2030:
2018:
2015:
2006:
1976:
1964:
1921:
1904:
1818:
1806:
1800:
1778:
1720:
1708:
1699:
1687:
1657:
1645:
1642:
1633:
1596:
1575:
1563:
1477:
1468:
1435:{\displaystyle {\hat {\psi }}}
1426:
1396:
1390:
1381:
1330:
1318:
1312:
1303:
1058:
671:
639:
630:
618:
571:{\displaystyle \mathbf {Cat} }
530:
508:
490:
478:
446:
429:
366:
343:
316:{\displaystyle \mathbf {Set} }
212:
13:
1:
2543:. Princeton University Press.
2504:
2142:
50:
2184:can be seen as a profunctor
580:category of small categories
7:
2492:
10:
2599:
2138:between those profunctors.
1182:{\displaystyle \psi \phi }
1162:Composition of profunctors
680:{\displaystyle {\hat {D}}}
2528:Borceux, Francis (1994).
2296:{\displaystyle \phi _{F}}
2105:Bicategory of profunctors
835:Profunctors as categories
602:can be seen as a functor
65:by the French school and
2434:{\displaystyle F:C\to D}
2177:{\displaystyle F:C\to D}
39:are a generalization of
2136:natural transformations
1553:associates the functor
1513:(which to every object
543:to denote the actions.
83:{\displaystyle \,\phi }
2483:
2435:
2403:
2379:
2329:
2297:
2266:
2217:
2178:
2095:
1983:
1928:
1865:
1845:
1825:
1765:
1742:
1614:
1547:
1527:
1507:
1487:
1436:
1403:
1340:
1250:
1218:
1183:
1152:
1132:
1109:
1076:
1025:
1005:
985:
965:
945:
925:
905:
885:
865:
825:
799:
779:
752:
728:
681:
649:
596:
572:
537:
453:
407:
381:
317:
289:
265:
230:
162:
127:
107:
84:
69:by the Sydney school)
2537:Lurie, Jacob (2009).
2484:
2436:
2404:
2380:
2330:
2298:
2267:
2218:
2179:
2096:
1984:
1929:
1866:
1846:
1826:
1766:
1764:{\displaystyle \sim }
1743:
1623:It can be shown that
1615:
1548:
1528:
1508:
1488:
1437:
1404:
1341:
1251:
1219:
1184:
1153:
1133:
1110:
1077:
1026:
1006:
986:
966:
946:
931:and the morphisms of
926:
906:
886:
866:
826:
800:
780:
753:
729:
687:denotes the category
682:
650:
597:
595:{\displaystyle \phi }
573:
538:
454:
408:
382:
318:
290:
266:
231:
163:
128:
108:
85:
2518:Distributors at Work
2445:
2413:
2393:
2385:factors through the
2339:
2307:
2280:
2230:
2188:
2156:
2003:
1938:
1878:
1855:
1835:
1775:
1755:
1630:
1557:
1537:
1517:
1497:
1449:
1417:
1353:
1266:
1228:
1196:
1170:
1142:
1122:
1086:
1039:
1015:
995:
975:
955:
935:
915:
895:
875:
843:
809:
789:
769:
742:
691:
662:
609:
586:
554:
463:
417:
391:
331:
299:
279:
243:
182:
140:
117:
97:
73:
2540:Higher Topos Theory
1189:of two profunctors
1138:and the objects of
891:and the objects of
406:{\displaystyle D,C}
172:is defined to be a
2479:
2431:
2399:
2375:
2325:
2293:
2262:
2213:
2174:
2091:
1979:
1924:
1861:
1841:
1821:
1761:
1738:
1683:
1610:
1543:
1523:
1503:
1483:
1432:
1399:
1336:
1246:
1214:
1179:
1148:
1128:
1105:
1072:
1021:
1001:
981:
961:
941:
921:
901:
881:
861:
821:
795:
775:
748:
724:
677:
645:
592:
568:
533:
449:
403:
377:
327:. Given morphisms
313:
285:
261:
226:
158:
123:
103:
80:
2457:
2402:{\displaystyle D}
2387:Cauchy completion
2372:
2351:
1864:{\displaystyle D}
1844:{\displaystyle v}
1668:
1546:{\displaystyle D}
1526:{\displaystyle d}
1506:{\displaystyle D}
1480:
1429:
1393:
1333:
1315:
1151:{\displaystyle D}
1131:{\displaystyle C}
1096:
1049:
1024:{\displaystyle C}
1004:{\displaystyle D}
984:{\displaystyle C}
964:{\displaystyle D}
944:{\displaystyle D}
924:{\displaystyle C}
904:{\displaystyle D}
884:{\displaystyle C}
798:{\displaystyle D}
778:{\displaystyle C}
751:{\displaystyle D}
674:
642:
621:
582:, the profunctor
548:cartesian closure
288:{\displaystyle D}
273:opposite category
126:{\displaystyle D}
106:{\displaystyle C}
16:(Redirected from
2590:
2544:
2533:
2524:
2523:
2488:
2486:
2485:
2480:
2472:
2471:
2459:
2458:
2450:
2440:
2438:
2437:
2432:
2408:
2406:
2405:
2400:
2384:
2382:
2381:
2376:
2374:
2373:
2365:
2353:
2352:
2344:
2334:
2332:
2331:
2326:
2302:
2300:
2299:
2294:
2292:
2291:
2271:
2269:
2268:
2263:
2255:
2254:
2242:
2241:
2222:
2220:
2219:
2214:
2200:
2199:
2183:
2181:
2180:
2175:
2126:small categories
2100:
2098:
2097:
2092:
2051:
2050:
1988:
1986:
1985:
1980:
1948:
1933:
1931:
1930:
1925:
1920:
1888:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1830:
1828:
1827:
1822:
1799:
1788:
1770:
1768:
1767:
1762:
1747:
1745:
1744:
1739:
1734:
1733:
1727:
1723:
1682:
1619:
1617:
1616:
1611:
1609:
1595:
1594:
1593:
1552:
1550:
1549:
1544:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1492:
1490:
1489:
1484:
1482:
1481:
1473:
1461:
1460:
1441:
1439:
1438:
1433:
1431:
1430:
1422:
1408:
1406:
1405:
1400:
1395:
1394:
1386:
1380:
1379:
1378:
1377:
1367:
1345:
1343:
1342:
1337:
1335:
1334:
1326:
1317:
1316:
1308:
1302:
1301:
1300:
1299:
1289:
1255:
1253:
1252:
1247:
1223:
1221:
1220:
1215:
1188:
1186:
1185:
1180:
1157:
1155:
1154:
1149:
1137:
1135:
1134:
1129:
1114:
1112:
1111:
1106:
1098:
1097:
1094:
1081:
1079:
1078:
1073:
1071:
1051:
1050:
1047:
1030:
1028:
1027:
1022:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
970:
968:
967:
962:
950:
948:
947:
942:
930:
928:
927:
922:
910:
908:
907:
902:
890:
888:
887:
882:
870:
868:
867:
862:
830:
828:
827:
822:
805:is a profunctor
804:
802:
801:
796:
784:
782:
781:
776:
757:
755:
754:
749:
733:
731:
730:
725:
723:
722:
721:
720:
719:
705:
686:
684:
683:
678:
676:
675:
667:
654:
652:
651:
646:
644:
643:
635:
623:
622:
614:
601:
599:
598:
593:
577:
575:
574:
569:
567:
542:
540:
539:
534:
529:
518:
458:
456:
455:
450:
439:
412:
410:
409:
404:
387:respectively in
386:
384:
383:
378:
376:
353:
325:category of sets
322:
320:
319:
314:
312:
294:
292:
291:
286:
270:
268:
267:
262:
260:
259:
258:
235:
233:
232:
227:
225:
205:
204:
203:
167:
165:
164:
159:
132:
130:
129:
124:
112:
110:
109:
104:
89:
87:
86:
81:
21:
2598:
2597:
2593:
2592:
2591:
2589:
2588:
2587:
2573:
2572:
2571:
2521:
2507:
2495:
2467:
2463:
2449:
2448:
2446:
2443:
2442:
2414:
2411:
2410:
2394:
2391:
2390:
2364:
2363:
2343:
2342:
2340:
2337:
2336:
2308:
2305:
2304:
2287:
2283:
2281:
2278:
2277:
2250:
2246:
2237:
2233:
2231:
2228:
2227:
2195:
2191:
2189:
2186:
2185:
2157:
2154:
2153:
2150:
2145:
2107:
2040:
2036:
2004:
2001:
2000:
1941:
1939:
1936:
1935:
1913:
1881:
1879:
1876:
1875:
1856:
1853:
1852:
1836:
1833:
1832:
1792:
1781:
1776:
1773:
1772:
1756:
1753:
1752:
1729:
1728:
1672:
1667:
1663:
1631:
1628:
1627:
1599:
1586:
1585:
1581:
1558:
1555:
1554:
1538:
1535:
1534:
1518:
1515:
1514:
1498:
1495:
1494:
1472:
1471:
1456:
1452:
1450:
1447:
1446:
1421:
1420:
1418:
1415:
1414:
1413:of the functor
1385:
1384:
1373:
1369:
1368:
1357:
1356:
1354:
1351:
1350:
1325:
1324:
1307:
1306:
1295:
1291:
1290:
1279:
1278:
1267:
1264:
1263:
1229:
1226:
1225:
1197:
1194:
1193:
1171:
1168:
1167:
1164:
1143:
1140:
1139:
1123:
1120:
1119:
1093:
1089:
1087:
1084:
1083:
1061:
1046:
1042:
1040:
1037:
1036:
1033:heteromorphisms
1016:
1013:
1012:
1011:and objects of
996:
993:
992:
976:
973:
972:
956:
953:
952:
936:
933:
932:
916:
913:
912:
896:
893:
892:
876:
873:
872:
844:
841:
840:
837:
810:
807:
806:
790:
787:
786:
770:
767:
766:
743:
740:
739:
712:
711:
707:
706:
695:
694:
692:
689:
688:
666:
665:
663:
660:
659:
634:
633:
613:
612:
610:
607:
606:
587:
584:
583:
557:
555:
552:
551:
522:
511:
464:
461:
460:
432:
418:
415:
414:
413:and an element
392:
389:
388:
369:
346:
332:
329:
328:
302:
300:
297:
296:
280:
277:
276:
251:
250:
246:
244:
241:
240:
215:
196:
195:
191:
183:
180:
179:
141:
138:
137:
118:
115:
114:
98:
95:
94:
74:
71:
70:
53:
29:category theory
23:
22:
15:
12:
11:
5:
2596:
2586:
2585:
2570:
2569:
2560:Heteromorphism
2557:
2545:
2534:
2525:
2508:
2506:
2503:
2502:
2501:
2494:
2491:
2478:
2475:
2470:
2466:
2462:
2456:
2453:
2430:
2427:
2424:
2421:
2418:
2398:
2371:
2368:
2362:
2359:
2356:
2350:
2347:
2324:
2321:
2318:
2315:
2312:
2290:
2286:
2274:
2273:
2261:
2258:
2253:
2249:
2245:
2240:
2236:
2212:
2209:
2206:
2203:
2198:
2194:
2173:
2170:
2167:
2164:
2161:
2149:
2146:
2144:
2141:
2140:
2139:
2132:
2129:
2106:
2103:
2102:
2101:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2049:
2046:
2043:
2039:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
1991:
1990:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1947:
1944:
1923:
1919:
1916:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1887:
1884:
1860:
1840:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1798:
1795:
1791:
1787:
1784:
1780:
1760:
1749:
1748:
1737:
1732:
1726:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1681:
1678:
1675:
1671:
1666:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1608:
1605:
1602:
1598:
1592:
1589:
1584:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1542:
1522:
1502:
1479:
1476:
1470:
1467:
1464:
1459:
1455:
1444:Yoneda functor
1428:
1425:
1398:
1392:
1389:
1383:
1376:
1372:
1366:
1363:
1360:
1347:
1346:
1332:
1329:
1323:
1320:
1314:
1311:
1305:
1298:
1294:
1288:
1285:
1282:
1277:
1274:
1271:
1257:
1256:
1245:
1242:
1239:
1236:
1233:
1213:
1210:
1207:
1204:
1201:
1178:
1175:
1166:The composite
1163:
1160:
1147:
1127:
1104:
1101:
1092:
1070:
1067:
1064:
1060:
1057:
1054:
1045:
1020:
1000:
980:
971:to objects of
960:
940:
920:
900:
880:
860:
857:
854:
851:
848:
836:
833:
820:
817:
814:
794:
774:
763:correspondence
747:
718:
715:
710:
704:
701:
698:
673:
670:
656:
655:
641:
638:
632:
629:
626:
620:
617:
591:
566:
563:
560:
532:
528:
525:
521:
517:
514:
510:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
448:
445:
442:
438:
435:
431:
428:
425:
422:
402:
399:
396:
375:
372:
368:
365:
362:
359:
356:
352:
349:
345:
342:
339:
336:
311:
308:
305:
284:
257:
254:
249:
237:
236:
224:
221:
218:
214:
211:
208:
202:
199:
194:
190:
187:
170:
169:
157:
154:
151:
148:
145:
122:
113:to a category
102:
79:
52:
49:
31:, a branch of
9:
6:
4:
3:
2:
2595:
2584:
2581:
2580:
2578:
2568:
2566:
2561:
2558:
2556:
2554:
2549:
2546:
2542:
2541:
2535:
2531:
2526:
2520:
2519:
2514:
2513:BĂ©nabou, Jean
2510:
2509:
2500:
2497:
2496:
2490:
2476:
2473:
2468:
2464:
2460:
2451:
2428:
2422:
2419:
2416:
2396:
2388:
2366:
2357:
2354:
2345:
2322:
2319:
2316:
2313:
2310:
2288:
2284:
2259:
2256:
2251:
2247:
2243:
2238:
2234:
2226:
2225:
2224:
2210:
2207:
2204:
2201:
2196:
2192:
2171:
2165:
2162:
2159:
2137:
2133:
2130:
2127:
2123:
2122:
2121:
2119:
2116:
2112:
2085:
2082:
2079:
2073:
2070:
2064:
2061:
2058:
2052:
2047:
2044:
2041:
2037:
2033:
2027:
2024:
2021:
2012:
2009:
1999:
1998:
1997:
1996:
1973:
1970:
1967:
1961:
1958:
1955:
1952:
1949:
1945:
1942:
1917:
1914:
1910:
1907:
1901:
1898:
1895:
1892:
1889:
1885:
1882:
1874:
1873:
1872:
1858:
1838:
1815:
1812:
1809:
1803:
1796:
1793:
1789:
1785:
1782:
1758:
1735:
1724:
1717:
1714:
1711:
1705:
1702:
1696:
1693:
1690:
1684:
1679:
1676:
1673:
1669:
1664:
1660:
1654:
1651:
1648:
1639:
1636:
1626:
1625:
1624:
1621:
1582:
1578:
1572:
1569:
1566:
1560:
1540:
1520:
1500:
1474:
1465:
1462:
1457:
1453:
1445:
1423:
1412:
1411:Kan extension
1387:
1374:
1370:
1327:
1321:
1309:
1296:
1292:
1275:
1272:
1269:
1262:
1261:
1260:
1243:
1240:
1237:
1234:
1231:
1211:
1208:
1205:
1202:
1199:
1192:
1191:
1190:
1176:
1173:
1159:
1145:
1125:
1116:
1102:
1099:
1090:
1055:
1052:
1043:
1034:
1018:
998:
978:
958:
938:
918:
898:
878:
858:
855:
852:
849:
846:
832:
818:
815:
812:
792:
772:
764:
759:
745:
737:
708:
668:
636:
627:
624:
615:
605:
604:
603:
589:
581:
549:
544:
526:
523:
519:
515:
512:
505:
502:
499:
496:
493:
487:
484:
481:
475:
472:
469:
466:
443:
440:
436:
433:
426:
423:
420:
400:
397:
394:
373:
370:
363:
360:
357:
354:
350:
347:
340:
337:
334:
326:
282:
274:
247:
209:
206:
192:
188:
185:
178:
177:
176:
175:
155:
152:
149:
146:
143:
136:
135:
134:
120:
100:
93:
77:
68:
64:
63:
58:
48:
46:
42:
38:
34:
30:
19:
2564:
2552:
2538:
2529:
2517:
2275:
2151:
2124:0-cells are
2117:
2110:
2108:
1992:
1750:
1622:
1409:is the left
1348:
1259:is given by
1258:
1165:
1117:
1032:
838:
762:
760:
657:
545:
323:denotes the
271:denotes the
238:
171:
66:
60:
59:(also named
56:
54:
43:and also of
36:
26:
459:, we write
62:distributor
37:profunctors
33:mathematics
2548:Profunctor
2505:References
2499:Anafunctor
2441:such that
2152:A functor
2143:Properties
2115:bicategory
1871:such that
1442:along the
736:presheaves
546:Using the
133:, written
57:profunctor
51:Definition
2474:∘
2455:^
2452:ϕ
2426:→
2370:^
2361:→
2349:^
2346:ϕ
2320:↛
2311:ϕ
2285:ϕ
2257:∘
2235:ϕ
2208:↛
2193:ϕ
2169:→
2074:ϕ
2071:×
2053:ψ
2045::
2038:∫
2013:ϕ
2010:ψ
1962:ϕ
1959:∈
1902:ψ
1899:∈
1804:∼
1759:∼
1736:∼
1706:ϕ
1703:×
1685:ψ
1677:∈
1670:∐
1640:ϕ
1637:ψ
1597:→
1567:−
1478:^
1469:→
1427:^
1424:ψ
1391:^
1388:ψ
1331:^
1328:ϕ
1322:∘
1313:^
1310:ψ
1273:ϕ
1270:ψ
1241:↛
1232:ψ
1209:↛
1200:ϕ
1177:ϕ
1174:ψ
1100:×
1059:→
1056:ϕ
1053:×
1044:ϕ
856:↛
847:ϕ
816:↛
672:^
640:^
631:→
619:^
616:ϕ
590:ϕ
506:ϕ
503:∈
476:ϕ
473:∈
427:ϕ
424:∈
367:→
344:→
213:→
207:×
186:ϕ
153:↛
144:ϕ
78:ϕ
45:bimodules
41:relations
2583:Functors
2577:Category
2515:(2000),
2493:See also
1946:′
1918:′
1886:′
1797:′
1786:′
527:′
516:′
437:′
374:′
351:′
92:category
2562:at the
2550:at the
174:functor
90:from a
2532:. CUP.
2120:whose
1751:where
1349:where
658:where
578:, the
239:where
67:module
2522:(PDF)
1995:coend
1934:and
765:from
738:over
2118:Prof
1224:and
295:and
2567:Lab
2555:Lab
2389:of
2111:Set
1851:in
1620:).
1533:of
1493:of
1082:to
785:to
734:of
550:of
275:of
27:In
2579::
2489:.
1115:.
1095:op
1048:op
831:.
761:A
758:.
55:A
47:.
35:,
2565:n
2553:n
2477:F
2469:D
2465:Y
2461:=
2429:D
2423:C
2420::
2417:F
2397:D
2367:D
2358:C
2355::
2323:D
2317:C
2314::
2289:F
2272:.
2260:F
2252:D
2248:Y
2244:=
2239:F
2211:D
2205:C
2202::
2197:F
2172:D
2166:C
2163::
2160:F
2128:,
2089:)
2086:c
2083:,
2080:d
2077:(
2068:)
2065:d
2062:,
2059:e
2056:(
2048:D
2042:d
2034:=
2031:)
2028:c
2025:,
2022:e
2019:(
2016:)
2007:(
1989:.
1977:)
1974:c
1971:,
1968:d
1965:(
1956:x
1953:=
1950:v
1943:x
1922:)
1915:d
1911:,
1908:e
1905:(
1896:y
1893:v
1890:=
1883:y
1859:D
1839:v
1819:)
1816:x
1813:,
1810:y
1807:(
1801:)
1794:x
1790:,
1783:y
1779:(
1731:/
1725:)
1721:)
1718:c
1715:,
1712:d
1709:(
1700:)
1697:d
1694:,
1691:e
1688:(
1680:D
1674:d
1665:(
1661:=
1658:)
1655:c
1652:,
1649:e
1646:(
1643:)
1634:(
1607:t
1604:e
1601:S
1591:p
1588:o
1583:D
1579::
1576:)
1573:d
1570:,
1564:(
1561:D
1541:D
1521:d
1501:D
1475:D
1466:D
1463::
1458:D
1454:Y
1397:)
1382:(
1375:D
1371:Y
1365:n
1362:a
1359:L
1319:)
1304:(
1297:D
1293:Y
1287:n
1284:a
1281:L
1276:=
1244:E
1238:D
1235::
1212:D
1206:C
1203::
1146:D
1126:C
1103:C
1091:D
1069:t
1066:e
1063:S
1019:C
999:D
979:C
959:D
939:D
919:C
899:D
879:C
859:D
853:C
850::
819:C
813:D
793:D
773:C
746:D
717:p
714:o
709:D
703:t
700:e
697:S
669:D
637:D
628:C
625::
565:t
562:a
559:C
531:)
524:c
520:,
513:d
509:(
500:x
497:g
494:,
491:)
488:c
485:,
482:d
479:(
470:f
467:x
447:)
444:c
441:,
434:d
430:(
421:x
401:C
398:,
395:D
371:c
364:c
361::
358:g
355:,
348:d
341:d
338::
335:f
310:t
307:e
304:S
283:D
256:p
253:o
248:D
223:t
220:e
217:S
210:C
201:p
198:o
193:D
189::
168:,
156:D
150:C
147::
121:D
101:C
20:)
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