1972:
900:
696:
224:
686:
617:
895:{\displaystyle (A\otimes B)\circ (C\otimes D)={\begin{bmatrix}A&0\\0&B\end{bmatrix}}\circ {\begin{bmatrix}C&0\\0&D\end{bmatrix}}={\begin{bmatrix}AC&0\\0&BD\end{bmatrix}}=(A\circ C)\otimes (B\circ D)}
276:
363:
1435:
87:
458:
1092:
546:
304:
1504:
1147:
1252:
1003:
957:
931:
137:
1048:
977:
508:
1626:
1578:
626:
396:
2013:
1730:
1706:
1686:
1666:
1646:
1602:
1555:
1524:
1475:
1455:
1395:
1375:
1355:
1335:
1311:
1288:
1115:
482:
427:
551:
1952:
1915:
1841:
2006:
128:”, which is a particular kind of PROP, for the object which Boardman and Vogt called the "category of operators in standard form".
1779:
244:
2037:
1117:); these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product.
313:
2032:
1999:
28:
1709:
1255:
1181:
1400:
1905:
89:
and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each
42:
1095:
432:
399:
1057:
1940:
1746:
1015:
110:
513:
285:
1487:
1130:
219:{\displaystyle {\mathsf {Operads}}\subset {\tfrac {1}{2}}{\mathsf {PROP}}\subset {\mathsf {PROP}}}
116:
The notion was introduced by Adams and Mac Lane; the topological version of it was later given by
1930:
1847:
1230:
102:
982:
936:
910:
681:{\displaystyle \alpha \otimes \beta ={\begin{bmatrix}\alpha &0\\0&\beta \end{bmatrix}}}
1948:
1911:
1866:
1837:
1291:
1125:
1051:
1033:
962:
493:
32:
1983:
1611:
1563:
1882:
1829:
1794:
1314:
1225:
959:
matrices) are allowed, and with respect to multiplication count as being zero matrices. The
381:
117:
1806:
1979:
1802:
461:
366:
94:
20:
1022:
of a permutation on a matrix (morphism of this PROP) is to permute the rows, whereas the
1944:
1741:
1715:
1691:
1671:
1651:
1631:
1587:
1540:
1509:
1460:
1440:
1380:
1360:
1340:
1320:
1296:
1273:
1100:
467:
412:
1833:
2026:
1897:
1581:
1534:
1887:
1870:
1798:
1901:
1851:
620:
121:
1971:
612:{\displaystyle {\mathcal {R}}^{m}\otimes {\mathcal {R}}^{n}={\mathcal {R}}^{m+n}}
1203:
282:
matrices (regardless of number of rows and columns) over some fixed ring
1176:
If the requirement “symmetric” is dropped, then one gets the notion of
1935:
1154:
125:
1477:
and whose morphisms are the natural transformations between them.
1054:, but in that class of PROPs the matrices must all be of the form
229:
where the first category is the category of (symmetric) operads.
691:
The compatibility of composition and product thus boils down to
365:(sets of vectors) or just as the plain natural numbers (since
592:
575:
558:
439:
323:
291:
251:
131:
There are the following inclusions of full subcategories:
271:{\displaystyle {\mathcal {R}}^{\bullet \times \bullet }}
1987:
1608:
More precisely, what we mean here by "the algebras of
819:
780:
741:
647:
358:{\displaystyle \{{\mathcal {R}}^{n}\}_{n=0}^{\infty }}
170:
1718:
1694:
1674:
1654:
1634:
1614:
1590:
1566:
1543:
1512:
1490:
1463:
1443:
1403:
1383:
1363:
1343:
1323:
1299:
1276:
1233:
1133:
1103:
1060:
1036:
985:
965:
939:
913:
699:
629:
619:) and on morphisms like an operation of constructing
554:
516:
496:
470:
435:
415:
384:
316:
288:
247:
140:
45:
1030:There are also PROPs of matrices where the product
109:. The name PROP is an abbreviation of "PROduct and
1724:
1700:
1680:
1668:" for example is that the category of algebras of
1660:
1640:
1620:
1596:
1572:
1549:
1518:
1498:
1469:
1449:
1429:
1389:
1369:
1349:
1329:
1305:
1282:
1246:
1141:
1109:
1086:
1042:
997:
971:
951:
925:
894:
680:
611:
540:
502:
476:
452:
421:
390:
357:
298:
270:
218:
81:
310:of the PROP; the objects can be taken as either
372:be sets with some structure). In this example:
1437:of algebras whose objects are the algebras of
1262:is an example of PRO that is not even a PROB.
1158:of natural numbers and functions between them,
2007:
1875:Bulletin of the American Mathematical Society
8:
335:
317:
76:
46:
16:Type of monoidal category in category theory
1820:Markl, Martin (2006). "Operads and PROPs".
2014:
2000:
306:. More concretely, these matrices are the
1934:
1886:
1717:
1693:
1673:
1653:
1633:
1613:
1589:
1565:
1542:
1511:
1492:
1491:
1489:
1462:
1442:
1421:
1416:
1405:
1402:
1382:
1362:
1342:
1322:
1298:
1275:
1238:
1232:
1135:
1134:
1132:
1102:
1078:
1065:
1059:
1035:
984:
964:
938:
912:
814:
775:
736:
698:
642:
628:
597:
591:
590:
580:
574:
573:
563:
557:
556:
553:
515:
495:
469:
444:
438:
437:
434:
414:
383:
349:
338:
328:
322:
321:
315:
290:
289:
287:
256:
250:
249:
246:
201:
200:
182:
181:
169:
142:
141:
139:
44:
1907:Operads in Algebra, Topology and Physics
1765:
1180:category. If “symmetric” is replaced by
907:As an edge case, matrices with no rows (
1758:
1430:{\displaystyle \mathrm {Alg} _{P}^{C}}
1202:of natural numbers, equipped with the
211:
208:
205:
202:
192:
189:
186:
183:
161:
158:
155:
152:
149:
146:
143:
101:letters is given as a subgroup of the
35:whose objects are the natural numbers
1094:(sides are all powers of some common
7:
1968:
1966:
1778:Boardman, J.M.; Vogt, R.M. (1968).
82:{\displaystyle \{0,1,\ldots ,n-1\}}
1986:. You can help Knowledge (XXG) by
1615:
1567:
1412:
1409:
1406:
1235:
1172:of natural numbers and injections.
1165:of natural numbers and bijections,
453:{\displaystyle {\mathcal {R}}^{n}}
350:
14:
1927:Higher Operads, Higher Categories
1910:. American Mathematical Society.
1087:{\displaystyle k^{m}\times k^{n}}
1970:
39:identified with the finite sets
1888:10.1090/S0002-9904-1965-11234-4
1799:10.1090/S0002-9904-1968-12070-1
1780:"Homotopy-everything H -spaces"
1187:, then one gets the notion of
510:acts on objects like addition (
1929:. Cambridge University Press.
1712:to the category of monoids in
889:
877:
871:
859:
730:
718:
712:
700:
541:{\displaystyle m\otimes n=m+n}
299:{\displaystyle {\mathcal {R}}}
1:
1834:10.1016/S1570-7954(07)05002-4
1212:as the automorphisms of each
23:, a branch of mathematics, a
1499:{\displaystyle \mathbb {N} }
1142:{\displaystyle \mathbb {N} }
241:class of PROPs are the sets
1247:{\displaystyle \Delta _{+}}
1120:Further examples of PROPs:
124:then introduced the term “
120:and Vogt. Following them,
2054:
1965:
1648:are the monoid objects in
1256:order-preserving functions
1226:augmented simplex category
1220:is a PROB but not a PROP.
1026:is to permute the columns.
1216:(and no other morphisms).
998:{\displaystyle 0\times 0}
952:{\displaystyle m\times 0}
933:matrices) or no columns (
926:{\displaystyle 0\times n}
398:of morphisms is ordinary
1397:give rise to a category
1043:{\displaystyle \otimes }
972:{\displaystyle \otimes }
503:{\displaystyle \otimes }
1621:{\displaystyle \Delta }
1573:{\displaystyle \Delta }
1254:of natural numbers and
621:block diagonal matrices
1982:-related article is a
1925:Leinster, Tom (2004).
1726:
1702:
1682:
1662:
1642:
1622:
1598:
1574:
1551:
1520:
1500:
1471:
1451:
1431:
1391:
1371:
1351:
1331:
1307:
1284:
1248:
1143:
1111:
1088:
1044:
999:
973:
953:
927:
896:
682:
613:
542:
504:
478:
454:
423:
392:
391:{\displaystyle \circ }
359:
300:
272:
220:
83:
2038:Category theory stubs
1871:"Categorical Algebra"
1787:Bull. Amer. Math. Soc
1727:
1703:
1683:
1663:
1643:
1623:
1599:
1575:
1552:
1521:
1506:is just an object of
1501:
1472:
1452:
1432:
1392:
1372:
1352:
1332:
1308:
1285:
1249:
1144:
1112:
1089:
1045:
1000:
974:
954:
928:
897:
683:
614:
543:
505:
479:
455:
424:
400:matrix multiplication
393:
360:
301:
273:
233:Examples and variants
221:
84:
1747:Permutation category
1716:
1692:
1672:
1652:
1632:
1612:
1588:
1564:
1541:
1510:
1488:
1461:
1441:
1401:
1381:
1361:
1341:
1321:
1297:
1274:
1270:An algebra of a PRO
1231:
1131:
1101:
1058:
1034:
1016:permutation matrices
1014:in the PROP are the
983:
963:
937:
911:
697:
627:
552:
514:
494:
468:
433:
413:
382:
314:
286:
245:
138:
111:Permutation category
43:
2033:Monoidal categories
1945:2004hohc.book.....L
1822:Handbook of Algebra
1426:
1149:of natural numbers,
354:
1867:Mac Lane, Saunders
1722:
1698:
1678:
1658:
1638:
1618:
1594:
1570:
1547:
1516:
1496:
1467:
1447:
1427:
1404:
1387:
1367:
1347:
1327:
1303:
1280:
1244:
1139:
1107:
1084:
1040:
995:
969:
949:
923:
892:
850:
805:
766:
678:
672:
609:
538:
500:
474:
450:
419:
388:
355:
334:
296:
268:
216:
179:
103:automorphism group
79:
1995:
1994:
1954:978-0-521-53215-0
1917:978-0-8218-4362-8
1843:978-0-444-53101-8
1725:{\displaystyle C}
1701:{\displaystyle C}
1681:{\displaystyle P}
1661:{\displaystyle C}
1641:{\displaystyle C}
1597:{\displaystyle C}
1550:{\displaystyle C}
1533:is a commutative
1519:{\displaystyle C}
1470:{\displaystyle C}
1450:{\displaystyle P}
1390:{\displaystyle C}
1370:{\displaystyle P}
1350:{\displaystyle C}
1330:{\displaystyle P}
1306:{\displaystyle C}
1292:monoidal category
1283:{\displaystyle P}
1266:Algebras of a PRO
1126:discrete category
1110:{\displaystyle k}
1052:Kronecker product
477:{\displaystyle n}
422:{\displaystyle n}
407:identity morphism
178:
33:monoidal category
2045:
2016:
2009:
2002:
1974:
1967:
1958:
1938:
1921:
1892:
1890:
1857:
1855:
1817:
1811:
1810:
1784:
1775:
1769:
1763:
1731:
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1723:
1707:
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1704:
1699:
1687:
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1659:
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1619:
1603:
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1600:
1595:
1579:
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1571:
1556:
1554:
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1548:
1525:
1523:
1522:
1517:
1505:
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1502:
1497:
1495:
1476:
1474:
1473:
1468:
1456:
1454:
1453:
1448:
1436:
1434:
1433:
1428:
1425:
1420:
1415:
1396:
1394:
1393:
1388:
1376:
1374:
1373:
1368:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1315:monoidal functor
1312:
1310:
1309:
1304:
1289:
1287:
1286:
1281:
1253:
1251:
1250:
1245:
1243:
1242:
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1146:
1145:
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1116:
1114:
1113:
1108:
1093:
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1090:
1085:
1083:
1082:
1070:
1069:
1049:
1047:
1046:
1041:
1004:
1002:
1001:
996:
979:identity is the
978:
976:
975:
970:
958:
956:
955:
950:
932:
930:
929:
924:
901:
899:
898:
893:
855:
854:
810:
809:
771:
770:
687:
685:
684:
679:
677:
676:
618:
616:
615:
610:
608:
607:
596:
595:
585:
584:
579:
578:
568:
567:
562:
561:
547:
545:
544:
539:
509:
507:
506:
501:
483:
481:
480:
475:
459:
457:
456:
451:
449:
448:
443:
442:
428:
426:
425:
420:
397:
395:
394:
389:
364:
362:
361:
356:
353:
348:
333:
332:
327:
326:
305:
303:
302:
297:
295:
294:
277:
275:
274:
269:
267:
266:
255:
254:
225:
223:
222:
217:
215:
214:
196:
195:
180:
171:
165:
164:
88:
86:
85:
80:
2053:
2052:
2048:
2047:
2046:
2044:
2043:
2042:
2023:
2022:
2021:
2020:
1980:category theory
1963:
1961:
1955:
1924:
1918:
1896:Markl, Martin;
1895:
1865:
1861:
1860:
1844:
1819:
1818:
1814:
1782:
1777:
1776:
1772:
1764:
1760:
1755:
1738:
1714:
1713:
1690:
1689:
1670:
1669:
1650:
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1630:
1629:
1610:
1609:
1586:
1585:
1562:
1561:
1539:
1538:
1508:
1507:
1486:
1485:
1459:
1458:
1439:
1438:
1399:
1398:
1379:
1378:
1359:
1358:
1339:
1338:
1319:
1318:
1295:
1294:
1272:
1271:
1268:
1234:
1229:
1228:
1210:
1200:
1129:
1128:
1099:
1098:
1074:
1061:
1056:
1055:
1032:
1031:
981:
980:
961:
960:
935:
934:
909:
908:
849:
848:
840:
834:
833:
828:
815:
804:
803:
798:
792:
791:
786:
776:
765:
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752:
747:
737:
695:
694:
671:
670:
665:
659:
658:
653:
643:
625:
624:
589:
572:
555:
550:
549:
512:
511:
492:
491:
466:
465:
462:identity matrix
436:
431:
430:
411:
410:
380:
379:
320:
312:
311:
284:
283:
248:
243:
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136:
135:
95:symmetric group
41:
40:
21:category theory
17:
12:
11:
5:
2051:
2049:
2041:
2040:
2035:
2025:
2024:
2019:
2018:
2011:
2004:
1996:
1993:
1992:
1975:
1960:
1959:
1953:
1922:
1916:
1898:Shnider, Steve
1893:
1862:
1859:
1858:
1842:
1812:
1793:(6): 1117–22.
1770:
1768:, Ch. V, § 24.
1757:
1756:
1754:
1751:
1750:
1749:
1744:
1742:Lawvere theory
1737:
1734:
1721:
1697:
1677:
1657:
1637:
1617:
1606:
1605:
1593:
1569:
1560:an algebra of
1558:
1546:
1529:an algebra of
1527:
1515:
1494:
1484:an algebra of
1466:
1446:
1424:
1419:
1414:
1411:
1408:
1386:
1366:
1346:
1326:
1302:
1279:
1267:
1264:
1260:
1259:
1241:
1237:
1218:
1217:
1208:
1198:
1174:
1173:
1166:
1159:
1150:
1137:
1106:
1081:
1077:
1073:
1068:
1064:
1039:
1028:
1027:
1008:
1007:
1006:
994:
991:
988:
968:
948:
945:
942:
922:
919:
916:
905:
904:
903:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
853:
847:
844:
841:
839:
836:
835:
832:
829:
827:
824:
821:
820:
818:
813:
808:
802:
799:
797:
794:
793:
790:
787:
785:
782:
781:
779:
774:
769:
763:
760:
758:
755:
754:
751:
748:
746:
743:
742:
740:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
675:
669:
666:
664:
661:
660:
657:
654:
652:
649:
648:
646:
641:
638:
635:
632:
606:
603:
600:
594:
588:
583:
577:
571:
566:
560:
537:
534:
531:
528:
525:
522:
519:
499:
485:
473:
447:
441:
418:
403:
387:
370:do not have to
352:
347:
344:
341:
337:
331:
325:
319:
293:
265:
262:
259:
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234:
231:
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213:
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199:
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148:
145:
78:
75:
72:
69:
66:
63:
60:
57:
54:
51:
48:
15:
13:
10:
9:
6:
4:
3:
2:
2050:
2039:
2036:
2034:
2031:
2030:
2028:
2017:
2012:
2010:
2005:
2003:
1998:
1997:
1991:
1989:
1985:
1981:
1976:
1973:
1969:
1964:
1956:
1950:
1946:
1942:
1937:
1932:
1928:
1923:
1919:
1913:
1909:
1908:
1903:
1902:Stasheff, Jim
1899:
1894:
1889:
1884:
1880:
1876:
1872:
1868:
1864:
1863:
1853:
1849:
1845:
1839:
1835:
1831:
1828:(1): 87–140.
1827:
1823:
1816:
1813:
1808:
1804:
1800:
1796:
1792:
1788:
1781:
1774:
1771:
1767:
1766:Mac Lane 1965
1762:
1759:
1752:
1748:
1745:
1743:
1740:
1739:
1735:
1733:
1719:
1711:
1695:
1675:
1655:
1635:
1591:
1583:
1582:monoid object
1559:
1544:
1536:
1535:monoid object
1532:
1528:
1513:
1483:
1482:
1481:
1480:For example:
1478:
1464:
1444:
1422:
1417:
1384:
1377:and category
1364:
1344:
1324:
1316:
1300:
1293:
1277:
1265:
1263:
1257:
1239:
1227:
1223:
1222:
1221:
1215:
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1753:References
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464:with side
239:elementary
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1072:×
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