1983:
911:
707:
235:
697:
628:
906:{\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)}
287:
374:
1446:
98:
469:
1103:
557:
315:
1515:
1158:
1263:
1014:
968:
942:
148:
1059:
988:
519:
1637:
1589:
637:
407:
1741:
1717:
1697:
1677:
1657:
1613:
1566:
1535:
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1466:
1406:
1386:
1366:
1346:
1322:
1299:
1126:
493:
438:
562:
2024:
139:”, which is a particular kind of PROP, for the object which Boardman and Vogt called the "category of operators in standard form".
1790:
255:
1963:
1926:
1852:
1128:); these are the coordinate counterparts of appropriate symmetric monoidal categories of vector spaces under tensor product.
17:
324:
2017:
2048:
1236:
1411:
2043:
2010:
1916:
113:
100:
and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each
53:
39:
443:
1068:
1720:
1266:
1192:
524:
296:
631:
1498:
1141:
230:{\displaystyle {\mathsf {Operads}}\subset {\tfrac {1}{2}}{\mathsf {PROP}}\subset {\mathsf {PROP}}}
127:
The notion was introduced by Adams and Mac Lane; the topological version of it was later given by
1241:
1998:
993:
947:
921:
692:{\displaystyle \alpha \otimes \beta ={\begin{bmatrix}\alpha &0\\0&\beta \end{bmatrix}}}
1044:
973:
504:
1622:
1574:
1106:
410:
970:
matrices) are allowed, and with respect to multiplication count as being zero matrices. The
392:
1951:
1817:
1757:
1026:
121:
8:
1033:
of a permutation on a matrix (morphism of this PROP) is to permute the rows, whereas the
1955:
1941:
1858:
1726:
1702:
1682:
1662:
1642:
1598:
1551:
1520:
1471:
1451:
1391:
1371:
1351:
1331:
1307:
1284:
1111:
478:
423:
1844:
1959:
1922:
1877:
1848:
1302:
1136:
1062:
43:
1898:
1881:
1809:
1893:
1862:
1840:
1805:
1325:
128:
1990:
1813:
623:{\displaystyle {\mathcal {R}}^{m}\otimes {\mathcal {R}}^{n}={\mathcal {R}}^{m+n}}
472:
377:
105:
31:
1994:
1752:
2037:
1908:
1592:
1545:
1912:
132:
1214:
293:
matrices (regardless of number of rows and columns) over some fixed ring
1187:
If the requirement “symmetric” is dropped, then one gets the notion of
1982:
1946:
1488:
and whose morphisms are the natural transformations between them.
1065:, but in that class of PROPs the matrices must all be of the form
240:
where the first category is the category of (symmetric) operads.
702:
The compatibility of composition and product thus boils down to
1165:
136:
376:(sets of vectors) or just as the plain natural numbers (since
142:
There are the following inclusions of full subcategories:
282:{\displaystyle {\mathcal {R}}^{\bullet \times \bullet }}
1619:
More precisely, what we mean here by "the algebras of
830:
791:
752:
658:
369:{\displaystyle \{{\mathcal {R}}^{n}\}_{n=0}^{\infty }}
181:
1729:
1705:
1685:
1665:
1645:
1625:
1601:
1577:
1554:
1523:
1501:
1474:
1454:
1414:
1394:
1374:
1354:
1334:
1310:
1287:
1244:
1144:
1114:
1071:
1047:
996:
976:
950:
924:
710:
640:
630:) and on morphisms like an operation of constructing
565:
527:
507:
481:
446:
426:
395:
327:
299:
258:
151:
56:
1906:
1041:There are also PROPs of matrices where the product
120:. The name PROP is an abbreviation of "PROduct and
1735:
1711:
1691:
1679:" for example is that the category of algebras of
1671:
1651:
1631:
1607:
1583:
1560:
1529:
1509:
1480:
1460:
1440:
1400:
1380:
1360:
1340:
1316:
1293:
1257:
1152:
1120:
1097:
1053:
1008:
982:
962:
936:
905:
691:
622:
551:
513:
487:
463:
432:
401:
368:
309:
281:
229:
92:
2035:
321:of the PROP; the objects can be taken as either
383:be sets with some structure). In this example:
1448:of algebras whose objects are the algebras of
1273:is an example of PRO that is not even a PROB.
1169:of natural numbers and functions between them,
2018:
1886:Bulletin of the American Mathematical Society
1788:
346:
328:
87:
57:
27:Type of monoidal category in category theory
1831:Markl, Martin (2006). "Operads and PROPs".
2025:
2011:
317:. More concretely, these matrices are the
1945:
1897:
1503:
1146:
1935:
1918:Operads in Algebra, Topology and Physics
1876:
1776:
1191:category. If “symmetric” is replaced by
918:As an edge case, matrices with no rows (
243:
14:
2036:
1441:{\displaystyle \mathrm {Alg} _{P}^{C}}
1213:of natural numbers, equipped with the
222:
219:
216:
213:
203:
200:
197:
194:
172:
169:
166:
163:
160:
157:
154:
112:letters is given as a subgroup of the
46:whose objects are the natural numbers
1830:
1105:(sides are all powers of some common
1977:
1276:
1789:Boardman, J.M.; Vogt, R.M. (1968).
93:{\displaystyle \{0,1,\ldots ,n-1\}}
24:
1626:
1578:
1423:
1420:
1417:
1246:
1183:of natural numbers and injections.
1176:of natural numbers and bijections,
603:
586:
569:
464:{\displaystyle {\mathcal {R}}^{n}}
450:
361:
334:
302:
262:
25:
2060:
1938:Higher Operads, Higher Categories
1921:. American Mathematical Society.
1098:{\displaystyle k^{m}\times k^{n}}
1981:
50:identified with the finite sets
1899:10.1090/S0002-9904-1965-11234-4
1810:10.1090/S0002-9904-1968-12070-1
1791:"Homotopy-everything H -spaces"
1198:, then one gets the notion of
521:acts on objects like addition (
1940:. Cambridge University Press.
1824:
1782:
1770:
1723:to the category of monoids in
900:
888:
882:
870:
741:
729:
723:
711:
552:{\displaystyle m\otimes n=m+n}
310:{\displaystyle {\mathcal {R}}}
13:
1:
1845:10.1016/S1570-7954(07)05002-4
1763:
1223:as the automorphisms of each
34:, a branch of mathematics, a
1997:. You can help Knowledge by
1510:{\displaystyle \mathbb {N} }
1153:{\displaystyle \mathbb {N} }
252:class of PROPs are the sets
7:
1746:
1258:{\displaystyle \Delta _{+}}
1131:Further examples of PROPs:
135:then introduced the term “
131:and Vogt. Following them,
10:
2065:
1976:
1659:are the monoid objects in
1267:order-preserving functions
1237:augmented simplex category
1231:is a PROB but not a PROP.
1037:is to permute the columns.
1227:(and no other morphisms).
1009:{\displaystyle 0\times 0}
963:{\displaystyle m\times 0}
944:matrices) or no columns (
937:{\displaystyle 0\times n}
409:of morphisms is ordinary
1408:give rise to a category
1054:{\displaystyle \otimes }
983:{\displaystyle \otimes }
514:{\displaystyle \otimes }
1632:{\displaystyle \Delta }
1584:{\displaystyle \Delta }
1265:of natural numbers and
632:block diagonal matrices
1993:-related article is a
1936:Leinster, Tom (2004).
1737:
1713:
1693:
1673:
1653:
1633:
1609:
1585:
1562:
1531:
1511:
1482:
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1402:
1382:
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1295:
1259:
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1055:
1010:
984:
964:
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907:
693:
624:
553:
515:
489:
465:
434:
403:
402:{\displaystyle \circ }
370:
311:
283:
231:
94:
2049:Category theory stubs
1882:"Categorical Algebra"
1798:Bull. Amer. Math. Soc
1738:
1714:
1694:
1674:
1654:
1634:
1610:
1586:
1563:
1532:
1517:is just an object of
1512:
1483:
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1403:
1383:
1363:
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1319:
1296:
1260:
1155:
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1011:
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965:
939:
908:
694:
625:
554:
516:
490:
466:
435:
411:matrix multiplication
404:
371:
312:
284:
244:Examples and variants
232:
95:
18:PRO (category theory)
1758:Permutation category
1727:
1703:
1683:
1663:
1643:
1623:
1599:
1575:
1552:
1521:
1499:
1472:
1452:
1412:
1392:
1372:
1352:
1332:
1308:
1285:
1281:An algebra of a PRO
1242:
1142:
1112:
1069:
1045:
1027:permutation matrices
1025:in the PROP are the
994:
974:
948:
922:
708:
638:
563:
525:
505:
479:
444:
424:
393:
325:
297:
256:
149:
122:Permutation category
54:
2044:Monoidal categories
1956:2004hohc.book.....L
1833:Handbook of Algebra
1437:
1160:of natural numbers,
365:
1878:Mac Lane, Saunders
1733:
1709:
1689:
1669:
1649:
1629:
1605:
1581:
1558:
1527:
1507:
1478:
1458:
1438:
1415:
1398:
1378:
1358:
1338:
1314:
1291:
1255:
1150:
1118:
1095:
1051:
1006:
980:
960:
934:
903:
861:
816:
777:
689:
683:
620:
549:
511:
485:
461:
430:
399:
366:
345:
307:
279:
227:
190:
114:automorphism group
90:
2006:
2005:
1965:978-0-521-53215-0
1928:978-0-8218-4362-8
1854:978-0-444-53101-8
1736:{\displaystyle C}
1712:{\displaystyle C}
1692:{\displaystyle P}
1672:{\displaystyle C}
1652:{\displaystyle C}
1608:{\displaystyle C}
1561:{\displaystyle C}
1544:is a commutative
1530:{\displaystyle C}
1481:{\displaystyle C}
1461:{\displaystyle P}
1401:{\displaystyle C}
1381:{\displaystyle P}
1361:{\displaystyle C}
1341:{\displaystyle P}
1317:{\displaystyle C}
1303:monoidal category
1294:{\displaystyle P}
1277:Algebras of a PRO
1137:discrete category
1121:{\displaystyle k}
1063:Kronecker product
488:{\displaystyle n}
433:{\displaystyle n}
418:identity morphism
189:
44:monoidal category
16:(Redirected from
2056:
2027:
2020:
2013:
1985:
1978:
1969:
1949:
1932:
1903:
1901:
1868:
1866:
1828:
1822:
1821:
1795:
1786:
1780:
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1742:
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1611:
1606:
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1582:
1567:
1565:
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1559:
1536:
1534:
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1516:
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1513:
1508:
1506:
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1484:
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1447:
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1444:
1439:
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1431:
1426:
1407:
1405:
1404:
1399:
1387:
1385:
1384:
1379:
1367:
1365:
1364:
1359:
1347:
1345:
1344:
1339:
1326:monoidal functor
1323:
1321:
1320:
1315:
1300:
1298:
1297:
1292:
1264:
1262:
1261:
1256:
1254:
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1094:
1093:
1081:
1080:
1060:
1058:
1057:
1052:
1015:
1013:
1012:
1007:
990:identity is the
989:
987:
986:
981:
969:
967:
966:
961:
943:
941:
940:
935:
912:
910:
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904:
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865:
821:
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781:
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308:
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266:
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236:
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207:
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191:
182:
176:
175:
99:
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96:
91:
21:
2064:
2063:
2059:
2058:
2057:
2055:
2054:
2053:
2034:
2033:
2032:
2031:
1991:category theory
1974:
1972:
1966:
1929:
1907:Markl, Martin;
1872:
1871:
1855:
1829:
1825:
1793:
1787:
1783:
1775:
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1728:
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1621:
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1449:
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1413:
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1373:
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1369:
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1333:
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1305:
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1046:
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1042:
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991:
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971:
949:
946:
945:
923:
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919:
860:
859:
851:
845:
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839:
826:
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815:
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809:
803:
802:
797:
787:
786:
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764:
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747:
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705:
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480:
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473:identity matrix
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360:
349:
339:
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212:
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153:
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106:symmetric group
55:
52:
51:
32:category theory
28:
23:
22:
15:
12:
11:
5:
2062:
2052:
2051:
2046:
2030:
2029:
2022:
2015:
2007:
2004:
2003:
1986:
1971:
1970:
1964:
1933:
1927:
1909:Shnider, Steve
1904:
1873:
1870:
1869:
1853:
1823:
1804:(6): 1117–22.
1781:
1779:, Ch. V, § 24.
1768:
1767:
1765:
1762:
1761:
1760:
1755:
1753:Lawvere theory
1748:
1745:
1732:
1708:
1688:
1668:
1648:
1628:
1617:
1616:
1604:
1580:
1571:an algebra of
1569:
1557:
1540:an algebra of
1538:
1526:
1505:
1495:an algebra of
1477:
1457:
1435:
1430:
1425:
1422:
1419:
1397:
1377:
1357:
1337:
1313:
1290:
1278:
1275:
1271:
1270:
1252:
1248:
1229:
1228:
1219:
1209:
1185:
1184:
1177:
1170:
1161:
1148:
1117:
1092:
1088:
1084:
1079:
1075:
1050:
1039:
1038:
1019:
1018:
1017:
1005:
1002:
999:
979:
959:
956:
953:
933:
930:
927:
916:
915:
914:
902:
899:
896:
893:
890:
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884:
881:
878:
875:
872:
869:
864:
858:
855:
852:
850:
847:
846:
843:
840:
838:
835:
832:
831:
829:
824:
819:
813:
810:
808:
805:
804:
801:
798:
796:
793:
792:
790:
785:
780:
774:
771:
769:
766:
765:
762:
759:
757:
754:
753:
751:
746:
743:
740:
737:
734:
731:
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719:
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713:
686:
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652:
649:
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617:
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611:
605:
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588:
582:
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571:
548:
545:
542:
539:
536:
533:
530:
510:
496:
484:
458:
452:
429:
414:
398:
381:do not have to
363:
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89:
86:
83:
80:
77:
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71:
68:
65:
62:
59:
26:
9:
6:
4:
3:
2:
2061:
2050:
2047:
2045:
2042:
2041:
2039:
2028:
2023:
2021:
2016:
2014:
2009:
2008:
2002:
2000:
1996:
1992:
1987:
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1980:
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1967:
1961:
1957:
1953:
1948:
1943:
1939:
1934:
1930:
1924:
1920:
1919:
1914:
1913:Stasheff, Jim
1910:
1905:
1900:
1895:
1891:
1887:
1883:
1879:
1875:
1874:
1864:
1860:
1856:
1850:
1846:
1842:
1839:(1): 87–140.
1838:
1834:
1827:
1819:
1815:
1811:
1807:
1803:
1799:
1792:
1785:
1778:
1777:Mac Lane 1965
1773:
1769:
1759:
1756:
1754:
1751:
1750:
1744:
1730:
1722:
1706:
1686:
1666:
1646:
1602:
1594:
1593:monoid object
1570:
1555:
1547:
1546:monoid object
1543:
1539:
1524:
1494:
1493:
1492:
1491:For example:
1489:
1475:
1455:
1433:
1428:
1395:
1388:and category
1375:
1355:
1335:
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419:
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1031:left action
1029:. Thus the
388:Composition
2038:Categories
1892:: 40–106.
1764:References
1721:equivalent
1202:category.
475:with side
250:elementary
1627:Δ
1579:Δ
1247:Δ
1083:×
1049:⊗
1001:×
978:⊗
955:×
929:×
895:∘
886:⊗
877:∘
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727:∘
718:⊗
679:β
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648:β
645:⊗
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397:∘
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319:morphisms
275:∙
272:×
269:∙
209:⊂
178:⊂
133:J. P. May
82:−
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1915:(2002).
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1747:See also
129:Boardman
1952:Bibcode
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1016:matrix.
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1989:This
1942:arXiv
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1859:S2CID
1794:(PDF)
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1995:stub
1960:ISBN
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1200:PROB
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1021:The
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440:(or
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