2070:
2317:
2337:
2327:
137:. Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first. Later, the concrete structure of sets was abstracted away â taken "only up to isomorphism", to produce the abstract theory of arithmetic. This is a "decategorification" â categorification reverses this step.
276:
124:
of finite sets (and any two sets with the same cardinality are isomorphic). In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about
1217:
428:
93:
over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.
1001:
795:
739:
1584:
613:
954:
882:
500:
1131:
909:
528:
467:
96:
Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like '
655:
1318:
1254:
213:
1345:
1281:
1059:
1032:
1079:
838:
818:
633:
551:
225:
1139:
299:
1714:
1999:
959:
744:
663:
81:. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the
564:
914:
434:
1707:
1686:
1911:
1866:
2366:
2340:
2280:
1564:, Contemp. Math., vol. 230, Providence, Rhode Island: American Mathematical Society, pp. 1â36,
184:
1989:
847:
2330:
2116:
1980:
1888:
1372:
472:
43:
1084:
112:
One form of categorification takes a structure described in terms of sets, and interprets the sets as
2289:
1933:
1871:
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130:
890:
509:
448:
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2320:
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134:
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1222:
2083:
1849:
1829:
1752:
1367:
78:
59:
47:
198:
1965:
1804:
1399:"Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases"
82:
1777:
1772:
1638:
1612:
1323:
1259:
1037:
1010:
841:
8:
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1995:
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290:
180:
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1616:
1975:
1970:
1952:
1834:
1809:
1665:
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1628:
1602:
1565:
1560:; Dolan, James (1998), "Categorification", in Getzler, Ezra; Kapranov, Mikhail (eds.),
1511:
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1985:
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1960:
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1420:
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885:
1579:
2157:
1723:
188:
153:
141:
35:
2194:
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2046:
433:
have the same decomposition numbers over their respective bases, both given by
169:
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101:
97:
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2184:
2016:
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1819:
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1938:
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1687:
https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html
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63:
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of functors satisfying additional properties. The term was coined by
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1916:
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271:{\displaystyle S^{\lambda }{\stackrel {\varphi }{\to }}s_{\lambda },}
126:
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1398:
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74:
55:
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1607:
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1814:
1212:{\displaystyle F_{i}F_{j}\cong \bigoplus _{k}F_{k}^{c_{ij}^{k}},}
423:{\displaystyle \left\qquad {\text{ and }}\qquad s_{\mu }s_{\nu }}
51:
31:
293:. This map reflects how the structures are similar; for example
2259:
38:
analogues. Categorification, when done successfully, replaces
2141:
1585:
Cahiers de
Topologie et Géométrie Différentielle Catégoriques
289:
to a representation-theoretic favorite basis of the ring of
1601:, QGM Master Class Series, European Mathematical Society,
187:
is categorified by the category of representations of the
1648:
1494:
1656:(2009), "A brief review of abelian categorifications",
1502:(2009), "A brief review of abelian categorifications",
1320:
decomposes as the linear combination of basis elements
996:{\displaystyle F_{i}:{\mathcal {B}}\to {\mathcal {B}}}
73:. Decategorification is a systematic process by which
1326:
1289:
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1538:
790:{\displaystyle c_{ij}^{k}\in \mathbb {Z} _{\geq 0}.}
734:{\displaystyle a_{i}a_{j}=\sum _{k}c_{ij}^{k}a_{k},}
1448:"Clock and category: Is quantum gravity algebraic?"
116:of objects in a category. For example, the set of
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270:
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69:The reverse of categorification is the process of
2353:
1685:A blog post by one of the above authors (Baez):
608:{\displaystyle \mathbf {a} =\{a_{i}\}_{i\in I}}
152:gave the modern formulation of homology as the
1708:
1529:
1397:Crane, Louis; Frenkel, Igor B. (1994-10-01).
1396:
949:{\displaystyle \phi :K({\mathcal {B}})\to B}
844:. Then a (weak) abelian categorification of
635:such that the multiplication is positive in
590:
576:
440:
285:map from a favorite basis of the associated
1577:
2336:
2326:
2082:
1715:
1701:
1669:
1632:
1606:
1596:
1569:
1556:
1544:
1515:
1463:
1414:
1256:decomposes as the direct sum of functors
771:
77:objects in a category are identified as
16:Connects set theory with category theory
1578:Crane, Louis; Yetter, David N. (1998),
1532:"What precisely Is "Categorification"?"
191:. The decategorification map sends the
2354:
1622:
1599:Lectures on Algebraic Categorification
2081:
1734:
1696:
1445:
1364:theorems by set-theoretic analogues.
1722:
13:
1679:
988:
978:
932:
896:
877:{\displaystyle (A,\mathbf {a} ,B)}
515:
484:
454:
435:LittlewoodâRichardson coefficients
14:
2378:
1283:in the same way that the product
495:{\displaystyle K({\mathcal {B}})}
160:by categorifying the notion of a
2335:
2325:
2316:
2315:
2068:
1735:
1625:Introduction to Categorification
1126:{\displaystyle \phi =a_{i}\phi }
861:
643:
569:
1452:Journal of Mathematical Physics
1403:Journal of Mathematical Physics
399:
393:
219:indexed by the same partition,
1580:"Examples of categorification"
1523:
1488:
1439:
1390:
1104:
1091:
983:
940:
937:
927:
904:{\displaystyle {\mathcal {B}}}
871:
851:
523:{\displaystyle {\mathcal {B}}}
489:
479:
462:{\displaystyle {\mathcal {B}}}
385:
359:
243:
1:
1597:Mazorchuk, Volodymyr (2010),
1383:
1530:Alex Hoffnung (2009-11-10).
650:{\displaystyle \mathbf {a} }
27:is the process of replacing
7:
2010:Constructions on categories
1446:Crane, Louis (1995-11-01).
1360:, the process of replacing
1351:
185:ring of symmetric functions
107:
10:
2383:
2117:Higher-dimensional algebra
1373:Higher-dimensional algebra
1313:{\displaystyle a_{i}a_{j}}
1249:{\displaystyle F_{i}F_{j}}
281:essentially following the
120:can be seen as the set of
2311:
2244:
2208:
2156:
2149:
2100:
2090:
2077:
2066:
2009:
1951:
1902:
1857:
1848:
1745:
1741:
1730:
1623:Savage, Alistair (2014),
956:, and exact endofunctors
441:Abelian categorifications
1652:; Mazorchuk, Volodymyr;
1498:; Mazorchuk, Volodymyr;
559:free as an abelian group
208:{\displaystyle \lambda }
1927:Cokernels and quotients
1850:Universal constructions
1219:, i.e. the composition
1136:there are isomorphisms
140:Other examples include
135:category of finite sets
2084:Higher category theory
1830:Natural transformation
1562:Higher Category Theory
1368:Higher category theory
1341:
1314:
1277:
1250:
1213:
1127:
1075:
1055:
1028:
997:
950:
905:
878:
834:
814:
791:
735:
651:
629:
609:
547:
524:
496:
463:
424:
272:
209:
1545:Baez & Dolan 1998
1342:
1340:{\displaystyle a_{k}}
1315:
1278:
1276:{\displaystyle F_{k}}
1251:
1214:
1128:
1076:
1056:
1054:{\displaystyle a_{i}}
1029:
1027:{\displaystyle F_{i}}
998:
951:
906:
879:
835:
815:
792:
736:
652:
630:
610:
548:
525:
497:
464:
425:
273:
210:
195:indexed by partition
83:representation theory
1953:Algebraic categories
1324:
1287:
1260:
1223:
1140:
1085:
1065:
1038:
1034:lifts the action of
1011:
960:
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891:
848:
824:
804:
745:
664:
639:
619:
565:
537:
510:
473:
449:
300:
226:
199:
60:natural isomorphisms
2122:Homotopy hypothesis
1800:Commutative diagram
1658:Theory Appl. Categ.
1654:Stroppel, Catharina
1643:2014arXiv1401.6037S
1617:2010arXiv1011.0144M
1504:Theory Appl. Categ.
1500:Stroppel, Catharina
1358:Combinatorial proof
1205:
1203:
765:
717:
355:
291:symmetric functions
181:finite group theory
158:free abelian groups
114:isomorphism classes
2367:Algebraic topology
1835:Universal property
1337:
1310:
1273:
1246:
1209:
1186:
1176:
1175:
1123:
1071:
1051:
1024:
993:
946:
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874:
830:
810:
787:
748:
731:
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699:
647:
625:
605:
543:
520:
504:Grothendieck group
492:
459:
420:
308:
287:Grothendieck group
268:
205:
71:decategorification
36:category-theoretic
2349:
2348:
2307:
2306:
2303:
2302:
2285:monoidal category
2240:
2239:
2112:Enriched category
2064:
2063:
2060:
2059:
2037:Quotient category
2032:Opposite category
1947:
1946:
1650:Khovanov, Mikhail
1496:Khovanov, Mikhail
1458:(11): 6180â6193.
1409:(10): 5136â5154.
1166:
1074:{\displaystyle B}
911:, an isomorphism
833:{\displaystyle A}
813:{\displaystyle B}
690:
628:{\displaystyle A}
546:{\displaystyle A}
397:
252:
166:Khovanov homology
142:homology theories
100:', and not like '
2374:
2339:
2338:
2329:
2328:
2319:
2318:
2154:
2153:
2132:Simplex category
2107:Categorification
2098:
2097:
2079:
2078:
2072:
2042:Product category
2027:Kleisli category
2022:Functor category
1867:Terminal objects
1855:
1854:
1790:Adjoint functors
1743:
1742:
1732:
1731:
1717:
1710:
1703:
1694:
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1674:
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1645:
1636:
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1548:
1542:
1536:
1535:
1527:
1521:
1520:
1519:
1492:
1486:
1485:
1474:10.1063/1.531240
1467:
1443:
1437:
1436:
1425:10.1063/1.530746
1418:
1394:
1378:Categorical ring
1362:number theoretic
1346:
1344:
1343:
1338:
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1335:
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1080:
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1060:
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1049:
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972:
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953:
952:
947:
936:
935:
910:
908:
907:
902:
900:
899:
886:abelian category
883:
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839:
837:
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572:
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214:
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211:
206:
25:categorification
2382:
2381:
2377:
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2375:
2373:
2372:
2371:
2362:Category theory
2352:
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2269:
2236:
2213:
2204:
2161:
2145:
2096:
2086:
2073:
2056:
2005:
1943:
1912:Initial objects
1898:
1844:
1737:
1726:
1724:Category theory
1721:
1682:
1680:Further reading
1677:
1671:math.RT/0702746
1664:(19): 479â508,
1571:math.QA/9802029
1552:
1551:
1543:
1539:
1528:
1524:
1517:math.RT/0702746
1510:(19): 479â508,
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884:consists of an
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849:
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482:
474:
471:
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453:
452:
450:
447:
446:
445:For a category
443:
414:
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396: and
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379:
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366:
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342:
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259:
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247:
242:
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233:
229:
227:
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200:
197:
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189:symmetric group
118:natural numbers
110:
17:
12:
11:
5:
2380:
2370:
2369:
2364:
2347:
2346:
2344:
2343:
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2312:
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2301:
2300:
2298:
2297:
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2287:
2273:
2267:
2262:
2257:
2251:
2249:
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2241:
2238:
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2235:
2234:
2229:
2218:
2216:
2211:
2206:
2205:
2203:
2202:
2197:
2192:
2187:
2182:
2177:
2166:
2164:
2159:
2151:
2147:
2146:
2144:
2139:
2137:String diagram
2134:
2129:
2127:Model category
2124:
2119:
2114:
2109:
2104:
2102:
2095:
2094:
2091:
2088:
2087:
2075:
2074:
2067:
2065:
2062:
2061:
2058:
2057:
2055:
2054:
2049:
2047:Comma category
2044:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2007:
2006:
2004:
2003:
1993:
1983:
1981:Abelian groups
1978:
1973:
1968:
1963:
1957:
1955:
1949:
1948:
1945:
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1941:
1936:
1931:
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1620:
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1549:
1537:
1522:
1487:
1438:
1416:hep-th/9405183
1388:
1387:
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1380:
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1146:
1134:
1122:
1117:
1113:
1109:
1106:
1101:
1097:
1093:
1090:
1070:
1061:on the module
1048:
1044:
1021:
1017:
990:
985:
980:
975:
970:
966:
945:
942:
939:
934:
929:
926:
923:
920:
898:
873:
870:
867:
863:
859:
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829:
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798:
797:
786:
781:
778:
773:
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763:
758:
755:
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730:
725:
721:
715:
710:
707:
703:
697:
693:
689:
684:
680:
674:
670:
645:
624:
615:be a basis of
602:
599:
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279:
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236:
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217:Schur function
204:
179:An example in
170:knot invariant
109:
106:
102:sheafification
98:generalization
15:
9:
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2185:Tetracategory
2183:
2181:
2178:
2175:
2174:pseudofunctor
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2018:
2017:Free category
2015:
2014:
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2000:Vector spaces
1997:
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1894:Inverse limit
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1820:Kan extension
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1465:gr-qc/9504038
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343:
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288:
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260:
256:
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230:
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193:Specht module
190:
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123:
122:cardinalities
119:
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105:
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30:
29:set-theoretic
26:
22:
2265:
2246:Categorified
2245:
2150:n-categories
2106:
2101:Key concepts
1939:Direct limit
1922:Coequalizers
1840:Yoneda lemma
1746:Key concepts
1736:Key concepts
1661:
1657:
1624:
1598:
1589:
1583:
1561:
1540:
1525:
1507:
1503:
1490:
1455:
1451:
1441:
1406:
1402:
1392:
1007:the functor
799:
532:
444:
432:
280:
183:is that the
178:
162:Betti number
150:Emmy Noether
139:
121:
113:
111:
95:
87:Lie algebras
70:
68:
24:
18:
2214:-categories
2190:Kan complex
2180:Tricategory
2162:-categories
2052:Subcategory
1810:Exponential
1778:Preadditive
1773:Pre-abelian
174:knot theory
164:. See also
156:of certain
64:Louis Crane
21:mathematics
2356:Categories
2232:3-category
2222:2-category
2195:â-groupoid
2170:Bicategory
1917:Coproducts
1877:Equalizers
1783:Bicategory
1558:Baez, John
1384:References
1003:such that
561:, and let
127:coproducts
75:isomorphic
44:categories
2281:Symmetric
2226:2-functor
1966:Relations
1889:Pullbacks
1634:1401.6037
1608:1011.0144
1592:(1): 3â25
1482:0022-2488
1433:0022-2488
1168:⨁
1164:≅
1121:ϕ
1089:ϕ
984:→
941:→
919:ϕ
777:≥
767:∈
692:∑
598:∈
557:which is
416:ν
406:μ
381:ν
373:⊗
368:μ
357:
324:⊗
283:character
261:λ
249:φ
244:→
235:λ
203:λ
56:equations
48:functions
2341:Glossary
2321:Category
2295:n-monoid
2248:concepts
1904:Colimits
1872:Products
1825:Morphism
1768:Concrete
1763:Additive
1753:Category
1352:See also
146:topology
131:products
108:Examples
52:functors
32:theorems
2331:Outline
2290:n-group
2255:2-group
2210:Strict
2200:â-topos
1996:Modules
1934:Pushout
1882:Kernels
1815:Functor
1758:Abelian
1639:Bibcode
1613:Bibcode
1081:, i.e.
657:, i.e.
502:be the
215:to the
133:of the
91:modules
2277:Traced
2260:2-ring
1990:Fields
1976:Groups
1971:Magmas
1859:Limits
1480:
1431:
842:module
820:be an
469:, let
54:, and
2271:-ring
2158:Weak
2142:Topos
1986:Rings
1666:arXiv
1629:arXiv
1603:arXiv
1566:arXiv
1512:arXiv
1460:arXiv
1411:arXiv
1133:, and
741:with
553:be a
168:as a
79:equal
58:with
50:with
42:with
34:with
1961:Sets
1478:ISSN
1429:ISSN
800:Let
555:ring
533:Let
154:rank
129:and
40:sets
1805:End
1795:CCC
1470:doi
1421:doi
506:of
310:Ind
172:in
144:in
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1100:i
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