1886:
866:
1694:
651:
1095:
2096:
1654:
639:
structures or multiplications, but this suggests they are assumed to be associative, a property that is not required for the proof. In fact, associativity follows. Likewise, we do not have to require that the two operations have the same neutral element; this is a consequence.
1466:
1232:
1881:{\displaystyle a\ b={\begin{matrix}a\ \ \bullet \\\bullet \ \ b\end{matrix}}={\begin{matrix}a\\b\end{matrix}}={\begin{matrix}\bullet \ \ a\\b\ \ \bullet \end{matrix}}=b\ a={\begin{matrix}b\ \ \bullet \\\bullet \ \ a\end{matrix}}={\begin{matrix}b\\a\end{matrix}}}
861:{\displaystyle 1_{\circ }=1_{\circ }\circ 1_{\circ }=(1_{\otimes }\otimes 1_{\circ })\circ (1_{\circ }\otimes 1_{\otimes })=(1_{\otimes }\circ 1_{\circ })\otimes (1_{\circ }\circ 1_{\otimes })=1_{\otimes }\otimes 1_{\otimes }=1_{\otimes }}
906:
1932:
1561:
2398:
497:
2265:
It is important that a similar argument does NOT give such a trivial result in the case of monoid objects in the categories of small categories or of groupoids. Instead the notion of group object in the category of
2403:
whenever both sides are defined. For an example of its use, and some discussion, see the paper of
Higgins referenced below. The interchange law implies that a double category contains a family of abelian monoids.
382:
2212:
1513:
1357:
324:
1103:
1279:
2285:
is a set, or class, equipped with two category structures, each of which is a morphism for the other structure. If the compositions in the two category structures are written
2309:
1352:
541:
246:
1927:
219:
2476:
attended this conference, and his first work on higher homotopy groups appeared in 1935. Thus the dreams of the early topologists have long been regarded as a mirage.
2144:
1689:
1536:
901:
613:
585:
188:
145:
2466:
1556:
633:
565:
408:
168:
125:
1090:{\displaystyle a\circ b=(1\otimes a)\circ (b\otimes 1)=(1\circ b)\otimes (a\circ 1)=b\otimes a=(b\circ 1)\otimes (1\circ a)=(b\otimes 1)\circ (1\otimes a)=b\circ a}
1305:
2091:{\displaystyle a\ (b\ c)=a\ {\begin{pmatrix}b\\c\end{pmatrix}}={\begin{matrix}a\ \ b\\\bullet \ \ c\end{matrix}}={\begin{matrix}(a\ b)\\c\end{matrix}}=(a\ b)\ c}
2260:
2232:
2124:
1649:{\displaystyle \bullet ={\begin{matrix}\bullet \\\bullet \end{matrix}}={\begin{matrix}\bullet \ \ \circ \\\circ \ \ \bullet \end{matrix}}=\circ \ \circ =\circ }
266:
101:
2317:
416:
2664:
329:
2149:
1461:{\displaystyle {\begin{matrix}(a\ b)\\(c\ d)\end{matrix}}={\begin{pmatrix}a\\c\end{pmatrix}}{\begin{pmatrix}b\\d\end{pmatrix}}}
2486:
1227:{\displaystyle (a\otimes b)\otimes c=(a\otimes b)\otimes (1\otimes c)=(a\otimes 1)\otimes (b\otimes c)=a\otimes (b\otimes c)}
2514:
Eckmann, B.; Hilton, P. J. (1962), "Group-like structures in general categories. I. Multiplications and comultiplications",
1471:
2644:
271:
2566:
2558:
2419:
could be defined in all dimensions; and that for a connected space, the first homology group was the fundamental group
1249:
17:
2659:
1246:. For this version of the proof, we write the two operations as vertical and horizontal juxtaposition, i.e.,
2468:, and on these grounds persuaded Čech to withdraw his paper, so that only a small paragraph appeared in the
1242:
The above proof also has a "two-dimensional" presentation that better illustrates the application to higher
2416:
2516:
2288:
2282:
2278:
1888:, so horizontal composition is the same as vertical composition and both operations are commutative.
2239:
1313:
502:
224:
2570:
2411:
is interesting. The workers in topology of the early 20th century were aware that the nonabelian
60:
1894:
197:
2490:
2412:
2129:
1662:
1521:
874:
598:
570:
173:
130:
48:
2563:
Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids
2525:
2445:
2435:
1541:
618:
550:
387:
153:
110:
104:
2609:
2584:
2537:
2423:. So there was a desire to generalise the nonabelian fundamental group to all dimensions.
2605:
2580:
2533:
2473:
1284:
56:
2485:
cited below, which develops basic algebraic topology, including higher analogues to the
2593:
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2408:
2271:
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2217:
2109:
1243:
251:
191:
86:
64:
2653:
2235:
2481:
2479:
Cubical higher homotopy groupoids are constructed for filtered spaces in the book
72:
68:
52:
44:
2274:. This leads to the idea of using multiple groupoid objects in homotopy theory.
2106:
If the operations are associative, each one defines the structure of a monoid on
2640:
Eugenia Cheng of 'the
Catsters' video team explains the Eckmann–Hilton argument.
2277:
More generally, the
Eckmann–Hilton argument is a special case of the use of the
24:
2552:, Nederl. Akad. Wetensch. Proc. Ser. A, vol. 38, pp. 112–119, 521–528
2639:
2439:
2126:, and the conditions above are equivalent to the more abstract condition that
2508:
2503:
2393:{\displaystyle (a\circ b)\otimes (c\circ d)=(a\otimes c)\circ (b\otimes d)}
492:{\displaystyle (a\otimes b)\circ (c\otimes d)=(a\circ c)\otimes (b\circ d)}
55:
for the other. Given this, the structures are the same, and the resulting
2545:
2267:
2214:(or vice versa). An even more abstract way of stating the theorem is: If
1097:. This establishes that the two operations coincide and are commutative.
40:
2594:"Thin elements and commutative shells in cubical $ \omega$ -categories"
1558:
be the units for vertical and horizontal composition respectively. Then
2529:
2575:
2281:
in the theory of (strict) double and multiple categories. A (strict)
636:
2626:
2434:
63:. This can then be used to prove the commutativity of the higher
648:
First, observe that the units of the two operations coincide:
1307:. The interchange property can then be expressed as follows:
377:{\displaystyle 1_{\otimes }\otimes a=a=a\otimes 1_{\otimes }}
2035:
1996:
1971:
1861:
1822:
1771:
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1711:
1593:
1572:
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1411:
1362:
1254:
2207:{\displaystyle (X,\circ )\times (X,\circ )\to (X,\circ )}
1508:{\displaystyle {\begin{matrix}a\ b\\c\ d\end{matrix}}}
587:
are the same and in fact commutative and associative.
2448:
2320:
2291:
2248:
2220:
2152:
2132:
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1935:
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909:
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654:
621:
601:
573:
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419:
390:
332:
319:{\displaystyle 1_{\circ }\circ a=a=a\circ 1_{\circ }}
274:
254:
227:
200:
176:
156:
133:
113:
89:
16:
For the notion of duality in algebraic topology, see
2569:Tracts in Mathematics, vol. 15, p. 703,
2460:
2415:was of use in geometry and analysis; that abelian
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491:
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213:
182:
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139:
119:
95:
1274:{\displaystyle {\begin{matrix}a\\b\end{matrix}}}
2509:John Baez: Eckmann–Hilton principle (week 100)
2504:John Baez: Eckmann–Hilton principle (week 89)
2442:quickly proved these groups were abelian for
8:
2627:Permutation of elements in double semigroups
2270:turns out to be equivalent to the notion of
2625:Murray Bremner and Sara Madariaga. (2014)
194:, meaning that there are identity elements
2574:
2561:, R.; Higgins, P. J.; Sivera, R. (2011),
2447:
2319:
2290:
2247:
2219:
2151:
2131:
2111:
2034:
1995:
1966:
1934:
1896:
1860:
1821:
1770:
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1543:
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88:
2550:Beitrage zur Topologie der Deformationen
7:
2598:Theory and Application of Categories
14:
2262:is in fact a commutative monoid.
2098:, so composition is associative.
2311:then the interchange law reads
2645:Higher dimensional group theory
2304:{\displaystyle \circ ,\otimes }
75:, who used it in a 1962 paper.
67:. The principle is named after
2387:
2375:
2369:
2357:
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2333:
2321:
2201:
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2079:
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2038:
1954:
1942:
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1377:
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720:
694:
486:
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468:
456:
450:
438:
432:
420:
1:
2567:European Mathematical Society
2493:or simplicial approximation.
2482:Nonabelian algebraic topology
2665:Theorems in abstract algebra
2430:submitted a paper on higher
1347:{\displaystyle a,b,c,d\in X}
536:{\displaystyle a,b,c,d\in X}
241:{\displaystyle 1_{\otimes }}
2407:The history in relation to
1656:, so both units are equal.
103:be a set equipped with two
2681:
2487:Seifert–Van Kampen theorem
1922:{\displaystyle a,b,c\in X}
214:{\displaystyle 1_{\circ }}
15:
2146:is a monoid homomorphism
635:are often referred to as
79:The Eckmann–Hilton result
2139:{\displaystyle \otimes }
1684:{\displaystyle a,b\in X}
1531:{\displaystyle \bullet }
896:{\displaystyle a,b\in X}
608:{\displaystyle \otimes }
580:{\displaystyle \otimes }
183:{\displaystyle \otimes }
140:{\displaystyle \otimes }
33:Eckmann–Hilton principle
2592:Higgins, P. J. (2005),
29:Eckmann–Hilton argument
2462:
2461:{\displaystyle n>1}
2394:
2305:
2256:
2228:
2208:
2140:
2120:
2092:
1923:
1882:
1685:
1650:
1552:
1551:{\displaystyle \circ }
1532:
1509:
1462:
1348:
1301:
1275:
1228:
1091:
897:
862:
629:
628:{\displaystyle \circ }
609:
581:
561:
560:{\displaystyle \circ }
537:
493:
404:
403:{\displaystyle a\in X}
378:
320:
262:
242:
215:
184:
164:
163:{\displaystyle \circ }
141:
121:
120:{\displaystyle \circ }
107:, which we will write
97:
37:Eckmann–Hilton theorem
18:Eckmann–Hilton duality
2517:Mathematische Annalen
2463:
2395:
2306:
2257:
2229:
2209:
2141:
2121:
2093:
1924:
1883:
1686:
1651:
1553:
1533:
1510:
1463:
1349:
1302:
1276:
1238:Two-dimensional proof
1229:
1092:
898:
863:
630:
610:
582:
562:
538:
494:
405:
379:
321:
263:
243:
216:
185:
165:
142:
122:
98:
2617:James, I.M. (1999),
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2289:
2246:
2218:
2150:
2130:
2110:
1933:
1895:
1695:
1663:
1562:
1542:
1522:
1472:
1358:
1314:
1300:{\displaystyle a\ b}
1285:
1250:
1104:
907:
875:
652:
619:
599:
571:
551:
503:
417:
388:
330:
272:
252:
225:
198:
174:
154:
131:
111:
87:
2619:History of Topology
2240:category of monoids
1515:without ambiguity.
1100:For associativity,
2530:10.1007/bf01451367
2472:. It is said that
2458:
2390:
2301:
2252:
2224:
2204:
2136:
2116:
2088:
2062:
2029:
1986:
1919:
1878:
1876:
1855:
1804:
1765:
1744:
1681:
1646:
1626:
1587:
1548:
1528:
1505:
1503:
1468:, so we can write
1458:
1452:
1426:
1401:
1344:
1297:
1271:
1269:
1224:
1087:
893:
858:
625:
605:
577:
557:
533:
489:
400:
374:
316:
258:
238:
211:
180:
160:
137:
117:
93:
61:commutative monoid
2491:singular homology
2436:Pavel Alexandroff
2413:fundamental group
2255:{\displaystyle X}
2227:{\displaystyle X}
2119:{\displaystyle X}
2084:
2075:
2046:
2023:
2020:
2007:
2004:
1965:
1950:
1941:
1891:Finally, for all
1849:
1846:
1833:
1830:
1814:
1798:
1795:
1782:
1779:
1738:
1735:
1722:
1719:
1703:
1636:
1620:
1617:
1604:
1601:
1497:
1484:
1392:
1373:
1293:
261:{\displaystyle X}
105:binary operations
96:{\displaystyle X}
2672:
2622:
2612:
2587:
2578:
2553:
2540:
2489:, without using
2467:
2465:
2464:
2459:
2399:
2397:
2396:
2391:
2310:
2308:
2307:
2302:
2261:
2259:
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2253:
2233:
2231:
2230:
2225:
2213:
2211:
2210:
2205:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2097:
2095:
2094:
2089:
2082:
2073:
2063:
2044:
2030:
2021:
2018:
2005:
2002:
1991:
1990:
1963:
1948:
1939:
1928:
1926:
1925:
1920:
1887:
1885:
1884:
1879:
1877:
1856:
1847:
1844:
1831:
1828:
1812:
1805:
1796:
1793:
1780:
1777:
1766:
1745:
1736:
1733:
1720:
1717:
1701:
1690:
1688:
1687:
1682:
1655:
1653:
1652:
1647:
1634:
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1602:
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1233:
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47:structures on a
2680:
2679:
2675:
2674:
2673:
2671:
2670:
2669:
2660:Category theory
2650:
2649:
2636:
2631:
2621:, North Holland
2616:
2591:
2557:
2544:
2513:
2499:
2474:Witold Hurewicz
2444:
2443:
2432:homotopy groups
2417:homology groups
2409:homotopy groups
2316:
2315:
2287:
2286:
2283:double category
2279:interchange law
2244:
2243:
2216:
2215:
2148:
2147:
2128:
2127:
2108:
2107:
2104:
2061:
2060:
2054:
2053:
2028:
2027:
2012:
2011:
1985:
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1978:
1977:
1967:
1931:
1930:
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1244:homotopy groups
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742:
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710:
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595:The operations
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147:, and suppose:
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65:homotopy groups
51:where one is a
21:
12:
11:
5:
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2676:
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2634:External links
2632:
2630:
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2542:
2524:(3): 227–255,
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2294:
2272:crossed module
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984:
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975:
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883:
880:
855:
851:
847:
842:
838:
834:
829:
825:
821:
818:
813:
809:
805:
800:
796:
792:
789:
786:
781:
777:
773:
768:
764:
760:
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754:
749:
745:
741:
736:
732:
728:
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693:
688:
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675:
671:
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658:
645:
642:
624:
604:
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589:
576:
556:
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544:
532:
529:
526:
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517:
514:
511:
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488:
485:
482:
479:
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470:
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428:
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412:
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371:
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357:
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351:
348:
345:
340:
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309:
305:
302:
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296:
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290:
287:
282:
278:
257:
235:
231:
208:
204:
179:
159:
136:
116:
92:
80:
77:
13:
10:
9:
6:
4:
3:
2:
2677:
2666:
2663:
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2658:
2657:
2655:
2646:
2643:
2641:
2638:
2637:
2633:
2628:
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2615:
2611:
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2603:
2599:
2595:
2590:
2586:
2582:
2577:
2572:
2568:
2564:
2560:
2556:
2551:
2548:, W. (1935),
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2518:
2512:
2510:
2507:
2505:
2502:
2501:
2496:
2494:
2492:
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2441:
2437:
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2422:
2418:
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2314:
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2298:
2295:
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2275:
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2269:
2263:
2249:
2241:
2237:
2236:monoid object
2221:
2198:
2195:
2192:
2180:
2177:
2174:
2168:
2162:
2159:
2156:
2133:
2113:
2101:
2099:
2085:
2076:
2070:
2064:
2057:
2047:
2041:
2031:
2024:
2015:
2008:
1999:
1992:
1987:
1981:
1974:
1968:
1960:
1957:
1951:
1945:
1936:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1889:
1871:
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1739:
1730:
1723:
1714:
1707:
1704:
1698:
1678:
1675:
1672:
1669:
1666:
1659:Now, for all
1657:
1643:
1640:
1637:
1631:
1628:
1621:
1612:
1605:
1596:
1589:
1582:
1575:
1568:
1565:
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1525:
1516:
1498:
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1485:
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1447:
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1434:
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1414:
1408:
1403:
1393:
1387:
1374:
1368:
1341:
1338:
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1332:
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1326:
1323:
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1317:
1308:
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1288:
1264:
1257:
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1218:
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1200:
1194:
1191:
1188:
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1128:
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994:
991:
985:
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931:
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840:
836:
832:
827:
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798:
794:
787:
779:
775:
771:
766:
762:
755:
747:
743:
739:
734:
730:
723:
715:
711:
707:
702:
698:
691:
686:
682:
678:
673:
669:
665:
660:
656:
643:
641:
638:
622:
602:
590:
588:
574:
554:
530:
527:
524:
521:
518:
515:
512:
509:
506:
483:
480:
477:
471:
465:
462:
459:
453:
447:
444:
441:
435:
429:
426:
423:
413:
397:
394:
391:
369:
365:
361:
358:
355:
352:
349:
346:
343:
338:
334:
311:
307:
303:
300:
297:
294:
291:
288:
285:
280:
276:
255:
233:
229:
206:
202:
193:
177:
157:
150:
149:
148:
134:
114:
106:
90:
78:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
19:
2618:
2601:
2597:
2576:math/0407275
2562:
2549:
2521:
2515:
2480:
2478:
2469:
2425:
2421:made abelian
2406:
2402:
2276:
2264:
2105:
1890:
1658:
1517:
1309:
1241:
1099:
870:
647:
594:
546:
82:
73:Peter Hilton
69:Beno Eckmann
53:homomorphism
45:unital magma
36:
32:
28:
22:
2470:Proceedings
2428:Eduard Čech
25:mathematics
2654:Categories
2497:References
2440:Heinz Hopf
384:, for all
268:such that
43:about two
2604:: 60–74,
2426:In 1932,
2382:⊗
2373:∘
2364:⊗
2346:∘
2337:⊗
2328:∘
2299:⊗
2293:∘
2268:groupoids
2199:∘
2187:→
2181:∘
2169:×
2163:∘
2134:⊗
2016:∙
1914:∈
1842:∙
1835:∙
1800:∙
1775:∙
1731:∙
1724:∙
1676:∈
1644:∘
1638:∘
1632:∘
1622:∙
1613:∘
1606:∘
1597:∙
1583:∙
1576:∙
1566:∙
1546:∘
1526:∙
1339:∈
1216:⊗
1207:⊗
1192:⊗
1183:⊗
1174:⊗
1156:⊗
1147:⊗
1138:⊗
1123:⊗
1114:⊗
1082:∘
1067:⊗
1058:∘
1049:⊗
1031:∘
1022:⊗
1013:∘
998:⊗
983:∘
974:⊗
965:∘
947:⊗
938:∘
929:⊗
914:∘
888:∈
871:Now, let
854:⊗
841:⊗
833:⊗
828:⊗
812:⊗
804:∘
799:∘
788:⊗
780:∘
772:∘
767:⊗
748:⊗
740:⊗
735:∘
724:∘
716:∘
708:⊗
703:⊗
687:∘
679:∘
674:∘
661:∘
623:∘
603:⊗
575:⊗
555:∘
528:∈
499:for all
481:∘
472:⊗
463:∘
445:⊗
436:∘
427:⊗
395:∈
370:⊗
362:⊗
344:⊗
339:⊗
312:∘
304:∘
286:∘
281:∘
234:⊗
207:∘
190:are both
178:⊗
158:∘
135:⊗
115:∘
2546:Hurewicz
1310:For all
41:argument
39:) is an
2610:2122826
2585:2841564
2538:0136642
2242:, then
2238:in the
2102:Remarks
903:. Then
591:Remarks
2608:
2583:
2536:
2083:
2074:
2045:
2022:
2019:
2006:
2003:
1964:
1949:
1940:
1848:
1845:
1832:
1829:
1813:
1797:
1794:
1781:
1778:
1737:
1734:
1721:
1718:
1702:
1635:
1619:
1616:
1603:
1600:
1496:
1483:
1391:
1372:
1292:
637:monoid
192:unital
27:, the
2571:arXiv
2559:Brown
2234:is a
644:Proof
547:Then
59:is a
57:magma
2453:>
2438:and
1538:and
1518:Let
1281:and
615:and
567:and
326:and
221:and
170:and
127:and
83:Let
71:and
31:(or
2526:doi
2522:145
248:of
49:set
35:or
23:In
2656::
2606:MR
2602:14
2600:,
2596:,
2581:MR
2579:,
2565:,
2534:MR
2532:,
2520:,
1929:,
1691:,
1354:,
1234:.
868:.
2613:.
2588:.
2573::
2554:.
2541:.
2528::
2456:1
2450:n
2388:)
2385:d
2379:b
2376:(
2370:)
2367:c
2361:a
2358:(
2355:=
2352:)
2349:d
2343:c
2340:(
2334:)
2331:b
2325:a
2322:(
2296:,
2250:X
2222:X
2202:)
2196:,
2193:X
2190:(
2184:)
2178:,
2175:X
2172:(
2166:)
2160:,
2157:X
2154:(
2114:X
2086:c
2080:)
2077:b
2071:a
2068:(
2065:=
2058:c
2051:)
2048:b
2042:a
2039:(
2032:=
2025:c
2009:b
2000:a
1993:=
1988:)
1982:c
1975:b
1969:(
1961:a
1958:=
1955:)
1952:c
1946:b
1943:(
1937:a
1917:X
1911:c
1908:,
1905:b
1902:,
1899:a
1872:a
1865:b
1858:=
1851:a
1826:b
1819:=
1816:a
1810:b
1807:=
1791:b
1784:a
1768:=
1761:b
1754:a
1747:=
1740:b
1715:a
1708:=
1705:b
1699:a
1679:X
1673:b
1670:,
1667:a
1641:=
1629:=
1590:=
1569:=
1499:d
1493:c
1486:b
1480:a
1454:)
1448:d
1441:b
1435:(
1428:)
1422:c
1415:a
1409:(
1404:=
1397:)
1394:d
1388:c
1385:(
1378:)
1375:b
1369:a
1366:(
1342:X
1336:d
1333:,
1330:c
1327:,
1324:b
1321:,
1318:a
1295:b
1289:a
1265:b
1258:a
1222:)
1219:c
1213:b
1210:(
1204:a
1201:=
1198:)
1195:c
1189:b
1186:(
1180:)
1177:1
1171:a
1168:(
1165:=
1162:)
1159:c
1153:1
1150:(
1144:)
1141:b
1135:a
1132:(
1129:=
1126:c
1120:)
1117:b
1111:a
1108:(
1085:a
1079:b
1076:=
1073:)
1070:a
1064:1
1061:(
1055:)
1052:1
1046:b
1043:(
1040:=
1037:)
1034:a
1028:1
1025:(
1019:)
1016:1
1010:b
1007:(
1004:=
1001:a
995:b
992:=
989:)
986:1
980:a
977:(
971:)
968:b
962:1
959:(
956:=
953:)
950:1
944:b
941:(
935:)
932:a
926:1
923:(
920:=
917:b
911:a
891:X
885:b
882:,
879:a
850:1
846:=
837:1
824:1
820:=
817:)
808:1
795:1
791:(
785:)
776:1
763:1
759:(
756:=
753:)
744:1
731:1
727:(
721:)
712:1
699:1
695:(
692:=
683:1
670:1
666:=
657:1
543:.
531:X
525:d
522:,
519:c
516:,
513:b
510:,
507:a
487:)
484:d
478:b
475:(
469:)
466:c
460:a
457:(
454:=
451:)
448:d
442:c
439:(
433:)
430:b
424:a
421:(
410:.
398:X
392:a
366:1
359:a
356:=
353:a
350:=
347:a
335:1
308:1
301:a
298:=
295:a
292:=
289:a
277:1
256:X
230:1
203:1
91:X
20:.
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