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On the homotopy theory of monoids

Part of: Semigroups Applied homological algebra and category theory

Published online by Cambridge University Press:  09 April 2009

Carol M. Hurwitz
Carol M. Hurwitz
Affiliation:
William Paterson CollegeDepartment of Mathematics, Wayne, New Jersey 07470, U.S.A.
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In this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

MSC classification

Secondary: 20M50: Connections of semigroups with homological algebra and category theory 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 171 - 185
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Baumslag, G., Dyer, E. and Heller, A., “The topology of discrete groups’, J. Pure Appl. Algebra 16 (1980), 1–47.CrossRefGoogle Scholar
[2]Cartan, H. and Eilenberg, S., Homological algebra (Princeton Univ. Press, Princeton, N.J., 1956, 116–119 and 149–150).Google Scholar
[3]Hilton, P. J. and Stammbach, U., A course in homological algebra (Springer-Verlag, New York, Heidelberg, Berlin, 1971).CrossRefGoogle Scholar
[4]Kan, D. M., “Adjoint functors”, Trans. Amer. Math. Soc. 87 (1958), 294–329.CrossRefGoogle Scholar
[5]Kan, D. M. and Thurston, W. P., “Every connected space has the homology of a K(π 1),” Topology 15 (1976), 235–258.CrossRefGoogle Scholar
[6]Kurosh, A. G., The theory of groups, transl. by Hirsch, K. A. (Chelsea, 1956).Google Scholar
[7]Latch, D. M., “The uniqueness of homology for the category of small categories,” J. Pure Appl. Algebra 9 (1977), 221–237.CrossRefGoogle Scholar
[8]Latch, D. M. and Fritsch, R., “Homotopy inverses for nerve,” Bull. Amer. Math. Soc. (N.S.) 1 (1979), 258–262.Google Scholar
[9]Lane, S. Mac, Categories for the working mathematician (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
[10]McDuff, D., “On the classifying spaces of discrete monoids,” Topology 18 (1979), 313–319.CrossRefGoogle Scholar
[11]Quillen, D. G., “Higher algebraic K-theory I,” Algebraic k-theory I, pp. 85–147 (Lecture Notes in Mathematics, 341, Springer, New York, Heidelberg, Berlin, 1973).Google Scholar
[12]Segal, G. G., “Classifying spaces and spectral sequences,” Publications in Mathematics 34, pp. 105–112 (Institute Haute Etudes Science, 1968).Google Scholar
[13]Whitehead, J. H. C., “On the asphericity of regions on a 3-sphere,” Fund Math. 32 (1939), 149–166.CrossRefGoogle Scholar
[14]Whitehead, J. H. C., “Combinatorial homotopy. I,” Bull. Amer. Math. Soc. 55 (1949), 213–245.CrossRefGoogle Scholar