Homotopy in functor categories
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- by Alex Heller PDF
- Trans. Amer. Math. Soc. 272 (1982), 185-202 Request permission
Erratum: Trans. Amer. Math. Soc. 279 (1983), 429.
Abstract:
If ${\mathbf {C}}$ is a small category enriched over topological spaces the category ${\mathcal {J}^{\mathbf {C}}}$ of continuous functors from ${\mathbf {C}}$ into topological spaces admits a family of homotopy theories associated with closed subcategories of ${\mathbf {C}}$. The categories ${\mathcal {J}^{\mathbf {C}}}$, for various ${\mathbf {C}}$, are connected to one another by a functor calculus analogous to the $\otimes$, Hom calculus for modules over rings. The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories. Among the functors so described are homotopy limits and colimits and, more generally, homotopy Kan extensions. A by-product of the method is a generalization to functor categories of E. H. Brown’s representability theorem.References
- D. W. Anderson, Axiomatic homotopy theory, Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978) Lecture Notes in Math., vol. 741, Springer, Berlin, 1979, pp. 520–547. MR 557184
- G. Baumslag, E. Dyer, and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980), no. 1, 1–47. MR 549702, DOI 10.1016/0022-4049(80)90040-7
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
- Alex Heller, Abstract homotopy in categories of fibrations and the spectral sequence of Eilenberg and Moore, Illinois J. Math. 16 (1972), 454–474. MR 310890
- Alex Heller, Adjoint functors and bar constructions, Advances in Math. 12 (1974), 8–31. MR 334266, DOI 10.1016/S0001-8708(74)80016-2
- Alex Heller, On the homotopy theory of topogenic groups and groupoids, Illinois J. Math. 24 (1980), no. 4, 576–605. MR 586797
- Alex Heller, On the representability of homotopy functors, J. London Math. Soc. (2) 23 (1981), no. 3, 551–562. MR 616562, DOI 10.1112/jlms/s2-23.3.551
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Dusa McDuff, On the classifying spaces of discrete monoids, Topology 18 (1979), no. 4, 313–320. MR 551013, DOI 10.1016/0040-9383(79)90022-3
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393
- N. E. Steenrod, Milgram’s classifying space of a topological group, Topology 7 (1968), 349–368. MR 233353, DOI 10.1016/0040-9383(68)90012-8
- Rainer M. Vogt, Homotopy limits and colimits, Math. Z. 134 (1973), 11–52. MR 331376, DOI 10.1007/BF01219090
- Rainer M. Vogt, Commuting homotopy limits, Math. Z. 153 (1977), no. 1, 59–82. MR 451237, DOI 10.1007/BF01214734
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 272 (1982), 185-202
- MSC: Primary 55U35; Secondary 18A25, 18G55
- DOI: https://doi.org/10.1090/S0002-9947-1982-0656485-2
- MathSciNet review: 656485