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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Homotopy in functor categories


Author: Alex Heller
Journal: Trans. Amer. Math. Soc. 272 (1982), 185-202
MSC: Primary 55U35; Secondary 18A25, 18G55
Erratum: Trans. Amer. Math. Soc. 279 (1983), 429.
MathSciNet review: 656485
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0656485-2
PII: S 0002-9947(1982)0656485-2
Keywords: Abstract homotopy, functor category, homotopy limit, homotopy Kan extension, half-exact functor
Article copyright: © Copyright 1982 American Mathematical Society