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Journal Article
Homotopy in Functor Categories
Alex Heller
Transactions of the American Mathematical Society
Vol. 272, No. 1 (Jul., 1982), pp. 185-202
Published
by: American Mathematical Society
DOI: 10.2307/1998955
https://www.jstor.org/stable/1998955
Page Count: 18
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Topics: Functors, Adjoints, Mathematical theorems, Homotopy theory, Topological spaces, Cylinders, Monoids, Algebraic topology, Numbers
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Abstract
If C is a small category enriched over topological spaces the category $\mathcal{J}^C$ of continuous functors from $C$ into topological spaces admits a family of homotopy theories associated with closed subcategories of C. The categories $\mathcal{J}^C$, for various 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.
Transactions of the American Mathematical Society
© 1982 American Mathematical Society