Homotopy in functor categories

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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.