I am currently reading the book by Bousfield and Kan, in particular Ch. XII, par. 2, and would like to understand why the functor $hocolim: Top_{+}^{I} \rightarrow Top_{+}$ is left adjoint to $hom(I \backslash —,—)$, where $Top_{+}^{I}$ denotes the category of $I$-diagrams of dotted topological spaces and $I \backslash —$ yields, for $i \in I$, the respective "under" category. Also, $hom(Z, W)$ is the simplicial space of which an n-simplex is a map $\Delta_n \times Z \rightarrow W$.

However, I do not understand what taking the product with the n-simplex is good for. The problem comes up when I try to explicitly write down the isomorphisms between the sets of maps in the adjunction.

  • $\begingroup$ The simplicial sets $hom(Z,W)$ they define are really just an instance of a general categorical phenomenon: simplicial objects in any category admit a canonical enrichment over simplicial sets. (Note that the category of $I$-diagrams in $sSet$ is equivalent to the category of simplicial objects in the category of $I$-diagrams in $Set$.) Of course, this has nothing to do with the simplex category either. $\endgroup$ – user314 May 26 '15 at 15:10

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