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In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this:$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\hocolim}{hocolim}\DeclareMathOperator{\Ho}{Ho}\DeclareMathOperator{\holim}{holim}$ $$\Hom\nolimits_\mathcal{C}(\colim A_i,B)=\lim \Hom\nolimits_\mathcal{C}(A_i,B)$$

I am looking for a corresponding statement for hocolims - lets say in simplicial sets, but if there are more general statements, that's even better.

E.g. I could imagine $$\Hom\nolimits_{\mathcal{C}}(\hocolim A_i,B)=\lim \Hom\nolimits_{\Ho(\mathcal{C})}(A_i,B)$$ - maybe one needs to have $B$ fibrant and the A_i cofibrant here, i.e. that the Homs on the right are $\mathbb{R}Homs$.

Using the internal Hom in simplicial sets I could also imagine versions like this: $$\Hom(\colim A_i,B)=\holim \Hom(A_i,B)$$ $$\Hom(\colim A_i,B)=\holim \mathbb{R}\!\Hom(A_i,B)$$

What is the right statement and what is the place to learn this hocolim-yoga?

Thanks! N.B.

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Here's another perspective: just as the colimit is the initial object receiving maps from all objects in the diagram making all triangles commute, the homotopy colimit is the initial object receiving maps from all objects in the diagram making all triangles commute up to homotopy, and such that those homotopies are coherent up to higher homotopies, etc., all the way up. You could therefore call it "the initial homotopy-coherent map to a constant diagram". This is formalized in Dan Dugger's notes on hocolims and things -- I forget the name exactly, but it should be easy to find. – Aaron Mazel-Gee Oct 23 '12 at 1:58
up vote 7 down vote accepted

A formula like the one you are asking for could be the following (Bousfield-Kan, "Homotopy limits, completions and localizations", chapter XII, proposition 4.1):

$$ \mathrm{hom}_* (\mathrm{hocolim}\ \mathbf{A}, B) \cong \mathrm{holim}\ \mathrm{hom}_* (\mathbf{A}, B) \ . $$

Here $B$ is a pointed simplicial set, $\mathbf{A} : I \longrightarrow \Delta^{\mathrm{o}}\mathbf{Set}_*$ a functor from a small category $I$ to the category of pointed simplicial sets, and for pointed simplicial sets $A, B$

$$ \mathrm{hom}_* (A,B) \in \Delta^{\mathrm{o}}\mathbf{Set}_* $$

is the pointed simplicial function space which $n$-simplices are maps in $\Delta^{\mathrm{o}}\mathbf{Set}_* $

$$ \left( \Delta [n] \times A \right) / \left( \Delta [n] \times * \right) \longrightarrow B $$

(op.cit., chapter VIII, 4.8).

I didn't go through the details, but, if it annoys you, it seems to me that you can drop the "pointed" thing everywhere just deleting "pointed", $*$, and $\Delta [n] \times * $ in what Bousfield-Kan say.

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Thank you! So B does not have to be fibrant, now I wonder why, but Bousfield-Kan seems like the thing to read for me... – N.B. Oct 6 '10 at 13:23

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