Fubini theorem for hocolim

Question

I wanted to ask the following question, Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it true that the map $\mathrm{hocolim}_{J}(F(i))\rightarrow (\mathrm{hocolim}_J~F)(i)$ is a weak equivalence in $\mathbf{M}$ ?

For more precision: $F(i)$ is the evaluation of the functor $F:J\times I\rightarrow \mathbf{M} $ at $i\in I$, and $\mathrm{hocolim}_J~F$ is an object in the model category $\mathbf{M}^{I}$. The categories $\mathbf{M}^{I}$, $\mathbf{M}^{J\times I} $ are equipped with the projective model structure.

Thank you.

Answer 1

This property holds actually for right derivable categories in the sense of:

MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 138 (2010), no. 3, 317–393.

At least under suitable finiteness assumptions on $I$ and $J$. It also holds for arbitrary small categories $I$ and $J$ when working on a homotopically complete right derivable category. Model categories are examples of these (cofibrant generation is not needed).

Cisinski shows that the homotopy categories of diagrams on a (homotopically complete) right derivable category $\mathcal{C}$ form a right derivator whose domain is the category of directed finite categories (or all small categories in the homotopically complete case). In particular, one of the axiom says that if $u\colon A\rightarrow B$ is any functor in the domain and $u_!\colon\operatorname{Ho}(\mathcal{C}^A)\rightarrow \operatorname{Ho}(\mathcal{C}^B)$ is the left adjoint of the restriction along $u$ functor $u^{*}\colon\operatorname{Ho}(\mathcal{C}^B)\rightarrow \operatorname{Ho}(\mathcal{C}^A)$ then for any $F\colon A^{\operatorname{op}}\rightarrow \mathcal{C}$ and any object $b\in B$ the formula $u_!(F)(b)=p_!(A\downarrow b\rightarrow A\stackrel{F}\rightarrow \mathcal{C})$ holds. Here $p_!\colon\operatorname{Ho}(\mathcal{C}^{A\downarrow b})\rightarrow \operatorname{Ho}(\mathcal{C})$ is the usual homotopy colimit.

In your case, take $u\colon I\times J\rightarrow I$ to be the projection onto the first factor. Then $(\operatorname{hocolim}_JF)(i)=u_{!}(F)(i)$ by definition, the functor $J\rightarrow(I\times J)\downarrow i\colon j\mapsto (i,j)$ is cofinal, and hence $(\operatorname{hocolim}_JF(i))=p_!((I\times J)\downarrow i\rightarrow I\times J\stackrel{F}\rightarrow \mathcal{C})$.

Answer 2

You might also be interested in chapter III of the paper Homotopy theory of diagrams, by Chacholski and Scherer.