# Fubini theorem for hocolim

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.

• A related (possibly identical?) question asked previously: mathoverflow.net/questions/33556/… Oct 25 '12 at 16:11
• Not identical at all, triangulated categories only have homotopy push-outs and countable sequential homotopy colimits, and they are not functorial. Oct 25 '12 at 19:03
• I don't think I would call that a Fubini theorem. Oct 25 '12 at 21:05
• well-- glad to have that cleared up! Oct 26 '12 at 11:27

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 $$(\operatorname{hocolim}_JF(i))=p_!((I\times J)\downarrow i\rightarrow I\times J\stackrel{F}\rightarrow \mathcal{C})$$.

• I guess you mean $u_!$ is the left adjoint of the restriction functor $u^*$? Dec 18 '20 at 9:54
• @Nikitas correct, I'll fix it. Dec 18 '20 at 11:38

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