# Homotopy (co)limits and model category structure on functor category

Let $$C$$ be a model category, $$I$$ a small category and $$C^{I}$$ the functor category. I was reading about homotopy (co)limits, and they define them the following way. First give a model category structure to $$C^{I}$$ such that $$\text{colim}: C^{I} \to C$$ is a left Quillen functor, and then we can construct its left derived Quillen functor, which gives a functor $$\text{hocolim}: C^{I} \to C$$ which is "well defined up to homotopy".

What is the desired model category structure on $$C^{I}$$? (which apparently is not unique). A natural definition is that $$\alpha: F \Rightarrow G$$ is a weak equivalence iff for all $$x \in I$$ $$\alpha_x : F(x) \to G(x)$$ is a weak equivalence. What about for (co)fibrations?

• Any model category structure on $C^I$ with weak equivalences being pointwise weak equivalences and so that $\text{colim}$ is a left Quillen functor works to define $\text{hocolim}$. An alternative way is using left deformations and model categories provide a good context in which to construct these. The cofibrant replacement "functor" $Q: C^I \rightarrow C^I$ is a left deformation for $\text{colim}$ in particular. Commented Jan 7, 2021 at 12:56

To give more details, if $$\mathsf{C}$$ is cofibrantly-generated, the projective model structure on $$\mathsf{C}^\mathsf{I}$$ has objectwise weak equivalences, objectwise fibrations, and cofibrations whatever they have to be. Dually for the injective model structure. The Reedy model structure takes a bit longer to describe.