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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?

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  • $\begingroup$ 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. $\endgroup$ Commented Jan 7, 2021 at 12:56

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As you say, you have choices. Normally for homotopy colimits you want the projective model structure (if that is available). Sometimes the Reedy model structure works too. There is also the injective model structure, which is used for homotopy limits.

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.

You may also be interested in techniques to transfer model structures along adjunctions.

FYI, I like Dugger's (unfinished) notes on homotopy colimits as a reference. See part 3 for model structures on diagram categories.

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