Homotopy (co)limits in oo-categories vs model categories

Question

In $\infty$-category theory one can define limits and colimits by analogues of the usual universal properties, but stated in terms of mapping spaces and homotopy equivalences instead of mapping sets and isomorphisms. In the framework of model categories there is also a notion of homotopy (co)limits, which I am less familiar with. Why do these two notions give the same result, say in the category $\mathscr{S}$ of spaces (Kan complexes), or in general when an $\infty$-category also admits a model structure compatible with the $\infty$-category structure?

Answer 1

The ∞-categorical limits (respectively colimits) are given by the right (respectively left) adjoint of the constant diagram functor $$C→C^I,$$ where $I$ is the indexing category and $C$ is the ∞-category of values.

For a model category, homotopy limits (respectively colimits) are given by the right derived functor of the right adjoint (respectively left derived functor of the left adjoint) of the constant diagram functor $$C→C^I,$$ where $I$ is the indexing category and $C$ is the model category of values. (Some mild conditions should be imposed on $C$ to ensure the existence of the injective (respectively projective) model structure on $C^I$, e.g., $C$ is combinatorial or cellular.)

Finally, if $C$ is a model category and $\def\cC{{\cal C}}\cC$ is its underlying ∞-category, then the underlying ∞-functor of $C→C^I$ is equivalent to $\cC→\cC^{N I}$. (Two proofs of this fact are found in the books by Lurie and Cisinski.) Combined with the fact that adjoint functors are defined uniquely up to a contractible choice, we arrive at the desired statement.