Let $p:I\to Cat\_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x\_i,y\_i\in C\_i=p(i)$ be the corresponding objects in $C\_i$. Then we have a new diagram $q\_{x,y}: I\to \mathcal{S}$ given by $q\_{x,y}(i)=Map\_{C\_i}(x\_i,y\_i)$, where $\mathcal{S}$ is the infinity category of spaces.


Question: Is it always true that $Map\_C(x,y)\simeq\lim q\_{x,y}$?



Limits of infinity categories and mapping spaces
Yes. To see this, let us make the preliminary observation that it suffices to prove that this holds for products and pullbacks since we can decompose a general limit into these two special cases.


Let us begin with products. If $I$ is discrete then we want to show that $\prod\_i \hom\_{\mathcal{C\_i}}(x\_i,y\_i) \cong \hom\_{\prod\_i \mathcal{C\_i}}((x\_i)\_{i \in I},(y\_i)\_{i \in I})$. To see this, note that we can realize the limit as the honest product of quasicategories in simplicial sets. By considering the model of $\hom\_{\mathcal{D}}(d\_0,d\_1)$ given by $\mathbf{1} \times\_{\mathcal{D}} \mathcal{D}^{\Delta^1} \times\_{\mathcal{D}} \mathbf{1}$ the conclusion follows immediately as the two $\infty$-groupoids admit isomorphic models as simplicial sets.


Next, we want to show that the same is true for pullbacks. Again, we can rectify our diagram, this time into cospan of $\infty$-categories $\mathcal{C}\_0 \to \mathcal{C}\_{01} \leftarrow \mathcal{C}\_1$. We can replace $\mathcal{C}\_0$ and $\mathcal{C}\_0 \to \mathcal{C}\_{01}$ such that the latter becomes a fibration by changing $\mathcal{C}\_0$ up to equivalence. In this case, the desired homotopy limit is realized by the actual pullback and exactly the same argument as for products applies (exponentials and limits commute with limits in simplicial sets).


