# Hom spaces in (∞, 1)-categories

In ordinary category theory it is a well-known and important fact that the $$\hom$$ bifunctor into $$\text{Set}$$ preserves limits.

I am unable to find a reference for the corresponding fact in infinity category theory nor am I able to write down a proof myself.

I am looking for a statement of the following form:

Let $$\mathcal{C}, \mathcal{D}$$ be $$(\infty, 1)$$-categories (considered as a weak Kan complex preferably) and let $$F: \mathcal{D} \to \mathcal{C}$$ be a diagram. Then, for any object $$X \in \mathcal{C}$$, we have a natural equivalence between $$\lim \hom(X, F(D))$$ and $$\hom (X, \lim F(D))$$ where each $$\hom$$ is considered as a weak Kan complex.

I would presume that HTT would have a statement to this effect, but I am unable to find one.

• I thought this was the definition of "limit". – Theo Johnson-Freyd Jan 8 '20 at 1:01
• @TheoJohnson-Freyd Limits can be defined as terminal cones, in which case something has to be proved about the relationship between $\infty$-categories of cones and the mapping space functor. They can be defined as certain Kan liftings, in which case the connection to hom is still less obvious. I am not sure whether anyone writing foundational material literally defines a limit cone as one sent to a limit in spaces under hom. This would depend on giving an explicit definition of limits in spaces, which is not so easy as in sets. – Kevin Arlin Jan 8 '20 at 1:10

This is Cisinski, Corollary 6.3.5. The proof is essentially to show that cocontinuous functors out of presheaf $$\infty$$-categories admit right adjoints given by the usual formula, so that $$Hom_C$$ has a left adjoint if $$C$$ is cocomplete, and then to use the Yoneda embedding for a general $$C$$.