Why is the internal hom of a Kan complex also a Kan complex?

Question

Let $X, Y : \Delta^{op} \to \text{Set}$ be simplicial sets. I've seen it stated that if $Y$ is a Kan complex, then the internal hom $Y^X$ is a Kan complex also. How does one show this?

Answer 1

Allegedly, one can prove this directly, but I have never seen the details. (For instance, the same fact is asserted without proof as Theorem 1.9 in [Moore, Semi-simplicial complexes and Postnikov systems].) Here is a more high-level proof.

Recall that the class of anodyne extensions of simplicial sets is defined by induction:

• Any horn inclusion is an anodyne extension.
• Any pushout of an anodyne extension is an anodyne extension.
• Any finite or transfinite composite of anodyne extensions is an anodyne extension.

It immediately follows that a simplicial set $K$ is a Kan complex if and only if the unique morphism $K \to \Delta^0$ has the right lifting property with respect to all anodyne extensions.

Let $X$ be a simplicial set and let $Y$ be a Kan complex. We wish to show that $Y^X$ is a Kan complex. By adjointness, $Y^X$ is a Kan complex if and only if $Y \to \Delta^0$ has the right lifting property with respect to all morphisms of the form $i \times \mathrm{id}_X$ where $i$ is an anodyne extension. But it is well known that a morphism is an anodyne extension if and only if it is both a monomorphism and a weak homotopy equivalence, so $i \times \mathrm{id}_X$ is an anodyne extension if $i$ is. Thus $Y^X$ is indeed a Kan complex.