# Hom-like functor to a topos other than Set?

The hom-set functor is often given as the bifunctor:

$$\mathbf{Hom}(-,-) : \cal C^{op} \times \cal C \to \mathbf{Set}$$

(This is of course under the assumption that $\cal C$ is locally small.)

Is there a precedent for using a topos other than $\mathbf{Set}$? Such as for instance the category of weak infinity/omega groupoids known from Homotopy Type Theory?

• There's precedence for using much weaker categories than toposes. An enriched category is the general notion. – Malice Vidrine Sep 27 '17 at 15:15
• Weak infinity groupoids don't really form a category, in the classical sense-the composition is only coherently, not strictly, associative. In particular, they aren't a topos, but an $\infty$-topos. – Kevin Carlson Sep 28 '17 at 6:04
• @MaliceVidrine yeah, I know enriched categories have internal homs, but I'm working towards avoiding $\mathbf{Set}$ in general. – Karl Damgaard Asmussen Sep 28 '17 at 9:09
• @KevinCarlson good point. I wasn't actually aware that there was a difference, since the internal language of (ω,1)-groupoids (i.e. homotopy type theory) cannot distinguish the two. – Karl Damgaard Asmussen Sep 28 '17 at 9:10