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?

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    $\begingroup$ There's precedence for using much weaker categories than toposes. An enriched category is the general notion. $\endgroup$ – Malice Vidrine Sep 27 '17 at 15:15
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    $\begingroup$ 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. $\endgroup$ – Kevin Carlson Sep 28 '17 at 6:04
  • $\begingroup$ @MaliceVidrine yeah, I know enriched categories have internal homs, but I'm working towards avoiding $\mathbf{Set}$ in general. $\endgroup$ – Karl Damgaard Asmussen Sep 28 '17 at 9:09
  • $\begingroup$ @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. $\endgroup$ – Karl Damgaard Asmussen Sep 28 '17 at 9:10

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