Consider a situation where there is a pseudofunctor from some category $C$. Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a closed category) and then this functor becomes $C$-enriched.

Now, to describe the construction of the internal hom there is no need for any property of the pseudofunctor.

Therefore we can understand exactly what geometrical properties of the pseudofunctor determine this internal hom structure.

This is the construction:

Let $C$ be a category equipped with a pseudofunctor $P:C^{op}\to Cat$

Then it induces another pseudofunctor: $$P^*:[C^{op}\times C, C] \to [C^{op} \times C, Cat]$$ There is also the pseudofunctor $$[P-, P-] : C^{op}\times C \to Cat$$ As an object in the 2-category $[C^{op}\times C, Cat]$ it can be uniquely identified as a 2-functor from the initial 2-category: $$i_{[P-, P-]}:I\to [C^{op}\times C, Cat]$$

The unique object in the image of the right 2-Kan lift of $i_{[P-, P-]}$ through $P^*$ is the **internal hom** of $C$ through $P$

Examples:

If $C=Poset$ and $P$ is the interpretation of an ordered set as a category then we get the usual hom of order preserving functions.

If $C=Group$ and $P=$

**B**then I have no idea how to describe this Kan extension - if we manage to have a closed category in this case, then the forgetful functor to $Set$ would give all the members of the codomain that induce a conjugacy relation between the image subgroups.If $C=Top$ with $P=\pi_0$ I think this reconstructs the usual internal hom in $Htpy$ but I haven't looked at this one deeply yet.

Questions:

Have you seen this construction elsewhere?

What is this structure in example 2 (does it even exist)?

**What are the properties of $P$ needed to make this an internal hom in a closed category?**Can this be easily generalized to $\infty$-category theory (the last example shows it might be interesting)?

whenit exists. $\endgroup$ – Omer Rosler Aug 18 '18 at 11:16