The n-Category Café

If we allow the functors starting at $X$ to be anafunctors, we can simply write

$$\begin{array}{ccccc}X& & \stackrel{\sigma }{\to }& & V/{/}_{\rho }G\\ & {}_g\searrow & & {\swarrow }_r\\ & & BG\end{array},$$

This formulation of sections of bundles has the great advantage that its arrow theory remains valid when we pass this from the world of Lie $n$-groupoids to the world of Lie $n$-algebras. That’s described in $L_{\mathrm{\infty }}$-associated bundles, sections and covariant derivatives.

I am claiming that in extended QFTs (= extended cobordism representations) of $\Sigma$-model type, those induced from an $n$-bundle with connection on $X$, the assignment of the representation to a $k$-dimensional piece $\Sigma$ of cobordism is the “collection of sections” of the transgressed $n$-bundle

$${\mathrm{hom}}_{n\mathrm{Cat}}(\Sigma ,--)\left(\begin{array}{ccccc}X& & & & V/{/}_{\rho }G\\ & {}_g\searrow & & {\swarrow }_r\\ & & BG\end{array}\right),$$

hence is the collection of completions $\sigma$

$$\begin{array}{ccccc}\mathrm{hom}(\Sigma ,X)& & \stackrel{\sigma }{\to }& & \mathrm{hom}(\Sigma ,V/{/}_{\rho }G)\\ & {}_{\mathrm{hom}(\Sigma ,g)}\searrow & & {\swarrow }_{\mathrm{hom}(\Sigma ,r)}\\ & & \mathrm{hom}(\Sigma ,BG)\end{array}$$

where this “collection” is to be read as an inner hom

$${\mathrm{hom}}_{\mathrm{hom}(\Sigma ,BG)}(\mathrm{hom}(\Sigma ,X),\mathrm{hom}(\Sigma ,V/{/}_{\rho }G)).$$

(See maybe Bruce’s description of this situation).

Question

What I need to better understand are those inner homs in $n$-categories over $BG$.

What in general can be said about inner homs in over categories of closed categories? What’s a good way to handle them?