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Stream: theory: category theory

Topic: externalisation of non-Cartesian internal categories

Emily (May 20 2021 at 23:27):

(This question comes from MathOverflow. Since not everyone here is there and conversely, I figured it might be a good idea to ask this here too.)

In the context of category theory internal to a category $(\mathcal{E},\times,\mathbf{1}_{\mathcal{E}})$ with pullbacks and a terminal object, the process of externalisation builds an indexed category $\mathbb{E}_{\mathcal{C}}\colon\mathcal{E}^{\mathsf{op}}\to\mathsf{Cats}_{\mathsf{2}}$ from an $\mathcal{E}$-internal category $\mathcal{C}$, defining a $2$-functor

$$\mathbb{E} \colon \mathsf{Cats}_{\mathcal{E}} \longrightarrow \mathsf{PseudoFun}(\mathcal{E}^{\mathsf{op}},\mathsf{Cats}_{\mathsf{2}})$$

from the $2$-category of $\mathcal{E}$-internal categories to the $2$-category of $\mathcal{E}$-indexed categories.

When one replaces $(\mathcal{E},\times,\mathbf{1}_{\mathcal{E}})$ by a monoidal category $(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$, one can still define internal categories to $\mathcal{V}$, provided $\mathcal{V}$ is regular. This is the subject of Aguiar's PhD thesis, where the notion is defined and studied.

Question: Can one define externalisation for categories internal to $\mathcal{V}$?