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It is well known that the notion of an internal category can be generalized to categories without pullbacks by considering cotensors of comodules in a monoidal category. I'm curious about the other direction, considering internal monoidal categories in a category with products.

Definition. Let $\mathcal{C}$ be a category with finite products. A strict internal monoidal category $\mathscr{C}$ in $\mathcal{C}$ consists of an internal category $\mathscr{C}$ in $\mathcal{C}$ together with internal functors $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, $I:{\bf 1}\to\mathscr{C}$ such that $$ \otimes\circ(\otimes\times1)=\otimes\circ(1\times\otimes), $$ $$ \otimes\circ\langle1,I\circ!\rangle=1_\mathscr{C}=\otimes\circ\langle I\circ!,1\rangle. $$

We can internalize weak monoidal categories in a bicategory:

Definition. Let $\mathcal{C}$ be a bicategory with products. An internal monoidal category $\mathscr{C}$ in $\mathcal{C}$ consists of an internal category $\mathscr{C}$ in $\mathcal{C}$ together with internal functors $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, $I:{\bf 1}\to\mathscr{C}$ together with internal natural isomorphisms $$ \alpha:\otimes\circ(\otimes\times1)\Rightarrow\otimes\circ(1\times\otimes), $$ $$ \iota^r:\otimes\circ\langle1,I\circ!\rangle\Rightarrow 1_\mathscr{C}, $$ $$ \iota^\ell:\otimes\circ\langle I\circ!,1\rangle\Rightarrow 1_\mathscr{C}, $$ such that the following diagrams commute: (top is the pentagon identity, bottom is the triangle identity)

Have these notions been studied in the literature anywhere? We can consider an internal braiding in a bicategory by considering an internal natural isomorphism $B^\otimes:\otimes\Rightarrow\otimes\circ\pi_1\times\pi_0$ such that the following diagrams commute

The braiding is symmetric iff ${B^\otimes}^{-1}=B^\otimes_{\pi_1\times\pi_0}$, so on and so forth. I can't find mention of these notions after a few months of playing around with them but I'm certain they've been written down before, I'm just wondering if there's anything that's been made publicly available. Any assistance is appreciated.

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  • $\begingroup$ Bicategories internal to 2-categories have been studied in Internal bicategories by Douglas–Henriques. Internal monoidal categories ought to be one-object internal bicategories. $\endgroup$
    – varkor
    Oct 19, 2021 at 14:29
  • $\begingroup$ @varkor Thanks for the pointer, I'll check it out. $\endgroup$
    – Alec Rhea
    Oct 19, 2021 at 15:30

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This has been done in Section 1.3 of Enrico Ghiorzi's PhD thesis:

Ghiorzi, E. (2019). Internal enriched categories (Doctoral thesis). https://doi.org/10.17863/CAM.45286. Link.

It has also lead to these two preprints:

Ghiorzi, E. (2020). Complete internal categories. arXiv:2004.08741 [math.CT].

Ghiorzi, E. (2020). Internal enriched categories. arXiv:2006.07997 [math.CT].

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    $\begingroup$ +1 because I was about to post Enrico's thesis $\endgroup$
    – fosco
    Oct 19, 2021 at 21:00

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