# Internal monoidal categories

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.

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