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Stream: theory: category theory

Topic: (Pseudo)monoids in a monoidal double category

Dylan Braithwaite (Jun 12 2023 at 13:14):

For a monoidal double category $\mathbb E$ I am interested in pseudomonoid objects in the monoidal 2-category of tight morphisms $\mathcal T(\mathbb E)$. For two such pseudomonoid objects, say $M$ and $N$, I could then define a notion of a 'monoidal proarrow' between them, as a loose arrow $M \to N$ which is a monoid object in $\mathbb E_1$ (and which composes compatibly with the pseudomonoid structure cells). So then we have monoid multiplication squares like so:
99873D14-6729-4937-81CB-7E3C71096DDB.jpg

I believe then these would assemble into a double category of pseudomonoids, pseudomonoid morphisms, monoidal proarrows, and monoid morphisms. I'm wondering has this structure been studied or written down somewhere in literature already?