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Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some inte …

In an email to the categories mailing list dated 21 August 2003, Street writes: Max reminded me of his old result (not in the LaJolla Proceedings, but known soon after) that a monoidal V-category is …

Expanding upon my comment: categories without units are called semicategories. You define a notion of semicategory enriched in a semigroupal category, which is what you describe. The Yoneda lemma is s …

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed monoi …

Let $(\mathscr V, \otimes, 1, \sigma)$ be a symmetric strict monoidal category whose unit is terminal. Suppose that every object $A$ is equipped with the structure of a cocommutative comonoid $1 \xlef …

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For …

The first question is the main topic of two recent papers: Reggio's Polyadic Sets and Homomorphism Counting, which is also a good reference for previous results in the literature (including that of P …

I don't have a counterexample to show that (3) is not superfluous, but I can provide some motivation for it. First, let us recall what is probably the most well known characterisation of cartesian mon …

A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-close …

In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal …

While relevant references have been mentioned in the comments and in john's answer, I don't believe these fully answer the question as asked, at least without doing a little further reading, so here's …

Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X …

I have not been able to find an explicit reference for this in the literature, so I will sketch a proof here. Steve Lack suggested that one could make use of Mac Lane's proof of strictification for mo …

Action versus module In modern category theory parlance, action and module are typically interchangeable, and their use comes down to author preference: this is a historical artefact of the differing …

A relevant reference is §3 of Kelly's Doctrinal adjunction: in particular, see Theorem 3.1, which states that, if the adjunction is reflective, then the subcategory inherits monoidal structure if and …