### What’s a One-Object Sesquicategory?

#### Posted by John Baez

A **sesquicategory**, or $1\frac{1}{2}$-category, is like a 2-category, but without the interchange law relating vertical and horizontal composition of 2-morphisms:

$(\alpha \cdot \beta)(\gamma \cdot \delta) = (\alpha \gamma) \cdot (\beta \delta)$

Better, sesquicategories are categories enriched over $(Cat,\square)$: the category of categories with its “white” tensor product. In the *cartesian* product of categories $C$ and $D$, namely $C \times D$, we have the law

and we can define $f \times g$ to be either of these. In the *white* tensor product $C \square D$ we do not have this law, and $f \times g$ makes no sense.

What’s a one-object sesquicategory?

A one-object sesquicategory is like a strict monoidal category, but without the law

$(f \otimes 1)(1 \otimes g) = (1 \otimes g)(f \otimes 1)$

I seem to have run into a bunch of interesting examples. Is there some name for these gadgets?

If not, I may take the “one-and-a-half” joke embedded in the word “sesquicategory”, and subtract one. That would make these things **semi-monoidal categories**.

(The name “white” tensor product is part of another string of jokes, involving the white, Gray, and black tensor products of 2-categories. The white tensor product is also called the “funny” tensor product.)

## Re: What’s a One-Object Sesquicategory?

John asked:

Yes! These (in the not-necessarily-strict version) were named premonoidal categories by John Power and collaborators. Now I’m curious what kind of examples you have run into.