A Giraud-Conduché condition for $T$-categories

Given a monad $T$ on a category $ℰ$, Burroni introduced the notion of $T$-category, also known as multicategories, generalized multicategories, and $T$-multicategories. For suitable choices of $ℰ$ and $T$, this notion incluces multicategories in the sense of Lambek, internal categories in the sense of Ehresmann, topological spaces, and virtual double categories in the sense of Cruttwell and Shulman (earlier studied by Leinster under the name of fc-multicategories). The notion of $T$-category has received increasing interest in recent years and has been studied by too many people to list here.

Since the notion of $T$-category includes that of internal category and so in turn that of category, we can seek to generealize various categorical notions to the $T$-categorical context, either in general, or for particular choices of $ℰ$ and $T$. In earlier work, we defined a notion of nerve for a $T$-category, and studied local presentability and other such properties of the category (or 2-category) of $T$-categories. In this talk, we look at the situation where a $T$-category or $T$-functor is powerful (also known as exponentiable).

Recall that an object $A$ of a category $𝒦$ with finite products is powerful when the functor $-\times A:𝒦\to 𝒦$ has a right adjoint; thus all objects are powerful just when $𝒦$ is cartesian closed (has internal homs or mapping spaces). A morphism $A\to X$ is powerful when it is powerful as an object of the slice category $𝒦/X$; again, all morphisms are powerful when $𝒦$ is locally cartesian closed.

As is well-known, $\mathrm{𝐂𝐚𝐭}$ is cartesian closed, but not locally cartesian closed. The powerful morphisms in $\mathrm{𝐂𝐚𝐭}$ were characterized by Giraud and by Conduché, and they include both the fibrations and the opfibrations. We study the powerful $T$-categories and the powerful $T$-functors (morphisms of $T$-categories) under assumptions on $ℰ$ and $T$: the category $ℰ$ should be something like a topos, the functor $T$ should be a parametric right adjoint (and so in particular preserve connected limits), and the multiplcation and unit of the monad should be cartesian natural transformations. Under these assumptions we give a sufficient condition for a $T$-category or $T$-functor to be powerful. Our condition agrees with that of Giraud-Conduché in the case of ordinary functors, and with the known generalization to internal functors. In the case where $ℰ=\mathrm{𝐒𝐞𝐭}$ and $T$ is the monoid monad, so that $T$-categories are multicategories in the sense of Lambek, we recover Pisani’s result that the powerful multicategories are the promonoidal categories.