The n-Category Café

We know now that various structure on a double category corresponds to similar structure on a bicategory. For instance, a monoidal structure on a (suitably well-behaved) double category induces a monoidal structure on its underlying bicategory. However, the monoidal double category is generally much stricter and easier to work with.

Aleiferi’s paper is about extending this to the cartesian monoidal case. A cartesian monoidal double category is easy to define: its diagonal $D\to D\times D$ and projection $D\to 1$ have right adjoints, just as for ordinary categories. It’s also easy to say what it means for a $Cat$-like bicategory to be cartesian monoidal: we can say that its diagonal and projection have right adjoints too, although that’s more complicated because the adjoints are generally only pseudofunctors living in a tricategory.

But it’s not at all obvious what it means for a $Span$-like bicategory to be “cartesian monoidal”. Intuitively, bicategories like $Span$ itself, or more generally $Span(E)$ for $E$ a category with finite limits, and $Prof(V)$ when $V$ is cartesian monoidal, should be “cartesian” — but they are not cartesian monoidal in the $Cat$-like way. The notion of cartesian bicategory was defined (by Carboni, Walters, Kelly, Verity, and Wood) to capture examples like these, but it is quite complicated. Moreover, to someone familiar with double categories, it is crying out to be reformulated in double-category language (e.g. it requires certain morphisms to have right adjoints, and induces $Cat$-like cartesian structure on the sub-bicategory of morphisms with right adjoints). In fact, it blows my mind that anyone was able to define the notion of cartesian bicategory without secretly having double categories in their head!

Aleiferi has now made a more careful study of cartesian double categories, and shown that they can be used for at least some (which I suspect will eventually become “nearly all”) of the same purposes as cartesian bicategories. For instance, here is a theorem from Lack-Walters-Wood Bicategories of spans as cartesian bicategories:

Theorem: A bicategory is equivalent to $Span(E)$, for some category $E$ with finite limits, if and only if it is cartesian, each comonad has an Eilenberg-Moore object, and every map is comonadic.

And here is a theorem from Aleiferi’s paper:

Theorem: A double category is equivalent to $Span(E)$, for some category $E$ with finite limits, if and only if it is cartesian, fibrant, unit-pure, and has strong Eilenberg-Moore objects for copointed endomorphisms.

Even without understanding all the words, the family resemblance should be clear, even if the technicalities are different. On a quick skim of Aleiferi’s paper it looks like there is no formal comparison yet between cartesian double categories and cartesian bicategories, but I’m sure that will come.