The n-Category Café

Does your dislike extend to lax monoidal functors and lax functors of double categories? I think lax functors of double categories are very analogous to lax monoidal functors. For lax monoidal functors, you have algebraic structure on a category which is being preserved laxly, but the functor is an honest functor. A double category can be thought of as “algebraic structure” on a Cat-graph, which is therefore preserved laxly by a lax functor, but the underlying “functor” on Cat-graphs is again an honest functor. In addition, I forgot to mention that $MonCat_{lax}$ is a full sub-2-category of $DblCat_{lax,vert}$, when you view a monoidal category as a double category with one object and one vertical arrow.

I don’t regard icons as a “fundamental” notion, but rather as a technical one that is sometimes convenient. As you probably know if you’ve read the paper, one way in which icons arise “naturally” is that when you write down a 2-monad $T$ on the 2-category of Cat-enriched graphs for which $T$-algebras are 2-categories, then pseudo, lax, and oplax functors are the pseudo, lax, and oplax $T$-morphisms, and icons are the $T$-transformations. This means that you can apply 2-monad theory.

For instance, there’s a general coherence theorem saying that if a 2-monad $T$ is well-behaved, then any pseudo $T$-algebra is equivalent to a strict one. I don’t know a version of this theorem that applies to this 2-monad exactly, but it does apply if you consider only Cat-enriched graphs with a fixed set of objects. Then the general theorem tells you that any (unbiased) bicategory is equivalent to a strict 2-category, where “equivalence” means in the 2-category of bicategories, pseudo functors, and icons. This implies an equivalence of bicategories in the usual sense, but is stronger, and also contains the fact that the equivalence is bijective on objects.

Of course at the end of the day, we are mostly interested in the “real” notion of equivalence. But the icons provide a useful technical tool allowing us to apply general theory to this situation.

Another way to think of icons is that if you regard bicategories as double categories “horizontally” with no nonidentity vertical arrows, then icons get identified with vertical transformations. In fact that embeds the 2-category $Bicat_{lax,icon}$ fully into $DblCat_{lax,vert}$. So icons are what you get out of the sensible notion of “vertical transformation of double categories” if you insist (often, unreasonably) that your bicategories really don’t have any vertical arrows.

This was actually precisely the reason I got interested in double categories. It’s just so nice that vertical transformations whisker compose with lax double functors, in contrast to (ordinary) lax transformations, which don’t whisker compose with (ordinary) lax functors between bicategories.

Of course, lax functors come up so naturally (in particular, as Mike mentions, when considering the change of base for enriched categories), that it’s important that they work well in the double categorical setting.

So I guess it’s something like each 2-morphism is a square with strings coming into various sides where when you isotope you’re not allowed to ever rotate squares. Does that sound right?

In particular, there’s probably not a good notion of double category with duals?

I guess it’s something like each 2-morphism is a square with strings coming into various sides where when you isotope you’re not allowed to ever rotate squares.

Yes, that sounds right. Although if your double category is a framed bicategory, then horizontally running strings can be bent into vertical ones. So in that case, the string diagrams look just like those for a bicategory but with a certain class of (dualizable) strings singled out as coming from the vertical arrows.

In particular, there’s probably not a good notion of double category with duals?

Why not?

Assuming the existence of null morphisms shouldn’t be necessary for categorification.

Well, to start with, the constraint cells of a lax 2-functor are 2-cells in a 2-category, so they are dualizable iff they are invertible. However, we could imagine considering lax functors between 3-categories, and ask whether they form a 4-category if their constraints have adjoints. Or we could try some other variation on lax 2-functors that comes with constraints in both directions, such as saying we have a functor that’s both lax and colax with the constraints satisfying some relationship other than adjointness, e.g. one being a retraction of the other, or maybe a “Frobenius” condition. Another possibility that some people sometimes think of is generalizing the notion of transformation so that $\alpha_f$ can be a zigzag rather than just a morphism in one direction or the other.

Unfortunately, none of these ideas give you a 3-category either, although they come closer. All these ideas are aimed at allowing us to whisker transformations by functors. Notice that in all cases, for this to work the notion of “transformation” involved has to be some sort of lax, since the 2-cell components of the whisker-composite $H\alpha$ involve the not-necessarily-invertible constraint cells of the functor $H$. If we require these cells to be dualizable, or have retractions, then we must consider at least lax transformations whose 2-cell components are dualizable, or have retractions, etc.—pseudo natural transformations aren’t general enough.

However, there is no 3-category of 2-categories whose 2-cells are ??? transformations for any value of ??? weaker than “pseudo,” no matter what the 1-morphisms are. For although we might be able to perform whiskering, the interchange law now fails. Namely, suppose we have 2-categories $C,D,E$, functors $F,G\colon C\to D$ and $H,K\colon D\to E$, and ??? transformations $\alpha\colon F\to G$ and $\beta\colon H\to K$. Then in a 3-category, the two composites $$(\beta G) . (H\alpha) \qquad\text{and}\qquad (K\alpha) . (\beta F)$$ must be isomorphic. (In a 2-category, they would be equal, while in a 4-category, they would be equivalent, and so on.) Now the components of these two transformations at an object $x\in C$ are, respectively, $$\beta_{G x} \circ H(\alpha_x) \qquad\text{and}\qquad K(\alpha_x) \circ \beta_{F x}$$ So for the interchange law to hold, we’d want an isomorphism in the following square: $$\array{H F x & \overset{H(\alpha_x)}{\to} & H G x\\ ^{\beta_{F x}}\downarrow & ?? & \downarrow^{\beta_{G x}}\\ K F x& \underset{K(\alpha_x)}{\to} & K G x}$$ This is precisely the naturality square for $\beta$ at $\alpha_x$, so the only thing we can put in there is the 2-cell component $\beta_{\alpha_x}$. But of course, this 2-cell is not invertible, in general, unless $\beta$ is pseudo natural.

However, this is now the only thing that fails. That means that $2Cat_{pseudo,lax}$, for instance, while not a 3-category, is a very 3-category-like structure which merely has “lax interchange.” There’s even a nice way to say this formally: while (weak) 3-categories can be described as categories weakly enriched over $2Cat$ with the cartesian product, or equivalently the pseudo version of the Gray tensor product (the two being equivalent for weak enrichment, though not for strict enrichment), $2Cat_{pseudo,lax}$ is a category weakly enriched over $2Cat$ with the lax version of the Gray tensor product. In fact, the lax Gray tensor product is closed on both sides (though not symmetric), and $2Cat_{pseudo,lax}$ is essentially its canonical self-enrichment arising from this closed structure. I haven’t written it down, but it seems possible that the above modifications of lax functors might give you categories weakly enriched over $2Cat$ with other versions of the Gray tensor product. So although you don’t get a 3-category, you don’t get nonsense either.

It’s certainly true that most of the double categories that arise in this way are framed bicategories, aka proarrow equipments on their vertical 2-categories. It’s also true that the “quick-and-dirty” definition of proarrow equipment as a certain kind of functor $K\to B$ may be easier to write down for $(\infty,2)$-categories. However, I still believe that developing a theory of “$\infty$-double categories” will be important in order to really understand the notion of “framed $(\infty,2)$-category”.

I still believe that developing a theory of “∞-double categories” will be important in order to really understand the notion of “framed (∞,2)-category”.

I see. But it worries me that next we’ll need $\infty$-triple categories and without a good systematic way to proceed the pattern that sounds like asking for headaches.

On the other hand, likely I am missing a good feeling for all the relevant structure involved here. Maybe you can help me with the following toy case:

given a monoidal category $C$ and with $\mathbf{B}C$ its incarnation as a 1-object bicategory, I suppose it becomes a framed bicategory in the trivial way with the identity functor

$$\mathbf{B}C \to \mathbf{B}C$$

exhibiting the (trivial) proarrow equipment.

With monoidal categories regarded as such a bifunctor this way, how do we naturally see lax 2-functors $\mathbf{B}C \to \mathbf{B}D$ arise, and hence lax monoidal functors $C \to D$? Is there a good way?

I guess what I really want is that if $C$ and $D$ are symmetric monoidal categories, the category $[C,D]$ of

• symmetric lax monoidal functors $F: C \to D$ and

• monoidal natural transformations between these

is again symmetric monoidal. And again, what I really want is a reference, not just someone saying it’s true!

(If it’s not true, okay, just convince me.)

Morally, this is a categorification of the fact that for abelian monoids $A$ and $B$, the set $Hom(A,B)$ of monoid homomorphisms is again an abelian monoid with a pointwise structure. Abstractly, the latter is because the theory of abelian monoids is commutative. A categorification of this abstract theory can be found in Hyland-Power, Pseudo-commutative monads and pseudo-closed 2-categories; but IIRC, in addition to omitting a lot of checking of axioms, they only deal with the case of pseudo morphisms. I don’t recall offhand seeing it done for lax ones anywhere, which now that I think about it is a surprising omission. One could of course trace through the proofs to see whether invertibility of the constraint morphisms is used anywhere.

In the special case of symmetric monoidal functors, however, I think there is a simpler approach. If $C$ and $D$ are symmetric monoidal, $C$ is small, and $D$ is cocomplete, then the category $Cat(C,D)$ of all functors $C\to D$ is itself symmetric monoidal under Day convolution. Moreover, the lax symmetric monoidal functors $C\to D$ are precisely the internal commutative monoids in $Cat(C,D)$, and hence are a symmetric monoidal category. (One should be able to deal with non-cocomplete $D$ by embedding it in its own presheaf category with its own Day convolution structure.) One reference for the identification of Day-convolution monoids with lax monoidal functors is Proposition 22.1 of Mandell-May-Schwede-Shipley Model categories of diagram spectra, although I don’t see them explicitly write down the additional conclusion that it is a symmetric monoidal category, nor do they mention the non-cocomplete case (or even the case of $D\neq Top_\ast$).

I know this isn’t quite what you want, but it’s the best I can think of right now.