Are Lax Functors Good for Anything?

So I’ve recently been thinking a lot about lax functors between n-categories, trying to get a better feel for what they are and why we should care. I have a few ideas about how certain lax functors could eventually be useful for TQFTs, but ever since I asked this question on MathOverflow I have started to doubt that lax functors in themselves are really good for anything.

Let’s set some terminology first. A functor between n-categories is supposed to have several pieces. First of all there are some maps which take objects of the source to objects of the target, 1-morphisms of the source to 1-morphisms of the target, and so on. Then there are some coherence morpshism, for example

F(f) \circ F(g) \to F(f \circ g)

for every pair of composible 1-morphisms f and g. These are required to satisfy some additional equations (e.g. the pentagon and triangle identities, and they should be natural in f and g).

In a pseudo- or strong functor these additional coherence morphisms are isomorphism/equivalences. In a lax functor, they don’t have to be (and so there are really two version, lax and oplax, depending on which way the arrow goes). I think lax functors arose by thinking about monoidal categories as one-object bicategories. The notion of a lax monoidal functor is very useful and there are lots of examples. A famous one has recently been discussed on the n-category café.

This leads to a “natural” generalization to lax functors between arbitrary bicategories. The main problem, which I learned from MathOverflow (much thanks goes to Mike Shulman!) is that the bicategory Fun(A,B) of lax functors between A and B is not functorial! If I have an equivalence of bicategories A~A’, that does not mean that Fun(A, B) is equivalent to Fun(A’, B). In fact these are almost never equivalent.

Something like this makes me sad and want to throw lax functors out the window (except maybe for lax monoidal functors). I wonder if I am wrong to feel this way? So I’m asking you, dear readers, are lax functors good for anything? What are the most useful/important applications of lax functors beyond the monoidal case? I thought a blog post/discussion would be a more suitable format than a MathOverflow question.

On a related note, I’ve recently been chatting with Nick Rozenblyum and Reid Barton about a variation on the notion of lax functor which has the following property: if A and A’ are two equivalent 2-groupoids, then the “laxish” functors Fun(A, B) and Fun(A’,B) are equivalent bicategories. I know “laxish” is a horrible name, but let’s use it for the time being to distinguish these from lax. Roughly the rule is that these are lax functors, but the coherence morphisms are required to be equivalences whenever the 1-morphisms f or g are equivalences.

For example, consider the monoidal category of spaces (Top, \times ). Then there are two obvious lax monoidal functors to the category of chain complexes. The first is the following composite process: take the singular simplicial set, form the free simplicial abelian group, then obtain a chain complex by the naive alternating sum formula. The second is exactly the same, only we take the normalized chain complex, i.e. we quotient out by the degenerate simplices.

If we view both of these monoidal categories as single object bicategories, then the only invertible 1-morphisms in “Top” are the singleton topological spaces.

X \times Y \cong Y \Leftrightarrow X \cong pt

So the first of these is not “laxish” because it does not send pt to the unit chain complex, but the second functor is. I’m not sure if these “laxish” functors are good idea or not. Has anyone seen something like this already in nature?

9 thoughts on “Are Lax Functors Good for Anything?”

  1. Lax functors of bicategories do appear in plenty of places. One important example is that if F:V→W is a lax functor between monoidal categories, then it induces a lax functor Prof(V) → Prof(W) between their corresponding bicategories of enriched categories and enriched profunctors.

    However, I believe that in almost all cases (including this one), what’s going on is that the bicategory in question is actually the horizontal bicategory of some double category, and the lax functor actually underlies a lax functor of double categories. For example, Prof(V) can be improved to a double category in which the vertical arrows are V-functors. Lax functors of double categories are just as well-behaved as lax functors of monoidal categories (they are both lax morphisms for some 2-monad). In particular, they form a very nice 2-category (or 3-category) and are invariant under equivalence of double categories.

    The double categories that usually arise in these situations are special: every vertical arrow induces two horizontal ones in a universal way. For instance, every functor f:A→B induces two “representable” profunctors B(1,f) and B(f,1). This makes them into “proarrow equipments” (http://ncatlab.org/nlab/show/equipment) or “framed bicategories” (arXiv:0706.1286). Since lax functors of double categories are required to be strong on the vertical arrows, this implies that lax functors between framed bicategories are automatically also strong on the “representable” horizontal arrows. This seems related to your “laxish” idea of requiring the lax constraints to be invertible on some of the 1-morphisms, but it is more general: you can specify exactly which 1-morphisms you want to single out.

  2. Lax monoidal functors are nice, and lax functors between bicategories are also useful at times.

    But I think Jonas Frey at the PPS group of Paris 7 told us last summer that bicategories, lax functors, transformations and modifications don’t form a tricategory.

    Does that sound right?

    This would make me very unhappy with them, unless they could be rehabilated in some manner as Mike suggests.

    By the way, Chris: the TeX in your post isn’t getting rendered correctly!

  3. My unease all started when I tried to work out a couple of easy examples of bicategories of Lax functors. I started seeing some weird behavior and it prompted my to ask the following question on MathOverflow:

    Question: If A and A’ are equivalent bicategories are the bicategories of lax functors Fun(A, B) and Fun(A’, B) equivalent?

    Mike Shulman then gave a comprehensive answer which basically was a big fat No.

    Here is part of Mikes answer:

    First of all, for any two bicategories A and B, there is a bicategory Fun_{xy}(A,B) where x can denote either strong, lax, or oplax functors, and y can denote either strong, lax, or oplax transformations. There’s no problem defining and composing lax and oplax transformations between lax or oplax functors, and the lax/oplax-ness doesn’t even have to match up. It’s also true that two x-functors are equivalent in one of these bicategories iff they’re equivalent in any other one. That is, any lax or oplax transformation that is an equivalence is actually strong/pseudo.

    Where you run into problems is when you try to compose the functors. You can compose two x-functors and get another x-functor, but in general you can’t whisker a y-transformation with an x-functor unless x = strong, no matter what y is, and moreover if y isn’t strong, then the interchange law fails. Thus you only get a tricategory with homs Fun_{xy}(A,B) if x=y=strong. (In particular, I think this means that there isn’t a good notion of “equivalence of bicategories” involving lax functors.)

    For a fixed strong functor F:A \to A', you can compose and whisker with it to get a functor

    Fun_{x,y}(A',B) \to Fun_{x,y}(A,B)

    for any x and y. However, the same is not true for transformations F \to F', and the answer to your question is (perhaps surprisingly) no! The two bicategories are not equivalent.

    So in particular there is no tricategory of bicategories and lax functors. If I’m reading this right, there isn’t even a 2-category. That’s when I started wondering if lax functors were useful for anything.

    btw: The latex seems to be rendering for me. Are other people having trouble?

  4. The latex is rendering fine for me.

    Regarding 2-categories of bicategories, there isn’t even a 2-category of bicategories and *strong* functors if you want the transformations to be any of the usual sorts — the interchange law will only be satisfied up to a modification. However, there is a 2-category of bicategories, lax functors, and a very restricted sort of transformation called an *icon*, see

  6. Who cares about a bicategory of bicategories? Sure, it’s fun to flirt with danger, so we all like to cut corners and seeing if the math gods punish us with a thunderbolt… but what counts is the tricategory of bicategories.

    So, Mike: do you know if there’s a tricategory of bicategories with lax functors as morphisms? I seem to remember that something doesn’t work…

  7. I thought that what Mike said more or less rules out the possibility of a tricategory of bicategories with lax functors as 1-morphisms. You can’t horizontally compose the 2-morphisms. You can’t even wisker the 2-morphisms by lax functors unless they are these very restrictive “icons”. Am I missing something?

  9. I didn’t mean to steal the whole discussion, rather just to describe why you don’t get a tricategory of lax functors, since that was taking up the discussion here but seemed a bit tangential to your actual question.

    I don’t know much about representation theory, but here’s somewhere that lax functors might be implicitly present. For any commutative ring R, we have a bicategory R mod of R-algebras and bimodules, which is a natural home for Morita theory (e.g. http://arxiv.org/abs/0805.3673). And for any ring homomorphism f:R\to S, we have extension and restriction of scalars functors f_!\colon R mod \to S mod and f^* S mod \to R mod, of which the former is strong but the latter is only lax.

    Of course, the bicategory R mod underlies a double category whose vertical arrows are R-algebra maps. Another argument for this POV is one would naturally like an adjunction f_! \dashv f^*, and this is indeed true with double categories in the “obvious” sense, but it’s hard to even make sense of what it would mean for just bicategories, since there is no 2-category or tricategory for an “adjunction” to live in.

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