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
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 
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.
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?