nLab lax functor

Idea

A lax functor or lax $n$-functor is a morphism of $n$-categories that is allowed to have structural cells – compositors, associators, etc – that need not be invertible (not even weakly).

This is to distinguish from pseudofunctor for which all these cells are required to be equivalences.

This means that the definition of lax functor involves a choice of orientation of these structural cells which is not visible for pseudofunctors. The choice is such that cells map composites of images to images of composite. With the opposite choice one speaks of an oplax (or sometimes colax) functor.

The term lax functor can be used for $n$-functors $F : C \to D$ whose domain $C$ is an ordinary category (regarded as an $n$-category with only trivial higher morphisms), while the codomain $D$ is often taken to be a 2-category.

A normal lax functor (sometimes called strictly unitary) preserves identities strictly.

There exist a similar concept for double and multiple categories.

Definition

See the definition at pseudofunctor, and let the natural isomorphisms in that definition be merely natural transformations.

Properties

One of the most important properties is that lax functors compose associatively and unitally: we have a category $\bf{BiCat}$ with bicategories as objects and lax functors as morphisms. If we add icons as 2-cells, this becomes a 2-category.

Examples

• Any lax monoidal functor gives an example. In fact monoidal categories can be presented as bicategories with one object (see delooping), and thus a lax functor $\mathbg{B}M \to \mathbf{B} N$ corresponds to a lax monoidal functor $M \to N$.

• For $D$ a bicategory, lax functors $F : {*} \to D$ from the point category to $D$ are equivalent to monads in $D$.

  The compositor of the lax functor is the monad product, the unitor is the monad unit.

  • So in particular for $V$ a monoidal category and $\mathbf{B}V$ its one-object delooping bicategory, lax functors ${*} \to \mathbf{B}V$ are equivalent to monoids in $V$.

• Similarly, oplax functors ${*} \to D$ are equivalent to comonads in $D$.

• If $C$ is the codiscrete category on a set $S$, and $D$ is a bicategory, lax functors $F : C \to D$ are the same as categories enriched in $D$ having $S$ as their set of objects.

  • In particular, if $C = {*}$, then this example reduces to the first one.

  • Another special case arises when $D = \mathbf{B}V$ for some monoidal category $V$. Then lax functors $F : C \to D$ are the same as categories enriched in the monoidal category $V$.

• It makes sense to ask that a functor is lax and oplax in a compatible way such that ${*} \to D$ yields Frobenius monads.

  This is of relevance in conformal field theory where Frobenius algebra objects in modular tensor categories and bimodules over them play a central role.

  Some old remarks on this case are in Note on lax functors and bimodules.

  This relation between lax-oplax functors and conformal field theory was developed in detail in

  • Liang Kong, Ingo Runkel, Cardy algebras and sewing constraints, I (arXiv)

  A general discussion of lax-oplax functors is in section 2.1 there.

Isn’t it odd not to require any extra condition at all on the coherence morphisms? I would have expected a definition where they are required to be split epi, or require that the codomain be a subobject of the domain. Is there a name for something like that?

Mike Shulman: One could certainly add that as a condition, but I don’t think I’ve ever heard of anyone having a use for it, or giving it a name. The interesting examples listed above (and others) don’t use any such condition.

See also

• lax double functor

References

• Niles Johnson, Donald Yau, Section 4.1 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

• R. Street, Two constructions on lax functors, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 13 (1972) no. 3 , p. 217-264 numdam