nLab lax functor

Contents

Context

2-Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

A lax functor or lax nn-functor is a morphism of nn-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 nn-functors F:CDF : C \to D whose domain CC is an ordinary category (regarded as an nn-category with only trivial higher morphisms), while the codomain DD 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 bfBiCat\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 mathbgBMBN\mathbg{B}M \to \mathbf{B} N corresponds to a lax monoidal functor MNM \to N.

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

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

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

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

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

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

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

References

Last revised on October 26, 2022 at 22:04:57. See the history of this page for a list of all contributions to it.