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Stream: theory: category theory

Topic: Monoidal functors and adjunctions

Joshua Meyers (Nov 14 2024 at 06:11):

I was looking at the string diagrams for monoidal functors and it occured to me that the left and right 'faces' of the coloring used to wrap the objects of the domain category could be considered strings in their own right, with the laws of the monoidal functor resembling the laws of an adjunction. This led to the following observation:

Let $B$ be a bicategory. Then an adjunction $0,1,L:1\to 0, R:0\to 1, R\vdash L$ in $B$ induces a lax monoidal functor $F:\text{End}(0)\to \text{End}(1)$ defined by

• $Fc \coloneqq R\circ c \circ L$
• $F(f:c\to c')\coloneqq R\circ f \circ L$
• $F_0 = \eta$, the unit of the adjunction
• $(F_2)_{c,c'}=R\circ c' \circ \epsilon \circ c \circ L$, where $\epsilon$ is the counit of the adjunction

This satisfies the axioms of a lax monoidal functor. Moreover, I conjecture that every lax monoidal functor is of this form for some adjunction in a bicategory.

By duality, the same adjunction induces a colax monoidal functor $G:\text{End}(1)\to \text{End}(0)$. So every lax monoidal functor induces a colax monoidal functor going the other way?

Does anyone know if these results are in the literature or folklore or are immediate consequences of known stuff? Or am I doing something wrong?

Mike Shulman (Nov 14 2024 at 06:15):

That's interesting. It doesn't immediately ring a bell. But I'm sure it's not true that every lax monoidal functor induces a colax one going the other way, so that suggests that your conjecture is false...

Mike Shulman (Nov 14 2024 at 06:15):

Perhaps it might hold if your lax monoidal functor has a left adjoint (which is then automatically colax monoidal)?

Joshua Meyers (Nov 14 2024 at 06:49):

Well one special property of lax monoidal functors of this form is that there exists a special object $\kappa\in C$ such that for all $c,c'\in C$ we have $F(c)\otimes F(c') = F(c\otimes \kappa\otimes c')$ --- I don't think that is a general feature of lax monoidal functors.

Mike Shulman (Nov 14 2024 at 07:36):

Mm, indeed. Not even the right adjoint ones.

Chad Nester (Nov 14 2024 at 08:21):

I think this paper --- "Collages of String Diagrams" by Dylan Braithwaite and Mario Román --- is relevant here, although confess that I haven't really engaged with it myself.

Nayan Rajesh (Nov 14 2024 at 11:02):

I'm not sure if there's a result that this follows from, but the known fact that monads can be transported across an adjunction follows from your observation. Any monad on 0 is a monoid in End(0), and since F is lax monoidal, gets sent to a monoid F0 in End(1) which is a monad on 1. Also, comonads get transported the other way since G is colax monoidal. This latter fact appears as Theorem 4.2 here.

Joshua Meyers (Nov 14 2024 at 11:27):

Chad Nester said:

I think this paper --- "Collages of String Diagrams" by Dylan Braithwaite and Mario Román --- is relevant here, although confess that I haven't really engaged with it myself.

Yes, thanks, this is relevant.

Joshua Meyers (Nov 17 2024 at 18:39):

New conjecture: Let $C$ be a monoidal category. Consider the category $\mathcal{V}$ whose objects are bicategories $B$ equipped with a strict 2-functor $I:BC\to B$ (where $BC$ is the delooping of $C$) and an adjunction $\alpha:I(\ast)\leftrightarrows b: \beta$, and whose morphisms are strict 2-functors preserving $I,\alpha,\beta$, as well as the unit and counit of the adjunction.

Every object of $\mathcal{V}$ induces a (co)lax monoidal functor $J: I(\ast)\to b$ defined by $c\mapsto \alpha\circ c\circ\beta$, with the direction of the laxity determined by the direction of the adjunction.

Let $(B,I,b, \alpha,\beta)$ be the initial object of $\mathcal{V}$, and let $J$ be the (co)lax monoidal functor induced by this object. Then $J\circ I: C\to\text{End}(b)$ is the free (co)lax monoidal functor out of $C$, in the following sense: Given any (co)lax monoidal functor $F:C\to D$, there is a strict monoidal functor $H:\text{End}(b)\to D$ such that $F = H\circ (J\circ I)$, and $H$ is unique up to unique isomorphism.