Contents

category theory

# Contents

## Idea

There are several notions of homomorphisms between pairs of adjoint functors, notably the notion of pairs of conjugate natural transformations (Def. below).

## Definition

There are several layers of generality at which one may consider a notion of homomorphism between adjoint functors.

Here is a basic but important notion:

###### Definition

Given a pair of pairs of adjoint functors between the same categories

(1)$\array{ \mathcal{C} \underoverset {\underset{R_1}{\longleftarrow}} {\overset{L_1}{\longrightarrow}} {\;\;\; \bot \;\;\;} \mathcal{D} \\ \mathcal{C} \underoverset {\underset{R_2}{\longleftarrow}} {\overset{L_2}{\longrightarrow}} {\;\;\; \bot \;\;\;} \mathcal{D} }$

then a pair of natural transformations between the adjoints of the same chirality, of this form

(2)$\lambda \,\colon\, L_1 \to L_2 \;\;\;\;\;\; \rho \,\colon\, R_2 \to R_1 \,,$

is called conjugate for [MacLane (1971), §IV.7 (5)] or a pseudo-transformation of [Harpaz & Prasma (2015), Sec. 2.2] the given adjunctions if they make the following diagram of natural transformations between hom-sets commute:

(3)$\array{ \mathcal{C}\big( L_2(-) ,\, (-) \big) &\overset{\sim}{\longrightarrow}& \mathcal{D}\big( (-) ,\, R_2(-) \big) \\ \mathllap{{}^{ \mathcal{C}\big(\lambda_{(-)},\,id_{(-)}\big) }} \Big\downarrow && \Big\downarrow \mathrlap{{}^{ \mathcal{C}\big(id_{(-)},\,\rho_{(-)}\big) }} \\ \mathcal{C}\big( L_1(-) ,\, (-) \big) &\overset{\sim}{\longrightarrow}& \mathcal{D}\big( (-) ,\, R_1(-) \big) \mathrlap{\,,} }$

where the horizontal maps are the given hom-isomorphisms (see there).

Such pairs of conjugate transformation compose via composition of their component natural transformations (cf. functor category) to yields a category

(4)$Cat_{adj}(\mathcal{C},\mathcal{D})$

whose

###### Proposition

Given a pair of pairs of adjunctions as in (1) and given (just) $\lambda$ as in (2) then there exists a unique $\rho$ as in (2) such that the conjugacy condition (3) holds.

In other words, the evident forgetful functor from $Cat_{adj}(\mathcal{C},\,\mathcal{D})$ (4) to the hom-category $\Cat(\mathcal{C},\,\mathcal{D})$ (i.e. the functor category) is a fully faithful functor exhibiting a full subcategory-inclusion:

(5)$\mathcal{C},\,\mathcal{D} \,\in\, Obj(Cat) \;\;\;\;\;\; \vdash \;\;\;\;\;\; Cat_{adj}(\mathcal{C},\,\mathcal{D}) \hookrightarrow Cat(\mathcal{C},\,\mathcal{D}) \,.$

[Mac Lane (1971), p. 98]

In fact:

###### Proposition

(conjugate pairs are mates)
The conjugacy condition (3) means equivalently that $\lambda$ and $\rho$ are mates in the sense of 2-category theory.

This is MacLane (1971) IV.7 Thm. 2 (6), p. 98 (not using the “mate”-terminology, though, which is due to Kelly & Street 2006).

Moreover:

###### Proposition

Conjugacy of transformations is compatible with horizontal composition $(-)\cdot(-)$ of natural transformations as 2-morphisms in Cat (“whiskering”), so that (5) extends to a horizontal composition-functor:

$\mathcal{C} ,\, \mathcal{D} ,\, \mathcal{E} \;\in\; Cat \;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\; \array{ Cat_{adj}(\mathcal{D},\,\mathcal{E}) \times Cat_{adj}(\mathcal{C},\,\mathcal{D}) &\overset{}{\longrightarrow}& Cat_{adj}(\mathcal{C},\,\mathcal{E}) \\ \big( (\lambda, \rho) ,\, (\lambda', \rho') \big) &\mapsto& \big( \lambda' \cdot \lambda ,\, \rho \cdot \rho' \big) } \,.$

[MacLane (1971), §IV.8 Thm. 2 & Exc. 1 (p. 102)]

Therefore, from Prop. and Prop. we have:

###### Definition
(6)$Cat_{adj} \longrightarrow Cat$

whose

[MacLane (1971), §IV.8 below (1), p. 102]

## Properties

### Relation to bifibrations

###### Proposition

Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{adj}$ (6) are equivalently the bifibrations.

###### Proof

A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. Jacobs (1998), Lem. 9.1.2). By the fact that $Cat_{adj} \to Cat$ is a locally full sub-2-category (Prop. ) this already means that the given pseudofunctor factors through $Cat_adj$, and essentially uniquely so.