nLab transformation of adjoints

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

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

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

• objects are the left adjoint functors $\mathcal{C} \to \mathcal{D}$

• morphisms are the conjugate transformations between these

• composition is the composition of component natural transformations.

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

The (very large) wide and locally full sub-2-category $Cat_{adj}$ of Cat

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

whose

• objects are categories,

• 1-morphisms are left adjoint functors

• 2-morphisms are natural transformations which are conjugate in the sense of Def. .

[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.

See also Harpaz & Prasma (2015), Prop. 2.2.1.

Related concepts

• monad transformation

• bifibration

• model structures on Grothendieck fibrations

• double category of model categories

References

• Saunders MacLane, Sections IV.7-8 of: Categories for the working mathematician, Graduate Texts in Mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

See also:

• Yonatan Harpaz, Matan Prasma, Section 2.2. of: The Grothendieck construction for model categories, Advances in Mathematics 281 (2015) 1306-1363 [arXiv:1404.1852, 10.1016/j.aim.2015.03.031]

  (in the context of model structures on Grothendieck constructions)