nLab transformation of adjoints




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


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:


(conjugate transformations of adjoints)
Given a pair of pairs of adjoint functors between the same categories

(1)๐’žโŠฅโŸตR 1โŸถL 1๐’Ÿ ๐’žโŠฅโŸตR 2โŸถL 2๐’Ÿ \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)ฮป:L 1โ†’L 2ฯ:R 2โ†’R 1, \lambda \,\colon\, L_1 \to L_2 \;\;\;\;\;\; \rho \,\colon\, R_2 \to R_1 \,,

is called conjugate for [MacLane (1971), ยงIV.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)๐’ž(L 2(โˆ’),(โˆ’)) โŸถโˆผ ๐’Ÿ((โˆ’),R 2(โˆ’)) ๐’ž(ฮป (โˆ’),id (โˆ’))โ†“ โ†“ ๐’Ÿ(id (โˆ’),ฯ (โˆ’)) ๐’ž(L 1(โˆ’),(โˆ’)) โŸถโˆผ ๐’Ÿ((โˆ’),R 1(โˆ’)), \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(๐’ž,๐’Ÿ) Cat_{adj}(\mathcal{C},\mathcal{D})



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

In other words, the evident forgetful functor from Cat adj(๐’ž,๐’Ÿ)Cat_{adj}(\mathcal{C},\,\mathcal{D}) (4) to the hom-category Cat(๐’ž,๐’Ÿ)\Cat(\mathcal{C},\,\mathcal{D}) (i.e. the functor category) is a fully faithful functor exhibiting a full subcategory-inclusion:

(5)๐’ž,๐’ŸโˆˆObj(Cat)โŠขCat adj(๐’ž,๐’Ÿ)โ†ชCat(๐’ž,๐’Ÿ). \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:


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

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



Conjugacy of transformations is compatible with horizontal composition (โˆ’)โ‹…(โˆ’)(-)\cdot(-) of natural transformations as 2-morphisms in Cat (โ€œwhiskeringโ€), so that (5) extends to a horizontal composition-functor:

๐’ž,๐’Ÿ,โ„ฐโˆˆCatโŠขCat adj(๐’Ÿ,โ„ฐ)ร—Cat adj(๐’ž,๐’Ÿ) โŸถ Cat adj(๐’ž,โ„ฐ) ((ฮป,ฯ),(ฮปโ€ฒ,ฯโ€ฒ)) โ†ฆ (ฮปโ€ฒโ‹…ฮป,ฯโ‹…ฯโ€ฒ). \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), ยงIV.8 Thm. 2 & Exc. 1 (p. 102)]

Therefore, from Prop. and Prop. we have:


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

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


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


Relation to bifibrations


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


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 adjโ†’CatCat_{adj} \to Cat is a locally full sub-2-category (Prop. ) this already means that the given pseudofunctor factors through Cat adjCat_adj, and essentially uniquely so.

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


See also:

Last revised on September 27, 2023 at 20:09:19. See the history of this page for a list of all contributions to it.