adjoint quadruple



An adjoint quadruple is a sequence of three adjunctions

f !f *f *f ! f_! \dashv f^* \dashv f_* \dashv f^!

between a quadruple of morphisms. That is, it is an adjoint string of length 4.



Every adjoint quadruple

(f !f *f *f !):Cf !f *f *f !D (f_! \dashv f^* \dashv f_* \dashv f^!) : C \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}} D

induces an adjoint triple on CC

(f *f !f *f *f !f *):CC, (f^* f_! \dashv f^* f_* \dashv f^! f_*) : C \to C \,,

(hence a monad left adjoint to a comonad left adjoint to a monad) and an adjoint triple

(f !f *f *f *f *f !):DD (f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D

on DD.

Since moreover every adjoint triple (FGH)(F \dashv G \dashv H) induces an adjoint pair (GFGH)(G F \dashv G H) and an adjoint pair (FGHG)(F G \dashv H G), the adjoint quadruple above induces four adjoint pairs, such as

(f *f *f *f !f *f *f !f *):CC. (f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C \,.

Canonical natural transformations

Let (p !p *p *p !):𝒮(p_! \dashv p^* \dashv p_*\dashv p^!) : \mathcal{E} \to \mathcal{S} be an adjoint quadruple of adjoint functors such that p *p^* and p !p^! are full and faithful functors. We record some general properties of such a setup.

We write

ι:idp *p ! \iota : id \to p^* p_!

etc. for units and

η:p !p *id \eta : p_! p^* \to id

etc. for counits.


We have commuting diagrams, natural in XX \in \mathcal{E}, S𝒮S \in \mathcal{S}

p *X η p *X 1 p !p *p *X p *(i X) θ X p !(η X) p *p *p !X ι p !X 1 p !X \array{ p_*X &\stackrel{\eta_{p^* X}^{-1}}{\to}& p_! p^* p_*X \\ {}^{\mathllap{p_*(i_X)}}\downarrow &\searrow^{\mathrlap{\theta_X}}& \downarrow^{\mathrlap{p_!(\eta_X)}} \\ p_* p^* p_! X &\stackrel{\iota_{p_!X}^{-1}}{\to}& p_! X }


p *S ι p *S p !p *p *S p *ϵ S 1 ϕ X p !(ι S 1) p *p *p !S ϵ p !S p !S. \array{ p^* S &\stackrel{\iota_{p^* S}}{\to}& p^! p_* p^* S \\ {}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow &\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\iota_S^{-1})}} \\ p^* p_* p^!S &\stackrel{{\epsilon}_{p_!S }}{\to}& p^!S } \,.

where the diagonal morphisms

θ X:p *Xp !X \theta_X : p_* X \to p_! X


ϕ S:p *Sp !S \phi_S : p^* S \to p^! S

are defined to be the equal composites of the sides of these diagrams.

This appears as (Johnstone, lemma 2.1, corollary 2.2).


The following conditions are equivalent:

  • for all XX \in \mathcal{E} the morphism θ X:p *Xp !X\theta_X : p_*X \to p_! X is an epimorphism;

  • for all S𝒮S \in \mathcal{S}, the morphism ϕ S:p *Sp !S\phi_S : p^*S \to p^! S is a monomorphism;

  • p *p_* is faithful on morphisms of the form Ap *SA \to p^* S.

This appears as (Johnstone, lemma 2.3).


By the above definition, ϕ S\phi_S is a monomorphism precisely if ι p *S:p *Sp !p *p *S\iota_{p^* S} : p^* S \to p^! p_* p^* S is. This in turn is so (see monomorphism) precisely if the first function in

(A,p *X)(ι p *X)()(A,p !p *p *S)𝒮(p *A,p *p *S) \mathcal{E}(A,p^* X) \stackrel{(\iota_{p^* X}) \circ (-)}{\to} \mathcal{E}(A, p^! p_* p^* S) \stackrel{\simeq}{\to} \mathcal{S}(p_* A, p_* p^* S)

and hence the composite is a monomorphism in Set.

By definition of adjunct and using the (p *p !)(p_* \dashv p^!)-zig-zag identity, this is equal to the action of p *p_* on morphisms

(ι p *X)():(Ap *S)p *(Ap *S). (\iota_{p^* X}) \circ (-) : (A \to p^* S) \mapsto p_*(A \to p^* S) \,.

Similarly, by the above definition the morphism θ X\theta_X is an epimorphism precisely if p !(η X):p !p *p *Xp !Xp_!(\eta_X) : p_! p^* p_* X \to p_! X is so, which is the case precisely if the top morphism in

𝒮(p !X,S) ()p !(η X) 𝒮(p !p *p *X,S) (p *p *X,p *S) (X,p *S) p * 𝒮(p *X,p *p *S) \array{ \mathcal{S}(p_! X, S) &\stackrel{(-) \circ p_!(\eta_X)}{\to} & \mathcal{S}(p_! p^* p_* X, S) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ && \mathcal{E}(p^* p_* X, p^* S) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \mathcal{E}(X, p^* S) &\stackrel{p_*}{\to}& \mathcal{S}(p_* X, p_* p^* S) }

and hence the bottom morphism is a monomorphism in Set, where again the commutativity of this diagram follows from the definition of adjunct and the (p !p *)(p_! \dashv p^*)-zig-zag identity.


  • For (LR):CD(L \dashv R): C \to D any pair of adjoint functors, there is induced on the presheaf categories a quadruple of adjoint functors

    LanL()L()RRanR, Lan L \dashv (-)\circ L \dashv (-) \circ R \dashv Ran R \,,

    where LanLan and RanRan denote left and right Kan extension, respectively.

  • For cohesive topos by definition the terminal geometric morphism extends to an adjoint quadruple.


Revised on October 6, 2017 00:36:35 by Mike Shulman (