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 transformation


(p !p *p *p !):𝒮 (p_! \dashv p^* \dashv p_*\dashv p^!) \;\colon\; \mathcal{E} \longrightarrow \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

η:idp *p ! \eta \;\colon\; id \to p^* p_!

etc. for units and

ϵ:p !p *id \epsilon \;\colon\; p_! p^* \to id

etc. for counits.


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

p *X ϵ p *X 1 p !p *p *X p *(η X) θ X p !(ϵ X) p *p *p !X η p !X 1 p !X \array{ p_*X &\underoverset{\simeq}{\epsilon_{p^* X}^{-1}}{\longrightarrow}& p_! p^* p_*X \\ {}^{\mathllap{p_*(\eta_X)}}\downarrow &\searrow^{\mathrlap{\theta_X}}& \downarrow^{\mathrlap{p_!(\epsilon_X)}} \\ p_* p^* p_! X &\stackrel{\eta_{p_!X}^{-1}}{\longrightarrow}& p_! X }


p *S η p *S p !p *p *S p *ϵ S 1 ϕ X p !(η S 1) p *p *p !S ϵ p !S p !S. \array{ p^* S &\stackrel{\eta_{p^* S}}{\longrightarrow}& p^! p_* p^* S \\ {}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow &\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\eta_S^{-1})}} \\ p^* p_* p^!S &\stackrel{{\epsilon}_{p_!S }}{\longrightarrow}& 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 11, 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 11, lemma 2.3).


By the above definition, ϕ S\phi_S is a monomorphism precisely if η p *S:p *Sp !p *p *S\eta_{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)(η p *X)()(A,p !p *p *S)𝒮(p *A,p *p *S) \mathcal{E}(A,p^* X) \stackrel{(\eta_{p^* X}) \circ (-)}{\longrightarrow} \mathcal{E}(A, p^! p_* p^* S) \stackrel{\simeq}{\longrightarrow} \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

(η p *X)():(Ap *S)p *(Ap *S). (\eta_{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 !(ϵ X):p !p *p *Xp !Xp_!(\epsilon_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 !(ϵ 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_!(\epsilon_X)}{\longrightarrow} & \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_*}{\longrightarrow}& \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.




(Kan extension of adjoint pair is adjoint quadruple)

For 𝒱\mathcal{V} a symmetric closed monoidal category with all limits and colimits, let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two small 𝒱\mathcal{V}-enriched categoriesand let

𝒞pq𝒟 \mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}

be a 𝒱\mathcal{V}-enriched adjunction. Then there are 𝒱\mathcal{V}-enriched natural isomorphisms

(q op) *Lan p op:[𝒞 op,𝒱][𝒟 op,𝒱] (q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]
(p op) *Ran q op:[𝒟 op,𝒱][𝒞 op,𝒱] (p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]

between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other.

By essential uniqueness of adjoint functors, this means that the two Kan extension adjoint triples of qq and pp

Lan q op (q op) * Ran q op Lan p op (p op) * Ran p op \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }

merge into an adjoint quadruple

Lan q op (q op) * (p op) * Ran p op:[𝒞 op,𝒱][𝒟 op,𝒱] \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]

For every enriched presheaf F:𝒞 op𝒱F \;\colon\; \mathcal{C}^{op} \to \mathcal{V} we have a sequence of 𝒱\mathcal{V}-enriched natural isomorphism as follows

(Lan p opF)(d) c𝒞𝒟(d,p(c))F(c) c𝒞𝒞(q(d),c)F(c) F(q(d)) =((q op) *F)(d). \begin{aligned} (Lan_{p^{op}} F)(d) & \simeq \int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c) \\ & \simeq \int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c) \\ & \simeq F(q(d)) \\ & = \left( (q^{op})^\ast F\right) (d) \,. \end{aligned}

Here the first step is the coend-formula for left Kan extension (here), the second step if the enriched adjunction-isomorphism for qpq \dashv p and the third step is the co-Yoneda lemma.

This shows the first statement, which, by essential uniqueness of adjoints, implies the following statements.


  • Peter Johnstone, Remarks on punctual local connectedness, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63. (tac)

Last revised on June 24, 2018 at 08:32:24. See the history of this page for a list of all contributions to it.