nLab
adjoint quadruple

Definition

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.

Properties

General

Every adjoint quadruple

$$(f_{!}⊣f^{*}⊣f_{*}⊣f^{!}):C\stackrel{\stackrel{f_{!}}{\to }}{\stackrel{\stackrel{f^{*}}{\leftarrow }}{\stackrel{\stackrel{f_{*}}{\to }}{\underset{f^{!}}{\leftarrow }}}}D$$(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 $C$

$$(f^{*}{f}_{!}⊣f^{*}{f}_{*}⊣f^{!}{f}_{*}):C\to C,$$(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

$$(f_{!}{f}^{*}⊣f_{*}{f}^{*}⊣f_{*}{f}^{!}):D\to D$$(f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D

on $D$.

Since moreover every adjoint triple $(F⊣G⊣H)$ induces an adjoint pair $(GF⊣GH)$ and an adjoint pair $(FG⊣HG)$, the adjoint quadruple above induces four adjoint pairs, such as

$$(f^{*}{f}_{*}{f}^{*}{f}_{!}⊣f^{*}{f}_{*}{f}^{!}{f}_{*}):C\to C.$$(f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C \,.

Canonical natural transformations

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

We write

$$\iota :\mathrm{id}\to p^{*}{p}_{!}$$\iota : id \to p^* p_!

etc. for units and

$$\eta :p_{!}{p}^{*}\to \mathrm{id}$$\eta : p_! p^* \to id

etc. for counits.

Proposition/Definition

We have commuting diagrams, natural in $X\in ℰ$, $S\in 𝒮$

$$\begin{array}{ccc}{p}_{*}X& \stackrel{{\eta }_{p^{*}X}^{-1}}{\to }& p_{!}{p}^{*}{p}_{*}X\\ {}^{p_{*}(i_X)}\downarrow & {\searrow }^{{\theta }_X}& {\downarrow }^{p_{!}({\eta }_X)}\\ p_{*}{p}^{*}{p}_{!}X& \stackrel{{\iota }_{p_{!}X}^{-1}}{\to }& p_{!}X\end{array}$$\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 }

and

$$\begin{array}{ccc}{p}^{*}S& \stackrel{{\iota }_{p^{*}S}}{\to }& p^{!}{p}_{*}{p}^{*}S\\ {}^{p^{*}{ϵ}_S^{-1}}\downarrow & {\searrow }^{{\varphi }_X}& {\downarrow }^{p^{!}({\iota }_S^{-1})}\\ p^{*}{p}_{*}{p}^{!}S& \stackrel{{ϵ}_{p_{!}S}}{\to }& p^{!}S\end{array}.$$\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

$${\theta }_X:p_{*}X\to p_{!}X$$\theta_X : p_* X \to p_! X

and

$${\varphi }_S:p^{*}S\to p^{!}S$$\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).

Proposition

The following conditions are equivalent:

• for all $X\in ℰ$ the morphism ${\theta }_X:p_{*}X\to p_{!}X$ is an epimorphism;

• for all $S\in 𝒮$, the morphism ${\varphi }_S:p^{*}S\to p^{!}S$ is a monomorphism;

• $p_{*}$ is faithful on morphisms of the form $A\to p^{*}S$.

This appears as (Johnstone, lemma 2.3).

Proof

By the above definition, ${\varphi }_S$ is a monomorphism precisely if ${\iota }_{p^{*}S}:p^{*}S\to p^{!}{p}_{*}{p}^{*}S$ is. This in turn is so (see monomorphism) precisely if the first function in

$$ℰ(A,p^{*}X)\stackrel{({\iota }_{p^{*}X})\circ (-)}{\to }ℰ(A,p^{!}{p}_{*}{p}^{*}S)\stackrel{\simeq }{\to }𝒮(p_{*}A,p_{*}{p}^{*}S)$$\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 $(p_{*}⊣p^{!})$-zig-zag identity, this is equal to the action of $p_{*}$ on morphisms

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

Similarly, by the above definition the morphism ${\theta }_X$ is an epimorphism precisely if $p_{!}({\eta }_X):p_{!}{p}^{*}{p}_{*}X\to p_{!}X$ is so, which is the case precisely if the top morphism in

$$\begin{array}{ccc}𝒮(p_{!}X,S)& \stackrel{(-)\circ p_{!}({\eta }_X)}{\to }& 𝒮(p_{!}{p}^{*}{p}_{*}X,S)\\ {}^{\simeq }\downarrow & & {\downarrow }^{\simeq }\\ & & ℰ(p^{*}{p}_{*}X,p^{*}S)\\ {}^{\simeq }\downarrow & & {\downarrow }^{\simeq }\\ ℰ(X,p^{*}S)& \stackrel{p_{*}}{\to }& 𝒮(p_{*}X,p_{*}{p}^{*}S)\end{array}$$\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 $(p_{!}⊣p^{*})$-zig-zag identity.

Examples

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

  $$\mathrm{Lan}L⊣(-)\circ L⊣(-)\circ R⊣\mathrm{Ran}R,$$Lan L \dashv (-)\circ L \dashv (-) \circ R \dashv Ran R \,,

  where $\mathrm{Lan}$ and $\mathrm{Ran}$ denote left and right Kan extension, respectively.

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

Related concepts

• adjunction, adjoint functor

• adjoint triple

References

• Peter Johnstone, Remarks on punctual local connectedness (tac)