category theory

# Contents

## Definition

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

## Properties

### General

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

induces an adjoint triple on $C$

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

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

on $D$.

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

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

### Canonical natural transformations

Let $(p_! \dashv p^* \dashv p_*\dashv p^!) : \mathcal{E} \to \mathcal{S}$ 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

$\iota : id \to p^* p_!$

etc. for units and

$\eta : p_! p^* \to id$

etc. for counits.

###### Proposition/Definition

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

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

and

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

where the diagonal morphisms

$\theta_X : p_* X \to p_! X$

and

$\phi_S : p^* S \to p^! S$

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 $X \in \mathcal{E}$ the morphism $\theta_X : p_*X \to p_! X$ is an epimorphism;

• for all $S \in \mathcal{S}$, the morphism $\phi_S : p^*S \to p^! S$ 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, $\phi_S$ 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

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

and hence the composite is a monomorphism in Set.

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

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

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

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

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 $(p_! \dashv p^*)$-zig-zag identity.

## Examples

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

$Lan L \dashv (-)\circ L \dashv (-) \circ R \dashv Ran R \,,$

where $Lan$ and $Ran$ denote left and right Kan extension, respectively.

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

## References

Revised on June 10, 2013 11:12:30 by Urs Schreiber (82.113.99.36)