category theory

# Contents

## Definition

${f}_{!}⊣{f}^{*}⊣{f}_{*}⊣{f}^{!}$f_! \dashv f^* \dashv f_* \dashv f^!

## Properties

### General

$\left({f}_{!}⊣{f}^{*}⊣{f}_{*}⊣{f}^{!}\right):C\stackrel{\stackrel{{f}_{!}}{\to }}{\stackrel{\stackrel{{f}^{*}}{←}}{\stackrel{\stackrel{{f}_{*}}{\to }}{\underset{{f}^{!}}{←}}}}D$(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 $C$

$\left({f}^{*}{f}_{!}⊣{f}^{*}{f}_{*}⊣{f}^{!}{f}_{*}\right):C\to C\phantom{\rule{thinmathspace}{0ex}},$(f^* f_! \dashv f^* f_* \dashv f^! f_*) : C \to C \,,

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

on $D$.

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

$\left({f}^{*}{f}_{*}{f}^{*}{f}_{!}⊣{f}^{*}{f}_{*}{f}^{!}{f}_{*}\right):C\to C\phantom{\rule{thinmathspace}{0ex}}.$(f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C \,.

### Canonical natural transformations

Let $\left({p}_{!}⊣{p}^{*}⊣{p}_{*}⊣{p}^{!}\right):ℰ\to 𝒮$ 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 $X\in ℰ$, $S\in 𝒮$

$\begin{array}{ccc}{p}_{*}X& \stackrel{{\eta }_{{p}^{*}X}^{-1}}{\to }& {p}_{!}{p}^{*}{p}_{*}X\\ {}^{{p}_{*}\left({i}_{X}\right)}↓& {↘}^{{\theta }_{X}}& {↓}^{{p}_{!}\left({\eta }_{X}\right)}\\ {p}_{*}{p}^{*}{p}_{!}X& \stackrel{{\iota }_{{p}_{!}X}^{-1}}{\to }& {p}_{!}X\end{array}$\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

$\begin{array}{ccc}{p}^{*}S& \stackrel{{\iota }_{{p}^{*}S}}{\to }& {p}^{!}{p}_{*}{p}^{*}S\\ {}^{{p}^{*}{ϵ}_{S}^{-1}}↓& {↘}^{{\varphi }_{X}}& {↓}^{{p}^{!}\left({\iota }_{S}^{-1}\right)}\\ {p}^{*}{p}_{*}{p}^{!}S& \stackrel{{ϵ}_{{p}_{!}S}}{\to }& {p}^{!}S\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

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

$\left({\iota }_{{p}^{*}X}\right)\circ \left(-\right):\left(A\to {p}^{*}S\right)↦{p}_{*}\left(A\to {p}^{*}S\right)\phantom{\rule{thinmathspace}{0ex}}.$(\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 ${p}_{!}\left({\eta }_{X}\right):{p}_{!}{p}^{*}{p}_{*}X\to {p}_{!}X$ is so, which is the case precisely if the top morphism in

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

## Examples

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

$\mathrm{Lan}L⊣\left(-\right)\circ L⊣\left(-\right)\circ R⊣\mathrm{Ran}R\phantom{\rule{thinmathspace}{0ex}},$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.

## References

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