Contents

category theory

# Contents

## Definition

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

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

## 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 transformation

Let

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

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

$\eta \;\colon\; id \to p^* p_!$

etc. for units and

$\epsilon \;\colon\; p_! p^* \to id$

etc. for counits.

###### Proposition/Definition

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

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

and

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

where the diagonal morphisms

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

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 11, 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 11, lemma 2.3).

###### Proof

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

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

and hence the composite is a monomorphism in Set.

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

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

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

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

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

$\,$

## Examples

###### Example

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

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

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

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

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 $q$ and $p$

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }$

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$
For every enriched presheaf $F \;\colon\; \mathcal{C}^{op} \to \mathcal{V}$ we have a sequence of $\mathcal{V}$-enriched natural isomorphism as follows
\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 $q \dashv p$ and the third step is the co-Yoneda lemma.