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Definition
An adjoint quadruple is a sequence of three adjunctions
f_! \dashv f^* \dashv f_* \dashv f^!
between four morphisms.
Properties
General
Every adjoint quadruple
(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
(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^* \dashv f_* f^* \dashv f_* f^!) : D \to D
on .
Since moreover every adjoint triple induces an adjoint pair and an adjoint pair , the adjoint quadruple above induces four adjoint pairs, such as
(f^* f_* f^* f_! \dashv f^* f_* f^! f_*) : C \to C
\,.
Let be an adjoint quadruple of adjoint functors such that and 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 ,
\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 the morphism is an epimorphism;
-
for all , the morphism is a monomorphism;
-
is faithful on morphisms of the form .
This appears as (Johnstone, lemma 2.3).
Proof
By the above definition, is a monomorphism precisely if 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 -zig-zag identity, this is equal to the action of on morphisms
(\iota_{p^* X}) \circ (-) :
(A \to p^* S) \mapsto p_*(A \to p^* S)
\,.
Similarly, by the above definition the morphism is an epimorphism precisely if 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 -zig-zag identity.
Examples
References