nLab adjoint quadruple

1. Definition

An adjoint quadruple is a sequence of three adjunctions

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

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

2. 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 $C$

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

(hence a monad left adjoint to a comonad left adjoint to a monad) and an adjoint triple

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

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 2.1. 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 2.2. 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.  ▮

$\,$

3. Examples

Via Kan extension of adjoint pairs

A rich source of adjoint quadruples arises form adjoint pairs between small categories by left/right Kan extension to their categories of presheaves.

More interesting examples of adjoint quadruples tend to arise from these presheaf constructions when the quadruple (co)restricts to sub-categories of sheaves.

We spell out two proofs of this fact, the first using coend-calculus in the generality of enriched category theory, the second using more elementary colimit-notation.

Proposition 3.1. (Kan extension of adjoint pair is adjoint quadruple)
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 Prop.), this means that the two Kan extension adjoint triples of $q$ and $p$

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

merge into an adjoint quadruple

$$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$$

Proof. 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 is the enriched adjunction-isomorphism for $q \dashv p$ and the third step is the co-Yoneda lemma.

This shows the first statement. By essential uniqueness of adjoints (this Prop.), the other statements follow.  ▮

The following is the same argument without using coend-calculus. This argument applies verbatim also, for instance, in $\infty$-category theory using results from standard sources:

Proposition 3.2. Given a pair of adjoint functors between small categories the induced operations of pre-composition on categories of presheaves are adjoint to each other, $\ell^\ast \,\dashv\, r^\ast$, and their adjoint triples of Kan extensions overlap:

Proof.

We already know that each functor $f$ by itself induces an adjoint triple $f_! \dashv f^\ast \dashv f_\ast$, by Kan extension. Due to essential uniqueness of adjoints (this Prop.) it is hence sufficient to show that these two adjoint triples “overlap”, in that ($\ell^\ast \simeq r_!$ and equivalently) $\ell_\ast \simeq r^\ast$, hence equivalently that $\ell^\ast \dashv r^\ast$.

Now the hom-isomorphism which is characteristic of the latter adjunction $\ell^\ast \dashv r^\ast$ may be obtained as the following sequence of natural bijections:

$$\begin{aligned} & \mathrm{PSh}(\mathcal{S}_1) \big( X_1 ,\, r^\ast(X_2) \big) \\ & \;\simeq\; \mathrm{PSh}(\mathcal{S}_1) \Big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} \, y(s_1) \, ,\, r^\ast \big( \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} \, y(s_2) \big) \Big) \\ & \;\simeq\; \underset{ \underset{ s_1 \to X_1 }{\longleftarrow} }{\lim} \mathrm{PSh}(\mathcal{S}_1) \Big( y(s_1) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} \, r^\ast \big( y(s_2) \big) \Big) \\ & \;\simeq\; \underset{ \underset{ s_1 \to X_1 }{\longleftarrow} }{\lim} \, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} \mathrm{PSh}(\mathcal{S}_1) \Big( y(s_1) ,\, r^\ast \big( y(s_2) \big) \Big) \\ & \;\simeq\; \underset{ \underset{ s_1 \to X_1 }{\longleftarrow} }{\lim} \; \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} \mathcal{S}_2 \big( r(s_1) ,\, s_2 \big) \\ & \;\simeq\; \underset{ \underset{ s_1 \to X_1 }{\longleftarrow} }{\lim} \; \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} \mathrm{PSh}(\mathcal{S}_2) \Big( y\big(r(s_1)\big) ,\, y(s_2) \Big) \\ & \;\simeq\; \underset{ \underset{ s_1 \to X_1 }{\longleftarrow} }{\lim} \mathrm{PSh}(\mathcal{S}_2) \Big( y\big(r(s_1)\big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; \mathrm{PSh}(\mathcal{S}_2) \Big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} y\big(r(s_1)\big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; \mathrm{PSh}(\mathcal{S}_2) \Big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} \mathcal{S}_2 \big( (-) ,\, r(s_1) \big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; \mathrm{PSh}(\mathcal{S}_2) \Big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} \mathcal{S}_2 \big( \ell(-) ,\, s_1 \big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; PSh(\mathcal{S}_2) \Big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} \ell^\ast \big( y(s_1) \big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; PSh(\mathcal{S}_2) \Big( \ell^\ast \big( \underset{ \underset{ s_1 \to X_1 }{\longrightarrow} }{\lim} y(s_1) \big) ,\, \underset{ \underset{ s_2 \to X_2 }{\longrightarrow} }{\lim} y(s_2) \Big) \\ & \;\simeq\; PSh(\mathcal{S}_2) \Big( \ell^\ast ( X_1 ) ,\, X_2 \Big) \end{aligned}$$

Here we used repeatedly

• the co-Yoneda lemma in the form

  $$X \;\simeq\; \underset{ \underset{y(s) \to X}{\longrightarrow} }{\lim} \,y(s) \,,$$

  expressing a presheaf $X \,\in\, PSh(S)$ as a colimit of representable functors $y(s)$ (the colimit is over the comma category $y \downarrow X$, but that does not even matter in the proof above),

• the fact that any hom-functor sends colimits in its first argument to limits,

• the strong Yoneda lemma which says that $PSh(\mathcal{S})\big(y(s), X\big) \,\simeq\, X(s)$,

• the fact that colimits of presheaves are computed objectwise.

As before, the adjunction $\ell^\ast \dashv r_!$ implies the overlapping adjoint triples by essential uniqueness of adjoint functors (this Prop.).  ▮

Cohesion

Proposition 3.3. (Kan extension of finite product preserving reflection is cohesive adjoint quadruple)
Let

be a pair of adjoint functors between small categories that have finite products (or at least after passing to their free coproduct completion), such that

1. the right adjoint $i$ is fully faithful,

2. the left adjoint $p$ preserves finite products (or at least its coproduct-preserving extension to free coproduct completions does).

Then the induced adjoint quadruple of Kan extensions from Prop. 3.2 is cohesive in that

1. the two reverse functors are fully faithful.

2. the leftmost adjoint $p_!$ preserves finite products;

Proof. The preservation of finite products by the leftmost adjoint follows by Prop. 3.10 below.

The fully faithfulness of $i_!$ follows by Prop. 3.11 below. This implies that also $i_\ast$ is fully faithful, by this Prop..  ▮

Example 3.4. Consider a site $\mathcal{S}$ with finite products, in particular with a terminal object. Then the inclusion of the full subcategory on this terminal object, which is the terminal category $\ast$ is an adjunction of the form

$$\mathcal{S} \underoverset {\underset{}{\hookleftarrow}} {\overset{}{\longrightarrow}} {\;\;\;\bot\;\;\;} \ast \,.$$

If $\mathcal{S}$ is a cohesive site then the induced adjoint quadruple from Prop. 3.3 (co)restricts to the category of sheaves $Sh(\mathcal{S}) \xhookrightarrow{\;} PSh(\mathcal{S})$ and exhibits it as a cohesive topos.

The proof of Prop. 3.3 as spelled out below applies verbatim also in $\infty$-category theory. Here, it has interesting examples even without passage to $\infty$-sheaves, when the $\infty$-sites are higher than 1-sites:

Example 3.5. (cohesion of global- over G-equivariant homotopy theory)
For $G$ a group, there is an adjunction of (2,1)-categories

between the slice of the global orbit category over the object corresponding to $G$, and the $G$-orbit category. The adjoint quadruple of $\infty$-functors between $\infty$-categories of $\infty$-presheaves induced by this via Prop. 3.3 exhibits slices of global homotopy theory as being cohesive over $G$-equivariant homotopy theory

For more on this see at cohesion of global- over G-equivariant homotopy theory.

This example is a special case of the following general class:

Example 3.6. For $n \,\in\, \mathbb{N}$ and for $\mathcal{S} \xhookrightarrow{\;\;} \mathbf{H}$ any small full sub-$\infty$-category of an $\infty$-topos which is

1. has all finite homotopy products, these being computed in $\mathbf{H}$,

2. is closed under $n$-truncation

then the $n$-truncation reflection restricts

and it preserves finite products (by this Prop.).

Moreover, this adjunction (co)restricts to connected objects $X \,\in\, \mathcal{S}_{cn} \xhookrightarrow{\;} \mathcal{S}$ (i.e. those for which $\mathcal{S}(X,-)$ preserves coproducts):

If all objects of $\mathcal{S}$ are coproducts of connected ones then coroduct-preserving extension of the (co)restriced left adjoint is the original left adjoint and hence preserves finite products.

Hence the Kan extensions according to Prop. 3.3 exhibit the $\infty$-category of $\infty$-presheaves $PSh_\infty(\mathcal{S})$ as being cohesive over $PSh_\infty(\mathcal{S}_{\tau_n})$.

We now spell out the proof of the lemmas used in the proof of Prop. 3.3.

Lemma 3.7. Given a functor $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ between small categories, its left Kan extension $f_! \;\colon\; PSh(\mathcal{S}_1) \xrightarrow{\;\;} PSh(\mathcal{S}_1)$ restricts to $f$ on representables, in that for $s_1 \,\in\, \mathcal{S}_1$ we have a natural isomorphism

$$f_!\big( y(s_1) \big) \;\simeq\; y\big( f(s_1) \big) \,.$$

Proof. For $X \,\in\, PSh(\mathcal{S}_2)$ we have the following sequence of natural isomorphism:

$$\begin{aligned} PSh(\mathcal{S}_2) \left( f_! \left( y(s_1) \right) ,\, X \right) & \;\simeq\; PSh(\mathcal{S}_1) \left( y(s_1) ,\, f^\ast(X) \right) \\ & \;\simeq\; X\left( f(s_1) \right) \\ & \;\simeq\; PSh(\mathcal{S}_2) \left( y\left(f(s_1)\right) ,\, X \right) \,. \end{aligned}$$

The first line is the defining adjointness of $f_!$, the second line the Yoneda lemma over $\mathcal{S}_1$ and the definition of $f^\ast$, while the last line is the Yoneda lemma over $\mathcal{S}_2$.

Since the composite of these isomorphisms is natural, the Yoneda lemma over $PSh(\mathcal{S}_2)^{op}$ (which is large but locally small, so that the lemma does apply) implies the claim.  ▮

Lemma 3.8. Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be small categories with binary products and $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ a functor which preserves these, in that for $s, s' \,\in\, \mathcal{S}_1$ there is a natural isomorphism $f(s \times s') \,\simeq\, f(X_1) \times f(X_2)$. Then also the left Kan extension $f_!$ preserves binary products, in that for $X, X' \,\in\, PSh(\mathcal{S}_1)$ there is a natural isomorphism

$$f_!(X \times X') \;\simeq\; f_!(X) \times f_!(X') \,.$$

Proof. This is the composite of the following sequence of natural isomorphisms:

$$\begin{aligned} f_!(X \times X') & \;\simeq\; f_! \Big(\! \big( \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \, y(s) \big) \times \big( \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, y(s') \big) \!\Big) \\ & \;\simeq\; f_! \Big( \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, \left( y(s) \times y(s') \right) \!\!\Big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \big( y(s) \times y(s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \big( y(s \times s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, y\big( f(s \times s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, y\big( f(s) \times f(s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, y\big( f(s) \big) \times y\big(f(s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s) \right) \times f_! \left( y(s') \right) \\ & \;\simeq\; \Big(\, \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s) \right) \!\!\Big) \times \Big(\, \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s') \right) \!\!\Big). \end{aligned}$$

Here

• the first line is the co-Yoneda lemma, expressing the presheaves as colimits of representables.

• The second line uses that in a topos (like Set or whatever the base topos may be) colimits are pullback-stable and in particular distribute over products.

• The third line uses that $f_!$ is a left adjoint and that left adjoints preserve colimits.

• Then inside the two colimits we use

  1. that the Yoneda embedding preserves limits, in particular products,

  2. Lem. 3.7, to evaluate $f_!$ on representables as $f$,

  3. the assumption that $f$ preserves products.

• The last step is the first two steps in reverse.

  ▮

The following Lem. 3.9 is an immediate variant of Lem. 3.8 obtained by relaxing the assumptions slightly to a form that is often still readily checked:

Lemma 3.9. Let be a functor between small categories such that

1. their free coproduct completions $PSh_{\sqcup}(-)$ have binary products,

2. the unique coproduct-preserving extension $f_!$ of $f$ to these completions preserves binary products, in that for $s, s' \,\in, \mathcal{S}_1$ there is a natural isomorphism:

   $$f_!\big( y(s) \times y(s') \big) \,\simeq\, f_!\big(y(s)\big) \times f_!\big( y(s') \big)$$

Then also the left Kan extension $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$ to the full categories of presheaves preserves products.

Proof. The proof starts and ends as the proof of Prop. 3.8, but the main step in between is now more immediate, as it just needs to invoke the assumption:

$$\begin{aligned} f_!(X \times X') & \;\simeq\; f_! \Big(\! \big( \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \, y(s) \big) \times \big( \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, y(s') \big) \!\Big) \\ & \;\simeq\; f_! \Big( \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, \left( y(s) \times y(s') \right) \!\!\Big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \big( y(s) \times y(s') \big) \\ & \;\simeq\; \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \; \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s) \right) \times f_! \left( y(s') \right) \\ & \;\simeq\; \Big(\, \underset{ \underset{ s \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s) \right) \!\!\Big) \times \Big(\, \underset{ \underset{ s' \to X}{\longrightarrow} }{\lim} \, f_! \left( y(s') \right) \!\!\Big). \end{aligned}$$

  ▮

In conclusion so far:

Proposition 3.10. If a pair of small categories $\mathcal{S}_1$, $\mathcal{S}_2$ has finite products and a functor $\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2$ preserves these, then so does its left Kan extension $f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2)$.

More generally this is the case if the free coproduct completions $PSh_{\sqcup}(\mathcal{S}_i)$ have finite products and the unique coproduct-preserving extension preserves these.

Proof. We need to show that $f_!$ preserves (1) the terminal object and (2) binary products. With the given assumption on $f$, the first follows with Lem. 3.7 while the second follows with Lem. 3.8 or Lem. 3.9, respectively.  ▮

Proposition 3.11. The left Kan extension $f_!$ of a fully faithful functor $f$ between small categories is itself fully faithful:

Proof. We need to show for $X, X' \,\in\, PSh(\mathcal{S}_2)$ the morphism

$$PSh(\mathcal{S}_2) (X,\, X') \xrightarrow{ \; (i_!)_{X, X'} \; } PSh(\mathcal{S}_`) \big( i_!(X) ,\, i_!( X') \big)$$

is an equivalence. But since $i_!$ is a left adjoint and since $X$ and $X'$ are colimits of representables, this morphism is the unique one which reduces to

(1)$$\mathcal{S}_2 (s,\, s') \xrightarrow{ \; i_{s, s'} \; } \mathcal{S}_1 \big( i(s) ,\, i( s') \big)$$

on representables, where this is an isomorphism by the assumption that $i$ is fully faithful. If follows that $(i_!)_{X, X'}$, is the compostite of the following isomorphisms, and hence an isomorphism:

$$\begin{aligned} & PSh(\mathcal{S}_1) \big( i_!(X_1) ,\, i_!(X_2) \big) \\ & \;\simeq\; PSh(\mathcal{S}_1) \Big( i_! \big( \underset{ \underset{s \to X}{\longrightarrow} }{\lim} y(s) \big) ,\, i_! \big( \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} y(s') \big) \Big) \\ & \;\simeq\; PSh(\mathcal{S}_1) \Big( \underset{ \underset{s \to X}{\longrightarrow} }{\lim} i_! \big( y(s) \big) ,\, \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} i_! \big( y(s') \big) \Big) \\ & \;\simeq\; PSh(\mathcal{S}_1) \Big( \underset{ \underset{s \to X}{\longrightarrow} }{\lim} y \big( i (s) \big) ,\, \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} y \big( i(s') \big) \Big) \\ & \;\simeq\; \underset{ }{\lim} \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} PSh(\mathcal{S}_1) \Big( y \big( i (s) \big) ,\, y \big( i(s') \big) \Big) \\ & \;\simeq\; \underset{ \underset{s \to X}{\longleftarrow} }{\lim} \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} \mathcal{S}_1 \big( i(s) ,\, i(s') \big) \\ & \;\simeq\; \underset{ \underset{s \to X}{\longleftarrow} }{\lim} \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} \mathcal{S}_2 \big( s ,\, s' \big) \\ & \;\simeq\; \underset{ \underset{s \to X}{\longleftarrow} }{\lim} \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} PSh(\mathcal{S}_2) \big( y(s) ,\, y(s') \big) \\ & \;\simeq\; PSh(\mathcal{S}_2) \Big( \underset{ \underset{s \to X}{\longrightarrow} }{\lim} y(s) ,\, \underset{ \underset{s' \to X'}{\longrightarrow} }{\lim} y(s') \Big) \\ & \;\simeq\; PSh(\mathcal{S}_2) \big( X ,\, X' \big) \end{aligned}$$

Here the first and last steps are the co-Yoneda lemma and the preservation of its colimits as in the proofs before. In the middle steps we are using Lem. 3.7 to evaluate $i_!$ on representables and then (1) inside the (co)limits.  ▮

4. Related concepts

• adjunction, adjoint functor

• adjoint triple, adjoint string

5. References

The above prop. 2.1 is from:

• Peter Johnstone, Remarks on punctual local connectedness, Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63. (tac)