nLab
adjoint quadruple

Contents

Contents

Definition

An adjoint quadruple is a sequence of three adjunctions

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

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

Properties

General

Every adjoint quadruple

(f !f *f *f !):Cf !f *f *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 CC

(f *f !f *f *f !f *):CC, (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 *f *f *f *f !):DD (f_! f^* \dashv f_* f^* \dashv f_* f^!) : D \to D

on DD.

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

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

\,

Canonical natural transformation

Let

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

be an adjoint quadruple of adjoint functors such that p *p^* and p !p^! are full and faithful functors. We record some general properties of such a setup.

We write

η:idp *p ! \eta \;\colon\; id \to p^* p_!

etc. for units and

ϵ:p !p *id \epsilon \;\colon\; p_! p^* \to id

etc. for counits.

Proposition/Definition

We have commuting diagrams, natural in XX \in \mathcal{E}, S𝒮S \in \mathcal{S}

p *X ϵ p *X 1 p !p *p *X p *(η X) θ X p !(ϵ X) p *p *p !X η p !X 1 p !X \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

p *S η p *S p !p *p *S p *ϵ S 1 ϕ X p !(η S 1) p *p *p !S ϵ p !S p !S. \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

θ X:p *Xp !X \theta_X : p_* X \to p_! X

and

ϕ S:p *Sp !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 11, lemma 2.1, corollary 2.2).

Proposition

The following conditions are equivalent:

  • for all XX \in \mathcal{E} the morphism θ X:p *Xp !X\theta_X : p_*X \to p_! X is an epimorphism;

  • for all S𝒮S \in \mathcal{S},, the morphism ϕ S:p *Sp !S\phi_S : p^*S \to p^! S is a monomorphism;

  • p *p_* is faithful on morphisms of the form Ap *SA \to p^* S.

This appears as (Johnstone 11, lemma 2.3).

Proof

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

(A,p *X)(η p *X)()(A,p !p *p *S)𝒮(p *A,p *p *S) \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 *p !)(p_* \dashv p^!)-zig-zag identity, this is equal to the action of p *p_* on morphisms

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

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

𝒮(p !X,S) ()p !(ϵ X) 𝒮(p !p *p *X,S) (p *p *X,p *S) (X,p *S) p * 𝒮(p *X,p *p *S) \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 !p *)(p_! \dashv p^*)-zig-zag identity.

\,

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

(Kan extension of adjoint pair is adjoint quadruple)
For 𝒱\mathcal{V} a symmetric closed monoidal category with all limits and colimits, let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two small 𝒱\mathcal{V}-enriched categoriesand let

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

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

(q op) *Lan p op:[𝒞 op,𝒱][𝒟 op,𝒱] (q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]
(p op) *Ran q op:[𝒟 op,𝒱][𝒞 op,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 qq and pp

Lan q op (q op) * Ran q op Lan p op (p op) * Ran p op \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

Lan q op (q op) * (p op) * Ran p op:[𝒞 op,𝒱][𝒟 op,𝒱] \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:𝒞 op𝒱F \;\colon\; \mathcal{C}^{op} \to \mathcal{V} we have a sequence of 𝒱\mathcal{V}-enriched natural isomorphism as follows

(Lan p opF)(d) c𝒞𝒟(d,p(c))F(c) c𝒞𝒞(q(d),c)F(c) F(q(d)) =((q op) *F)(d). \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 qpq \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

Given a pair of adjoint functors between small categories

the induced operations of pre-composition on categories of presheaves are adjoint to each other, *r *\ell^\ast \,\dashv\, r^\ast, and their adjoint triples of Kan extensions overlap:

Proof

The hom-isomorphism which is characteristic of the adjunction *r *\ell^\ast \dashv r^\ast is the following sequence of natural bijections:

PSh(𝒮 1)(X 1,r *(X 2)) PSh(𝒮 1)(lims 1X 1y(s 1),r *(lims 2X 2y(s 2))) lims 1X 1PSh(𝒮 1)(y(s 1),lims 2X 2r *(y(s 2))) lims 1X 1lims 2X 2PSh(𝒮 1)(y(s 1),r *(y(s 2))) lims 1X 1lims 2X 2𝒮 2(r(s 1),s 2) lims 1X 1lims 2X 2PSh(𝒮 2)(y(r(s 1)),y(s 2)) lims 1X 1PSh(𝒮 2)(y(r(s 1)),lims 2X 2y(s 2)) PSh(𝒮 2)(lims 1X 1y(r(s 1)),lims 2X 2y(s 2)) PSh(𝒮 2)(lims 1X 1𝒮 2((),r(s 1)),lims 2X 2y(s 2)) PSh(𝒮 2)(lims 1X 1𝒮 2((),s 1),lims 2X 2y(s 2)) PSh(𝒮 2)(lims 1X 1 *(y(s 1)),lims 2X 2y(s 2)) PSh(𝒮 2)( *(lims 1X 1y(s 1)),lims 2X 2y(s 2)) PSh(𝒮 2)( *(X 1),X 2) \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

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

Cohesion

Proposition

(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 ii is fully faithful,

  2. the left adjoint pp 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. is cohesive in that

  1. the two reverse functors are fully faithful.

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

Proof

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

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

Example

Consider a site 𝒮\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

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

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

The proof of Prop. 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

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

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

For nn \,\in\, \mathbb{N} and for 𝒮H\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 H\mathbf{H},

  2. is closed under n n -truncation

then the nn-truncation reflection restricts

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

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

If all objects of 𝒮\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. exhibit the \infty -category of \infty -presheaves PSh (𝒮)PSh_\infty(\mathcal{S}) as being cohesive over PSh (𝒮 τ n)PSh_\infty(\mathcal{S}_{\tau_n}).


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

Lemma

Given a functor 𝒮 1f𝒮 2\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2 between small categories, its left Kan extension f !:PSh(𝒮 1)PSh(𝒮 1)f_! \;\colon\; PSh(\mathcal{S}_1) \xrightarrow{\;\;} PSh(\mathcal{S}_1) restricts to ff on representables, in that for s 1𝒮 1s_1 \,\in\, \mathcal{S}_1 we have a natural isomorphism

f !(y(s 1))y(f(s 1)). f_!\big( y(s_1) \big) \;\simeq\; y\big( f(s_1) \big) \,.

Proof

For XPSh(𝒮 2)X \,\in\, PSh(\mathcal{S}_2) we have the following sequence of natural isomorphism:

PSh(𝒮 2)(f !(y(s 1)),X) PSh(𝒮 1)(y(s 1),f *(X)) X(f(s 1)) PSh(𝒮 2)(y(f(s 1)),X). \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 !f_!, the second line the Yoneda lemma over 𝒮 1\mathcal{S}_1 and the definition of f *f^\ast, while the last line is the Yoneda lemma over 𝒮 2\mathcal{S}_2.

Since the composite of these isomorphisms is natural, the Yoneda lemma over PSh(𝒮 2) opPSh(\mathcal{S}_2)^{op} (which is large but locally small, so that the lemma does apply) implies the claim.

Lemma

Let 𝒮 1\mathcal{S}_1 and 𝒮 2\mathcal{S}_2 be small categories with binary products and 𝒮 1f𝒮 2\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2 a functor which preserves these, in that for s,s𝒮 1s, s' \,\in\, \mathcal{S}_1 there is a natural isomorphism f(s×s)f(X 1)×f(X 2)f(s \times s') \,\simeq\, f(X_1) \times f(X_2). Then also the left Kan extension f !f_! preserves binary products, in that for X,XPSh(𝒮 1)X, X' \,\in\, PSh(\mathcal{S}_1) there is a natural isomorphism

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

Proof

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

f !(X×X) f !((limsXy(s))×(limsXy(s))) f !(limsXlimsX(y(s)×y(s))) limsXlimsXf !(y(s)×y(s)) limsXlimsXf !(y(s×s)) limsXlimsXy(f(s×s)) limsXlimsXy(f(s)×f(s)) limsXlimsXy(f(s))×y(f(s)) limsXlimsXf !(y(s))×f !(y(s)) (limsXf !(y(s)))×(limsXf !(y(s))). \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 following Lem. is an immediate variant of Lem. obtained by relaxing the assumptions slightly to a form that is often still readily checked:

Lemma

Let

be a functor between small categories such that

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

  2. the unique coproduct-preserving extension f !f_! of ff to these completions preserves binary products, in that for s,s,𝒮 1s, s' \,\in, \mathcal{S}_1 there is a natural isomorphism:

    f !(y(s)×y(s))f !(y(s))×f !(y(s)) 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 !:PSh(𝒮 1)PSh(𝒮 2)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. , but the main step in between is now more immediate, as it just needs to invoke the assumption:

f !(X×X) f !((limsXy(s))×(limsXy(s))) f !(limsXlimsX(y(s)×y(s))) limsXlimsXf !(y(s)×y(s)) limsXlimsXf !(y(s))×f !(y(s)) (limsXf !(y(s)))×(limsXf !(y(s))). \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

If a pair of small categories 𝒮 1\mathcal{S}_1, 𝒮 2\mathcal{S}_2 has finite products and a functor 𝒮 1f𝒮 2\mathcal{S}_1 \xrightarrow{\;f\;} \mathcal{S}_2 preserves these, then so does its left Kan extension f !:PSh(𝒮 1)PSh(𝒮 2)f_! \,\colon\, PSh(\mathcal{S}_1) \xrightarrow{\;} PSh(\mathcal{S}_2).

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

Proof

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

Proposition

The left Kan extension f !f_! of a fully faithful functor ff between small categories is itself fully faithful:

Proof

We need to show for X,XPSh(𝒮 2)X, X' \,\in\, PSh(\mathcal{S}_2) the morphism

PSh(𝒮 2)(X,X)(i !) X,XPSh(𝒮 `)(i !(X),i !(X)) 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 !i_! is a left adjoint and since XX and XX' are colimits of representables, this morphism is the unique one which reduces to

(1)𝒮 2(s,s)i s,s𝒮 1(i(s),i(s)) \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 ii is fully faithful. If follows that (i !) X,X(i_!)_{X, X'}, is the compostite of the following isomorphisms, and hence an isomorphism:

PSh(𝒮 1)(i !(X 1),i !(X 2)) PSh(𝒮 1)(i !(limsXy(s)),i !(limsXy(s))) PSh(𝒮 1)(limsXi !(y(s)),limsXi !(y(s))) PSh(𝒮 1)(limsXy(i(s)),limsXy(i(s))) limlimsXPSh(𝒮 1)(y(i(s)),y(i(s))) limsXlimsX𝒮 1(i(s),i(s)) limsXlimsX𝒮 2(s,s) limsXlimsXPSh(𝒮 2)(y(s),y(s)) PSh(𝒮 2)(limsXy(s),limsXy(s)) PSh(𝒮 2)(X,X) \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. to evaluate i !i_! on representables and then (1) inside the (co)limits.

References

The above prop. is from:

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

Last revised on October 13, 2021 at 03:49:34. See the history of this page for a list of all contributions to it.