nLab
closed monoidal structure on presheaves

Contents

Context

Monoidal categories

Category theory

Contents

Idea

As every topos, a category of presheaves is cartesian closed monoidal.

Definition

Proposition

(cartesian closure of categories of presheaves)

Let 𝒞\mathcal{C} be a category and write [𝒞 op,Set][\mathcal{C}^{op}, Set] for its category of presheaves.

This is

  1. a cartesian monoidal category, whose Cartesian product is given objectwise in 𝒞\mathcal{C} by the Cartesian product in Set:

    for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set], their Cartesian product X×Y\mathbf{X} \times \mathbf{Y} exists and is given by

    X×Y:Ac 1 X(c 1)×Y(c 1) f X(f)×Y(f) c 2 X(c 2)×Y(c 2) \mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) }
  2. a cartesian closed category, whose internal hom is given for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] by

    [X,Y]:Ac 1 Hom [𝒞 op,Set](y(c 1)×X,y) f Hom [𝒞 op,Set](y(f)×X,y) c 2 Hom [𝒞 op,Set](y(c 2)×X,y) [\mathbf{X}, \mathbf{Y}] \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} ) \\ {}^{ \mathllap{ f } }\big\downarrow && \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } } \\ c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} ) }

    Here y:𝒞[𝒞 op,Set]y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set] denotes the Yoneda embedding and Hom [𝒞 op,Set](,)Hom_{[\mathcal{C}^{op}, Set]}(-,-) is the hom-functor on the category of presheaves.

(e.g. MacLane-Moerdijk, pages 46-47).

Proof

The first statement is a special case of the general fact that limits of presheaves are computed objectwise.

For the second statement, first assume that [X,Y][\mathbf{X}, \mathbf{Y}] does exist. Then by the hom-adjunction isomorphism we have for any other presheaf Z\mathbf{Z} a natural isomorphism of the form

(1)Hom [𝒞 op,Set](Z,[X,Y])Hom [𝒞 op,Set](Z×X,Y). Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) \;\simeq\; Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y}) \,.

This holds in particular for Z=y(c)\mathbf{Z} = y(c) a representable functor (i.e. in the essential image of the Yoneda embedding) and so the Yoneda lemma implies that if it exists, then [X,Y][\mathbf{X}, \mathbf{Y}] must have the claimed form:

[X,Y](c) Hom [𝒞 op,Set](y(c),[X,Y]) Hom [𝒞 op,Set](y(c)×X,Y). \begin{aligned} [\mathbf{X}, \mathbf{Y}](c) & \simeq Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] ) \\ & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} ) \,. \end{aligned}

Hence it remains to show that this formula does make (1) hold generally.

For this we use the equivalent definition of adjoint functors in terms of the adjunction counit providing a system of universal arrows.

Define a would-be adjunction counit, which here is called an evaluation morphism, by

X×[X,Y] ev Y X(c)×Hom [𝒞 op,Set](y(c)×X,Y) ev c Y(c) (x,ϕ) ϕ c(id c,x) \array{ \mathbf{X} \times [\mathbf{X} , \mathbf{Y}] &\overset{ev}{\longrightarrow}& \mathbf{Y} \\ \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y}) &\overset{ev_c}{\longrightarrow}& \mathbf{Y}(c) \\ (x, \phi) &\mapsto& \phi_c( id_c, x ) }

Then it remains to show that for every morphism of presheaves of the form X×AAfAY \mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y} there is a unique morphism f˜:A[X,Y]\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}] such that

(2)X×A X×f˜ X×[X,Y] f ev Y \array{ \mathbf{X} \times \mathbf{A} && \overset{ \mathbf{X} \times \widetilde f }{\longrightarrow} && \mathbf{X} \times [\mathbf{X}, \mathbf{Y}] \\ & {}_{\mathllap{ \mathrlap{f} }}\searrow && \swarrow_{ \mathrlap{ ev } } \\ && \mathbf{Y} }

The commutativity of this diagram means in components at c𝒞c \in \mathcal{C} that, that for all xX(c)x \in \mathbf{X}(c) and aA(c)a \in \mathbf{A}(c) we have

ev c(x,f˜ c(a)) (f˜ c(a)) c(id c,x) =f c(x,a) \begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned}

Hence this fixes the component f˜ c(a) c\widetilde f_c(a)_c when its first argument is the identity morphism id cid_c. But let g:dcg \;\colon\; d \to c be any morphism and chase (id c,x)(id_c, x ) through the naturality diagram for f˜ c(a)\widetilde f_c(a):

Hom 𝒞(c,c)×X(c) (f˜ c(a)) c Y(c) g * g * Hom 𝒞(d,c)×X(d) (f˜ c(a)) d Y(d)AAAA{(id c,x)} {f c(x,a)} {(g,g *(x))} {f d(g *(x),g *(a))} \array{ Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c) &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}& \mathbf{Y}(c) \\ {}^{\mathllap{ g^\ast }}\big\downarrow && \big\downarrow^{\mathrlap{ g^\ast }} \\ Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d) &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}& \mathbf{Y}(d) } \phantom{AAAA} \array{ \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \} \\ \big\downarrow && \big\downarrow \\ \{ (g, g^\ast(x)) \} &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \} }

This shows that (f˜ c(a)) d(\widetilde f_c(a))_d is fixed to be given by

(3)(f˜ c(a)) d(g,x)=f d(x,g *(a)) (\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) )

at least on those pairs (g,x)(g,x') such that xx' is in the image of g *g^\ast.

But, finally, (f˜ c(a)) d(\widetilde f_c(a))_d is also natural in cc

A(c) f˜ c [X,Y](c) g * g * A(d) f˜ d [X,Y](d) \array{ \mathbf{A}(c) &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c) \\ {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}} \\ \mathbf{A}(d) &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d) }

which implies that (3) must hold generally. Hence naturality implies that (2) indeed has a unique solution.

Definition in terms of homs of direct images

Often another, equivalent, expression is used to express the internal hom of presheaves:

Let XX be a pre-site with underlying category S XS_X. Recall from the discussion at site that just means that we have a category S XS_X on which we consider presheaves FPSh(S X):=[S X op,Set]F \in PSh(S_X) := [S_X^{op}, Set], but that it suggests that

  • to each object UPSh(X)U \in PSh(X) and in particular to each US XPSh(X)U \in S_X \hookrightarrow PSh(X) there is naturally associated the pre-site UU with underlying category the comma category S U=(Y/Y(U))S_U = (Y/Y(U));

  • that the canonical forgetful functor j UX t:S US Xj^t_{U \to X} : S_U \to S_X, which can be thought of as a morphism of pre-sites j UX:XUj_{U \to X} : X \to U induces the direct image functor (j UX) *:PSh(X)PSh(U)(j_{U \to X})_* : PSh(X) \to PSh(U) which we write FF| UF \mapsto F|_U.

Then in these terms the above internal hom for presheaves

hom:PSh(X) op×PSh(X)PSh(X) hom : PSh(X)^{op} \times PSh(X) \to PSh(X)

is expressed for all F,GPSh(X)F,G \in PSh(X) by

hom(F,G)=UHom PSh(U)(F| U,G| U). hom(F,G) = U \mapsto Hom_{PSh(U)}(F|_U, G|_U) \,.

Relation of the two definitions

To see the equivalence of the two definitions, notice

  • that by the Yoneda lemma we have that S US_U is simply the over category S U=S X/US_U = S_X/U;
  • by the general properties of functors and comma categories there is an equivalence PSh(S X/U)PSh(S X)/y(U)PSh(S_X/U) \simeq PSh(S_X)/y(U);
  • which identifies the functor ()| U:PSh(S X)PSh(S U)(-)|_U : PSh(S_X) \to PSh(S_U) with the functor (()×y(U)p 2y(U)):PSh(S X)PSh(S X)/y(U)((-)\times y(U) \stackrel{p_2}{\to} y(U)) : PSh(S_X) \to PSh(S_X)/y(U);
  • and that Hom PSh(S X)/y(U)(y(U)×F,y(U)×G)Hom PSh(S X)(y(U)×F,G)Hom_{PSh(S_X)/y(U)}(y(U) \times F, y(U) \times G) \simeq Hom_{PSh(S_X)}(y(U) \times F, G).

Presheaves over a monoidal category

It is worth noting that in the case where XX is itself a monoidal category (X,,I)(X, \otimes, I), Psh(X)Psh(X) is equipped with another (bi)closed monoidal structure given by the Day convolution product and its componentwise right adjoints. Let FF and GG be two presheaves over XX. Their tensor product FGF \star G can be defined by the following coend formula:

FG=U U 1,U 2XHom X(U,U 1U 2)×F(U 1)×G(U 2)F\star G = U \mapsto \int^{U_1,U_2\in X} Hom_X(U, U_1\otimes U_2) \times F(U_1) \times G(U_2)

Then we can define two right adjoints

FF\G/GF\star - \dashv F \backslash - \qquad -\star G \dashv - / G

by the following end formulas:

F\H=V UXF(U)H(UV)F \backslash H = V \mapsto \int_{U\in X} F(U) \to H(U\otimes V)
H/G=U VXG(V)H(UV)H / G = U \mapsto \int_{V\in X} G(V) \to H(U\otimes V)

In the case where the monoidal structure on XX is cartesian, the induced closed monoidal structure on Psh(X)Psh(X) coincides with the cartesian closed structure described in the previous sections.

References

The first definition is discussed for instance in section I.6 of

The second definition is discussed for instance in section 17.1 of

Last revised on June 20, 2018 at 23:54:12. See the history of this page for a list of all contributions to it.