nLab closed monoidal structure on presheaves

Idea

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

Definition

Proposition

(cartesian closure of categories of presheaves)

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

This is

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

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

   $$\mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) }$$

2. a cartesian closed category, whose internal hom is given for $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ by

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

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

(e.g. MacLane-Moerdijk, section I.6, 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 $[\mathbf{X}, \mathbf{Y}]$ does exist. Then by the hom-adjunction isomorphism we have for any other presheaf $\mathbf{Z}$ a natural isomorphism of the form

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

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

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

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

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

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

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

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

$$\begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned}$$

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

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

This shows that $(\widetilde f_c(a))_d$ is fixed to be given by

(3)$$(\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) )$$

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

But, finally, $(\widetilde f_c(a))_d$ is also natural in $c$

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

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

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

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

Then in these terms the above internal hom for presheaves

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

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

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

Relation of the two definitions

To see the equivalence of the two definitions, notice

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

Presheaves over a monoidal category

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

$$F\star G = U \mapsto \int^{U_1,U_2\in X} Hom_X(U, U_1\otimes U_2) \times F(U_1) \times G(U_2)$$

Then we can define two right adjoints

$$F\star - \dashv F \backslash - \qquad -\star G \dashv - / G$$

by the following end formulas:

$$F \backslash H = V \mapsto \int_{U\in X} F(U) \to H(U\otimes V)$$

$$H / G = U \mapsto \int_{V\in X} G(V) \to H(U\otimes V)$$

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

Closed monoidal structure on sheaves

• The cartesian closure restricts from presheaves to categories of sheaves (e.g. MacLane-Moerdijk, section III.6, p. 136-138)

Let $X$ be a site with underlying category $S_X$. Write $PSh(X) = [S^{op}, Set]$ for the presheaf category of $X$ and $Sh(X)$ for the corresponding subcategory of sheaves.

The closed monoidal structure on presheaves restricts under the inclusion $Sh(X) \hookrightarrow PSh(X)$ to a closed monoidal structure on sheaves.

For $f : X \to Y$ a left exact morphism of sites, there is for $F \in Sh(X)$, $F \in Sh(Y)$ a natural isomorphism

$$f_* hom(f^{-1}G, F) \simeq hom(G, f_* F).$$

Here $f_*$ is the direct image and $f^{-1}$ the inverse image operation.

References

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

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

The second definition is discussed for instance in section 17.1 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves