nLab
closed monoidal structure on presheaves

Idea

As every topos, the category $\mathrm{PSh}(X)$ of presheaves is cartesian closed monoidal.

Definition

Let $S$ be a category.

The standard monoidal structure on presheaves $\mathrm{PSh}:=[S^{\mathrm{op}},\mathrm{Set}]$ is the cartesian monoidal structure.

Recalling that limits of presheaves are computed objectwise, this is the pointwise cartesian product in Set: for two presheaves $F,G$ their product presheaf $F\times G$ is given by

$$F\times G:U\mapsto F(U)\times G(U),$$F \times G : U \mapsto F(U) \times G(U) \,,

where on the right the product is in Set.

Proposition

The corresponding internal hom

$$[-,-]:{\mathrm{PSh}}^{\mathrm{op}}\times \mathrm{PSh}\to \mathrm{PSh}$$[-,-] : PSh^{op} \times PSh \to PSh

exists and is given by

$$[F,G]:U\mapsto {\mathrm{Hom}}_{\mathrm{PSh}}(Y(U)\times F,G),$$[F,G] : U \mapsto Hom_{PSh}( Y(U)\times F, G ) \,,

where $Y:S\to [S^{\mathrm{op}},\mathrm{Set}]$ is the Yoneda embedding.

Proof

First assume that $[F,G]$ exists, so that by the hom-adjunction isomorphism we have $\mathrm{Hom}(R,[F,G])\simeq \mathrm{Hom}(R\times F,G)$. In particular, for each representable functor $R=Y(U)$ (with $Y$ the Yoneda embedding) and using the Yoneda lemma we get

$$\begin{array}{rl}[F,G](U)& \simeq \mathrm{Hom}(Y(U),[F,G])\\ & \simeq \mathrm{Hom}(Y(U)\times F,G)\end{array}.$$\begin{aligned} [F,G](U) & \simeq Hom(Y(U), [F,G]) \\ & \simeq Hom(Y(U) \times F, G) \end{aligned} \,.

So if the internal hom exists, it has to be of the form given. It remains to show that with this definition $[F,-]$ really is right adjoint to $-\otimes F$.

See pages 46, 47 of

• MacLane-Moerdijk, Sheaves in Geometry and Logic.

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 $F\in \mathrm{PSh}(S_X):=[S_X^{\mathrm{op}},\mathrm{Set}]$, but that it suggests that

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

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

Then in these terms the above internal hom for presheaves

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

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

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

References

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

• MacLane-Moerdijk, Sheaves in Geometry and Logic.

The second definition is discussed for instance in section 17.1 of

• Kashiwara-Shapira, Categories and Sheaves