nLab
closed monoidal structure on presheaves

Idea

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

Definition

Let $S$ be a category.

The standard monoidal structure on presheaves $PSh := [S^{op}, Set]$ 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

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

where on the right the product is in Set.

Proposition

The corresponding internal hom

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

exists and is given by

$$[F,G] = U \mapsto Hom_{PSh}( Y(U)\times F, G ) \,,$$

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

Proof

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

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

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 (MacLane-Moerdijk, pages 46, 47).

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.

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