closed monoidal structure on presheaves


Monoidal categories

Category theory



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


Let SS be a category.

The standard monoidal structure on presheaves PSh:=[S op,Set]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,GF, G their product presheaf F×GF \times G is given by

F×G=UF(U)×G(U), F \times G = U \mapsto F(U) \times G(U) \,,

where on the right the product is in Set.


The corresponding internal hom

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

exists and is given by

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

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


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

[F,G](U) Hom(Y(U),[F,G]) Hom(Y(U)×F,G). \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,][F,-] really is right adjoint to F-\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 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.


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

The second definition is discussed for instance in section 17.1 of

Revised on May 4, 2016 03:03:48 by Noam Zeilberger (