With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
As every topos, the category of presheaves is cartesian closed monoidal.
Let be a category.
The standard monoidal structure on presheaves is the cartesian monoidal structure.
Recalling that limits of presheaves are computed objectwise, this is the pointwise cartesian product in Set: for two presheaves their product presheaf is given by
where on the right the product is in Set.
The corresponding internal hom
exists and is given by
where is the Yoneda embedding.
First assume that exists, so that by the hom-adjunction isomorphism we have . In particular, for each representable functor (with the Yoneda embedding) and using the Yoneda lemma we get
So if the internal hom exists, it has to be of the form given. It remains to show that with this definition really is right adjoint to .
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 be a pre-site with underlying category . Recall from the discussion at site that just means that we have a category on which we consider presheaves , but that it suggests that
to each object and in particular to each there is naturally associated the pre-site with underlying category the comma category ;
that the canonical forgetful functor , which can be thought of as a morphism of pre-sites induces the direct image functor which we write .
Then in these terms the above internal hom for presheaves
is expressed for all by
Relation of the two definitions
To see the equivalence of the two definitions, notice
- that by the Yoneda lemma we have that is simply the over category ;
- by the general properties of functors and comma categories there is an equivalence ;
- which identifies the functor with the functor ;
- and that .
Presheaves over a monoidal category
It is worth noting that in the case where is itself a monoidal category , is equipped with another (bi)closed monoidal structure given by the Day convolution product and its componentwise right adjoints. Let and be two presheaves over . Their tensor product can be defined by the following coend formula:
Then we can define two right adjoints
by the following end formulas:
In the case where the monoidal structure on is cartesian, the induced closed monoidal structure on 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