category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
As every topos, the category $PSh(X)$ of presheaves is cartesian closed monoidal.
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
where on the right the product is in Set.
The corresponding internal hom
exists and is given by
where $Y : S \to [S^{op}, Set]$ is the Yoneda embedding.
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
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).
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
is expressed for all $F,G \in PSh(X)$ by
To see the equivalence of the two definitions, notice
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:
Then we can define two right adjoints
by the following end formulas:
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.
The first definition is discussed for instance in section I.6 of
The second definition is discussed for instance in section 17.1 of