category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
An exponential object $X^Y$ is an internal hom $[Y,X]$ in a cartesian closed category. It generalises the notion of function set, which is an exponential object in Set.
The above is actually a complete definition, but here we spell it out.
Let $X$ and $Y$ be objects of a category $C$ such that all binary products with $Y$ exist. (Usually, $C$ actually has all binary products.) Then an exponential object is an object $X^Y$ equipped with an evaluation map $ev: X^Y \times Y \to X$ which is universal in the sense that, given any object $Z$ and map $e: Z \times Y \to X$, there exists a unique map $u: Z \to X^Y$ such that
equals $e$.
As with other universal constructions, an exponential object, if any exists, is unique up to unique isomorphism. It can also be characterized as a distributivity pullback.
As before, let $C$ be a category and $X,Y\in C$.
If $X^Y$ exists, then we say that $X$ exponentiates $Y$.
If $Y$ is such that $X^Y$ exists for all $X$, we say that $Y$ is exponentiable (or powerful, cf. Street-Verity pdf). Then $C$ is cartesian closed if it has a terminal object and every object is exponentiable.
More generally, a morphism $f\colon Y \to A$ is exponentiable (or powerful) when it is exponentiable in the over category $C/A$. This is equivalent to saying that the base change functor $f^*$ has a right adjoint, usually denoted $\Pi_f$ and called a dependent product. In particular, $C$ is locally cartesian closed iff every morphism is exponentiable, iff all pullback functors have right adjoints.
Conversely, if $X$ is such that $X^Y$ exists for all $Y$, we say that $X$ is exponentiating. Again, $C$ is cartesian closed if it has a terminal object and every object is exponentiating. (The reader should beware that some authors say “exponentiable” for what is here called “exponentiating.”)
Dually, a coexponential object in $C$ is an exponential object in the opposite category $C^{op}$. A cocartesian coclosed category? has all of these (and an initial object). Some coexponential objects occur naturally in algebraic categories (such as rings or frames) whose opposites are viewed as categories of spaces (such as schemes or locales). Cf. also cocartesian closed category.
Of course, in any cartesian closed category every object is exponentiable and exponentiating. In general, exponentiable objects are more common and important than exponentiating ones, since the existence of $X^Y$ is usually more related to properties of $Y$ than properties of $X$.
In the cartesian closed category Set of sets, for $X,S \in Set$ to sets, their exponentiation $X^S$ is the set of functions $S\to X$.
Restricted to finite sets and under the cardinality operation $|-| : FinSet \to \mathbb{N}$ this induces an exponentiation operation on natural numbers
This exponentiation operation on numbers $(-)^{(-)} : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is therefore the decategorification of the canonically defined internal hom of sets. It sends numbers $a,b \in \mathbb{N}$ to the product
If $b = 0$ is zero, the expression on the right is 1, reflecting the fact that $0$ is the cardinality of the empty set, which is the initial object in Set.
When the natural numbers are embedded into larger rigs or rings, the operation of exponentiation may extend to these larger context. It yields for instance an exponentiation operation on the positive real numbers.
In Top (the category of all topological spaces), the exponentiable spaces are precisely the core-compact spaces. In particular, this includes locally compact Hausdorff spaces. However, most nice categories of spaces are cartesian closed, so that all objects are exponentiable; note that usually the cartesian product in such categories has a slightly different topology than it does in $Top$.
There are similar characterizations of exponentiable locales (see locally compact locale and continuous poset and (in the 2-categorical sense) toposes (see metastably locally compact locale? and continuous category).
In algebraic set theory one often assumes that only small objects (and morphisms) are exponentiable. This is analogous to how in material set theory one can talk about the class of functions $Y\to X$ when $Y$ is a set and $X$ a class, but not the other way round.
In a type theory with dependent products, every display morphism is exponentiable in the category of contexts —even in a type theory without identity types, so that not every morphism is display and the relevant slice category need not have all products.
However, exponentiating objects do matter sometimes.
In Abstract Stone Duality, Sierpinski space is exponentiating.
Toby Bartels has argued that predicative mathematics can have a set of truth values as long as this set is not exponentiating (or even exponentiates only finite sets).
As with other internal homs, the currying isomorphism
is a natural isomorphism of sets. By the usual Yoneda arguments, this isomorphism can be internalized to an isomorphism in $C$:
Similarly, $X \cong X^1$, where $1$ is a terminal object. Thus, a product of exponentiable objects is exponentiable.
Other natural isomorphisms that match equations from ordinary algebra include:
These show that, in a cartesian monoidal category, a product of exponentiating objects is also exponentiating.
Now suppose that $C$ is a distributive category. Then we have these isomorphisms:
Here $Y + Z$ is a coproduct of $Y$ and $Z$, while $0$ is an initial object. Thus in a distributive category, the exponentiable objects are closed under coproducts.
Note that any cartesian closed category with finite coproducts must be distributive, so all of the isomorphisms above hold in any closed 2-rig (such as Set, of course).