category with duals (list of them)
dualizable object (what they have)
The above is actually a complete definition, but here we spell it out.
Let and be objects of a category such that all binary products with exist. (Usually, actually has all binary products.) Then an exponential object is an object equipped with an evaluation map which is universal in the sense that, given any object and map , there exists a unique map such that
As before, let be a category and .
If exists, then we say that exponentiates .
More generally, a morphism is exponentiable (or powerful) when it is exponentiable in the over category . This is equivalent to saying that the base change functor has a right adjoint, usually denoted and called a dependent product. In particular, is locally cartesian closed iff every morphism is exponentiable, iff all pullback functors have right adjoints.
Conversely, if is such that exists for all , we say that is exponentiating. Again, 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 is an exponential object in the opposite category . 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 is usually more related to properties of than properties of .
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 .
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 when is a set and 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.
As with other internal homs, the currying isomorphism
Similarly, , where 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 is a distributive category. Then we have these isomorphisms: