This entry is about the concept in category theory. For exponential functions see at exponential map.
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 is an internal hom 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 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
equals .
The map is known by various names, such as the exponential transpose or currying of . It is sometimes denoted in a hat tip to the lambda-calculus, since in the internal logic of a cartesian closed category this is the operation corresponding to -abstraction. It is also sometimes denoted (as in music notation), being an instance of the more general notion of adjunct or mate.
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 be a category and .
If exists, then we say that exponentiates .
If is such that exists for all , we say that is exponentiable (or powerful, cf. Street-Verity pdf). Then is cartesian closed if it has a terminal object and every object is exponentiable.
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.
When is not cartesian but merely monoidal, then the analogous notion is that of a left/right residual.
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 .
In the cartesian closed category Set of sets, for to sets, their exponentiation is the set of functions .
Restricted to finite sets and under the cardinality operation this induces an exponentiation operation on natural numbers
This exponentiation operation on numbers is therefore the decategorification of the canonically defined internal hom of sets. It sends numbers to the product
If is zero, the expression on the right is 1, reflecting the fact that 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.
The condition that a topological space be core-compact (i.e. exponentiable) is in fact a condition on its underlying locale. More precisely, a topological space is core-compact if and only if its underlying locale is a continuous poset. In fact, a topological space is exponentiable if and only if its underlying locale is exponentiable:
A locale is exponentiable if and only if it is a continuous poset (see Hyland) – this is sometimes taken as the definition of a locally compact locale). The notion of continuous poset generalizes straightforward to that of continuous category and continuous ∞-category?. Using these notions, one has analogous characterization of those toposes and (∞,1)-toposes which are exponentiable (see metastably locally compact locale? and continuous category as well as exponentiable topos).
In algebraic set theory (see category with class structure for one example) one often assumes that only small objects (and morphisms) are exponentiable. 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.
In a functor category , a natural transformation is exponentiable if (though probably not “only if”) it is cartesian and each component is exponentiable in . Given , we define ; then for to obtain a map we need a map . But since is cartesian, , so we have the counit that we can compose with .
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).
A dialogue category? is a symmetric monoidal category equipped with an exponentiating object.
Let be a category with finite limits and a morphism in . Then is exponentiable as an object of the slice category if and only if the base change functor has a right adjoint. In this case, we say that is an exponentiable morphism in .
The exponentiable morphisms in were characterized by Niefield. In particular, a subspace inclusion is exponentiable if and only if it is locally closed?.
The exponentiable morphisms in and which are embeddings were also characterized by Niefield. It seems(?) that no complete characterization of exponentiable morphisms in or appears in the literature.
The exponentiable morphisms in are the Conduché functors.
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 :
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:
Here is a coproduct of and , while 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).
Susan Niefield, Cartesianness: topological spaces, uniform spaces, and affine schemes., Journal of Pure and Applied Algebra, 23.2, 1982, pp. 147-167.(doi)
Susan Niefield, Cartesian inclusion: locales and toposes., Communications in Algebra, 9.16, 1981, pp. 1639-1671. doi
Martin Hyland, Function Spaces in the Category of Locales., available from Hyland’s website
Brian Day and G. Max Kelly?, On topological quotients preserved by pullback or
products_, doi
Last revised on February 22, 2021 at 08:44:00. See the history of this page for a list of all contributions to it.