nLab
exponential object

Context

Category theory

Monoidal categories

Mapping space

Contents

Idea

An exponential object X YX^Y is an internal hom [Y,X][Y,X] in a cartesian closed category. It generalises the notion of function set, which is an exponential object in Set.

Definition

The above is actually a complete definition, but here we spell it out.

Let XX and YY be objects of a category CC such that all binary products with YY exist. (Usually, CC actually has all binary products.) Then an exponential object is an object X YX^Y equipped with an evaluation map ev:X Y×YXev: X^Y \times Y \to X which is universal in the sense that, given any object ZZ and map e:Z×YXe: Z \times Y \to X, there exists a unique map u:ZX Yu: Z \to X^Y such that

Z×Yu×id YX Y×YevX Z \times Y \stackrel{u \times id_Y}\to X^Y \times Y \stackrel{ev}\to X

equals ee.

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 CC be a category and X,YCX,Y\in C.

  • If X YX^Y exists, then we say that XX exponentiates YY.

  • If YY is such that X YX^Y exists for all XX, we say that YY is exponentiable (or powerful, cf. Street-Verity pdf). Then CC is cartesian closed if it has a terminal object and every object is exponentiable.

  • More generally, a morphism f:YAf\colon Y \to A is exponentiable (or powerful) when it is exponentiable in the over category C/AC/A. This is equivalent to saying that the base change functor f *f^* has a right adjoint, usually denoted Π f\Pi_f and called a dependent product. In particular, CC is locally cartesian closed iff every morphism is exponentiable, iff all pullback functors have right adjoints.

  • Conversely, if XX is such that X YX^Y exists for all YY, we say that XX is exponentiating. Again, CC 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 CC is an exponential object in the opposite category C opC^{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.

Examples

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 YX^Y is usually more related to properties of YY than properties of XX.

Exponentiation of sets and of numbers

In the cartesian closed category Set of sets, for X,SSetX,S \in Set to sets, their exponentiation X SX^S is the set of functions SXS\to X.

Restricted to finite sets and under the cardinality operation ||:FinSet|-| : FinSet \to \mathbb{N} this induces an exponentiation operation on natural numbers

|X S|=|X| |S|. |X^S| = |X|^{|S|} \,.

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,ba,b \in \mathbb{N} to the product

a b=a×a××a(bfactors). a^b = a \times a \times \cdots \times a \;\; (b \; factors) \,.

If b=0b = 0 is zero, the expression on the right is 1, reflecting the fact that 00 is the cardinality of the empty set, which is the initial object in Set.

When the natural numbers are embedding into larger rigs or rings, the operation of exponentiation may extend to these larger context. It yields for instance an exponentiation operation on the real numbers \mathbb{R}, and on the complex numbers \mathbb{C}.

More examples

  • 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 TopTop.

  • 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 YXY\to X when YY is a set and XX 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.

Properties

As with other internal homs, the currying isomorphism

hom C(Z,X Y)hom C(Z×Y,X) hom_C(Z,X^Y) \cong hom_C(Z \times Y,X)

is a natural isomorphism of sets. By the usual Yoneda arguments, this isomorphism can be internalized to an isomorphism in CC:

(X Y) ZX Y×Z. (X^Y)^Z \cong X^{Y\times Z}.

Similarly, XX 1X \cong X^1, where 11 is a terminal object. Thus, a product of exponentiable objects is exponentiable.

Other natural isomorphisms that match equations from ordinary algebra include: * (X×Y) Z=X Z×Y Z(X \times Y)^Z = X^Z \times Y^Z; * 1 Z11^Z \cong 1.

These show that, in a cartesian monoidal category, a product of exponentiating objects is also exponentiating.

Now suppose that CC is a distributive category. Then we have these isomorphisms: * X Y+ZX Y×X ZX^{Y + Z} \cong X^Y \times X^Z; * X 01X^0 \cong 1.

Here Y+ZY + Z is a coproduct of YY and ZZ, while 00 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).

Revised on July 9, 2013 10:40:13 by Mike Shulman (203.191.192.138)