nLab exponential object

This entry is about the concept in category theory. For (co)exponential functions see at exponential map and coexponential map.

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Idea

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.

More generally, in a category with finite products, an exponential object $X^Y$ is a representing object for the functor $\hom(- \times Y, X)$.

Yet more generally, an exponential object $X^Y$ in a category $\mathcal{C}$ is a representing object for the functor $y(X)^{y(Y)}$, where $y \,\colon\, \mathcal{C} \longrightarrow PSh(\mathcal{C})$ denotes the Yoneda embedding and the latter exponentiation takes place in the category of presheaves $PSh(\mathcal{C})$, where it always exists (see at closed monoidal structure on presheaves).

Definition

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 $\mathrm{ev}\colon X^Y \times Y \to X$ which is universal in the sense that, given any object $Z$ and map $e\colon Z \times Y \to X$, there exists a unique map $u\colon Z \to X^Y$ such that

$$Z \times Y \stackrel{u \times \mathrm{id}_Y}\to X^Y \times Y \stackrel{\mathrm{ev}}\to X$$

equals $e$.

Equivalently, this data can be repackaged as a natural isomorphism $\hom_C(-, X^Y) \cong \hom_C(- \times Y, X)$ (where $\mathrm{ev}$ is the image of the identity arrow on $X^Y$), so that exponential objects are representations of the latter as a representable functor.

The map $u$ is known by various names, such as the exponential transpose or currying of $e$. It is sometimes denoted $\lambda(e)$ in a hat tip to the lambda-calculus, since in the internal logic of a cartesian closed category this is the operation corresponding to $\lambda$-abstraction. It is also sometimes denoted $e^\flat$ (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.

Without finite products

In a category without finite products, an exponential object is given by an object $X^Y$ and a universal natural transformation $C(-, X^Y) \times C(-, X) \to C(-, Y)$. See, for instance, this categories mailing list post by Richard Garner.

Related notions

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 (which requires that all binary products exist too).

• 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 all pullbacks along $f$ exist and that the resulting base change functor $f^* : C/A \to C/Y$ 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. (Sometimes locally cartesian closed categories are also required to have a terminal object, and hence to also be cartesian closed.)

• Conversely, if $X$ is such that $X^Y$ exists for all $Y$, we say that $X$ is exponentiating. (This requires that $C$ have all binary products.) 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.

When $C$ is not cartesian but merely monoidal, then the analogous notion is that of a left/right internal hom.

Properties

• If $C$ has equalizers of coreflexive pairs, then any pullback of an exponentiable morphism is exponentiable. This follows from the adjoint triangle theorem, since the left adjoint $\Sigma_f$ of pullback is comonadic.

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 $X^Y$ is usually more related to properties of $Y$ than properties of $X$.

Exponentiation of sets and of numbers

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

$$\left|X^S\right| = |X|^{|S|} \,.$$

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 $a,b \in \mathbb{N}$ to the product

$$a^b = a \times a \times \cdots \times a \;\; (b \; \text{factors}) \,.$$

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.

More examples

• In Top (the category of all topological spaces), the exponentiable spaces are precisely the core-compact spaces (see Day and Kelly). 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 $Top$.

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 $Y\to X$ 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.

• In a functor category $D^C$, a natural transformation $\alpha:F\to G$ is exponentiable if it is cartesian and each component $\alpha_c:F c \to G c$ is exponentiable in $D$. Given $H\to F$, we define $\Pi_\alpha(H)(c) = \Pi_{\alpha_c}(H c)$; then for $u:c\to c'$ to obtain a map $\Pi_{\alpha_c}(H c) \to \Pi_{\alpha_{c'}}(H c')$ we need a map $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \to H c'$. But since $\alpha$ is cartesian, $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \cong \alpha_c^* (\Pi_{\alpha_c}(H c))$, so we have the counit $\alpha_c^* (\Pi_{\alpha_c}(H c)) \to H c$ that we can compose with $H u$. (This is certainly not an if-and-only-if, however: for instance, if $C$ is small, then all morphisms of $Set^C$ are exponentiable, whether or not they are cartesian.)

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 the non-cartesian monoidal analogue of an exponentiating object.

Relative Examples

Let $\mathcal{C}$ be a category with finite limits and $f: C \to D$ a morphism in $\mathcal{C}$. Then $f$ is exponentiable as an object of the slice category $\mathcal{C}\downarrow D$ if and only if the base change functor $f^\ast: \mathcal{C} \downarrow D \to \mathcal{C} \downarrow C$ has a right adjoint. In this case, we say that $f$ is an exponentiable morphism in $\mathcal{C}$.

• The exponentiable morphisms in $Top$ were characterized by Niefield. In particular, a subspace inclusion $C \to D$ is exponentiable if and only if it is locally closed?.

• The exponentiable morphisms in $Locale$ and $Topos$ which are embeddings were also characterized by Niefield. It seems(?) that no complete characterization of exponentiable morphisms in $Locale$ or $Topos$ appears in the literature.

• The exponentiable morphisms in $Cat$ are the Conduché functors.

Properties

As with other internal homs, the currying isomorphism

$$hom_C(Z,X^Y) \cong hom_C(Z \times Y,X)$$

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

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

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:

• $(X \times Y)^Z = X^Z \times Y^Z$;
• $1^Z \cong 1$.

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:

• $X^{Y + Z} \cong X^Y \times X^Z$;
• $X^0 \cong 1$.

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).

Related concepts

• exponential ideal

• exponentiable topos

• reflexive object

References

Discussion with focus on application to topological spaces and compactly generated topological spaces:

• Brian Day, G. Max Kelly, On topological quotients preserved by pullback or products, Mathematical Proceedings of the Cambridge Philosophical Society 67 3 (1970) 553 - 558 (doi:10.1017/S0305004100045850)

• Susan Niefield, Cartesianness, PhD thesis, Rutgers 1978 (proquest:302920643)

• Susan Niefield, Cartesianness: topological spaces, uniform spaces, and affine schemes., Journal of Pure and Applied Algebra, 23.2, 1982, pp. 147-167.(doi:10.1016/0022-4049(82)90004-4, pdf)

• Susan Niefield, Cartesian inclusion: locales and toposes., Communications in Algebra, 9.16, 1981, pp. 1639-1671 (doi:10.1080/00927878108822672)

• Francis Borceux, Section 7.1 of: Categories and Structures, Vol. 2 of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

Discussion for locales:

• Martin Hyland, Function Spaces in the Category of Locales. (doi:10.1080/00927878108822672)

Discussion for internal groupoids in Top (topological groupoids):

• Susan B. Niefield, Dorette A. Pronk, Internal groupoids and exponentiability, Cahiers de topologie et géométrie différentielle catégoriques, LX 4 (2019) (pdf)

Exponentiable relational structures are considered in

• Jason Parker, Exponentiability in categories of relational structures (2022) [arXiv:2208:07350]