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exponential object in nLab

Contents

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

Context

Category theory

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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.

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

Yet more generally, an exponential object X YX^Y in a category 𝒞\mathcal{C} is a representing object for the functor y(X) y(Y)y(X)^{y(Y)}, where y:𝒞⟶PSh(𝒞)y \,\colon\, \mathcal{C} \longrightarrow PSh(\mathcal{C}) denotes the Yoneda embedding and the latter exponentiation takes place in the category of presheaves PSh(𝒞)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 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×Y→X\mathrm{ev}\colon X^Y \times Y \to X which is universal in the sense that, given any object ZZ and map e:Z×Y→Xe\colon Z \times Y \to X, there exists a unique map u:Z→X Yu\colon Z \to X^Y such that

Z×Y→u×id YX Y×Y→evX Z \times Y \stackrel{u \times \mathrm{id}_Y}\to X^Y \times Y \stackrel{\mathrm{ev}}\to X

equals ee.

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

The map uu is known by various names, such as the exponential transpose or currying of ee. It is sometimes denoted λ(e)\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 λ\lambda-abstraction. It is also sometimes denoted e ♭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 YX^Y and a universal natural transformation C(−,X Y)×C(−,X)→C(−,Y)C(-, X^Y) \times C(-, X) \to C(-, Y). See, for instance, this categories mailing list post by Richard Garner.

As before, let CC be a category and X,Y∈CX,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 (which requires that all binary products exist too).

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

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

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

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,S∈SetX,S \in Set to sets, their exponentiation X SX^S is the set of functions S→XS\to X.

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

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

This exponentiation operation on numbers (−) (−):ℕ×ℕ→ℕ(-)^{(-)} : \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∈ℕa,b \in \mathbb{N} to the product

a b=a×a×⋯×a(bfactors). a^b = a \times a \times \cdots \times a \;\; (b \; \text{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 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

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→XY\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.

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

However, exponentiating objects do matter sometimes.

Relative Examples

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

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

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

  • The exponentiable morphisms in CatCat are the Conduché functors.

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

Similarly, X≅X 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 Z≅11^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+Z≅X Y×X ZX^{Y + Z} \cong X^Y \times X^Z;
  • X 0≅1X^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).

If CC 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 Σ f\Sigma_f of pullback is comonadic.

References

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

Discussion for locales:

Discussion for internal groupoids in Top (topological groupoids):

Exponentiable relational structures are considered in

Last revised on December 6, 2024 at 20:37:39. See the history of this page for a list of all contributions to it.