Cartesian closed category

A category with finite products is cartesian closed when every product functor $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

A category with finite limits is locally Cartesian closed when every slice category is a Cartesian closed category. Covariant set-valued functor categories on small categories have this property, using the equivalence between their slices and functor categories on categories of elements.

A monoid encoding a one-object product and exponential structure: pairing is inverse to the map $z\mapsto (\pi z,{\pi }^{\prime }z)$, and currying is inverse to $y\mapsto \epsilon ⟨y\pi ,{\pi }^{\prime }⟩$. These bijections imply pairing naturality $⟨x,y⟩z=⟨xz,yz⟩$ and currying naturality $(x⟨z\pi ,{\pi }^{\prime }⟩{)}^{\ast }=x^{\ast }z$. The element $e=({\pi }^{\prime }{)}^{\ast }$ then satisfies $ez=e$ for every $z$, hence is idempotent. Splitting $e$ supplies a terminal object and a two-object Cartesian closed category whose endomorphisms at the original object are $M$. For the original object to be nonterminal, $M$ must be nontrivial.

An object isomorphic to its own exponential supplies mutually inverse folding and unfolding maps. Evaluation defines application, and folding after currying defines abstraction. The exponential universal property gives the beta rule, while the full isomorphism gives eta conversion. A mere retraction of the function object generally suffices for beta but not automatically eta.

An exponential object $B^A$ represents maps out of a product with $A$: there are natural bijectionsThe corresponding operations are currying and uncurrying, and the identity of $B^A$ corresponds to the evaluation morphism $B^A\times A\to B$.

For a small category, this formula defines the exponential object in its covariant set-valued functor category. An arrow $c\to d$ acts by precomposition with $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. Evaluation applies a natural transformation to $(1_c,z)$; currying a map $P\times G\to H$ sends $p\in P(c)$ to the transformation $(f,z)\mapsto \theta (P(f)p,z)$. The formulas prove the universal property directly, without an adjoint existence theorem.

Currying transforms a morphism of two variables into a morphism taking values in an exponential object. The evaluation map of an exponential object recovers the original morphism. Its categorical universal property requires naturality and uniqueness, not just an underlying set bijection.

The evaluation map is the counit representing application in an exponential object. For every $Z$, composition with evaluation gives the natural bijection $\mathrm{Hom}(Z,Y^X)\cong \mathrm{Hom}(Z\times X,Y)$. Its inverse is currying. Evaluation must be a morphism in the ambient category, such as an equivariant map in a category of group actions.

For left M-sets, $B^A$ consists of equivariant maps of monoid sets $M\times A\to B$ for the diagonal action on the domain. Its action is $(m\cdot f)(w,a)=f(wm,a)$ and evaluation is $f(1,a)$. The curry of an equivariant $h:C\times A\to B$ is $\hat{h}(c)(w,a)=h(w\cdot c,a)$. This construction can have noninjective action maps even when both $A$ and $B$ are decidable.

For the free monoid on $x,y$, act on $\mathbb{N}$ by adding word length and on $\{0,1\}$ trivially. Both actions are injective. The equivariant function $f(w,n)$ indicating words of length strictly greater than $n$ ending in $x$ is nonzero, but $y\cdot f=0$. Thus the exponential of monoid sets is not decidable. Strict inequality ensures invariance when the original word is empty.

If each $m$ admits $p,q$ with $pmq=p$, then $B^A$ is decidable whenever the left M-set $B$ is. For equivariant $f,g$, equality of all values at $(1,a)$ implies equality at $(m,a)$: apply $p$ and use $f(pm,pa)=pm\cdot f(1,qa)$, then cancel the injective action of $p$ on $B$. Equality after the exponential action of $m$ then gives equal traces by cancelling its action on $B$.

A functor $\mathcal{D}\times \mathcal{C}\to \mathcal{E}$ is uniquely a functor $\mathcal{D}\to [\mathcal{C},\mathcal{E}]$: fix its first coordinate and use its first-coordinate arrows as natural transformations. Uncurrying evaluates those transformations and second-coordinate arrows. The adjunction unit inserts a fixed first coordinate, and the adjunction counit is evaluation. This makes the category of small categories cartesian closed.

A full subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$ is an exponential ideal when $B^A$ belongs to $\mathcal{D}$ for every $B\in \mathcal{D}$ and every $A\in \mathcal{C}$.

For a reflective subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$, with reflector $L$, the subcategory $\mathcal{D}$ is an exponential ideal if and only if the canonical comparisonis an isomorphism for every $A,B$. The proof repeatedly curries a map into an object of $\mathcal{D}$ and factors it through the unit of the reflection.

An object $X$ in a category with finite products is exponentiable when $-\times X$ has a right adjoint $[X,-]$. The terminal object is exponentiable, and exponentiable objects are closed under binary products because right adjoints compose:

In a complete category that is a locally small category with a coseparator $S$ and all monomorphisms regular, an object $A$ is an exponentiable object exactly when $\mathcal{C}(-\times A,S)$ is a representable functor. A representing object $E$ gives representations $E^I$ for maps into powers $S^I$. An equalizer presentation by powers of a coseparator then constructs a representation for maps into every target. The Yoneda lemma makes these representations functorial and supplies the right adjoint.

If $A$ and $B$ are exponentiable objects, thenhas the composite of their exponential functors as a right adjoint. The terminal object supplies the empty product, so exponentiable objects are closed under finite products.

If the terminal object is also initial and $E$ is exponentiable, the left adjoint $-\times E$ preserves the initial object. But $0\times E\cong E$ because $0$ is terminal, while preservation of initiality gives $0\times E\cong 0$. Hence $E\cong 0$.

A $T_0$ space $E$ is exponentiable in ${\mathbf{Top}}_0$ if and only if the functor ${\mathbf{Top}}_0(-\times E,S)$ to sets is representable, where $S$ is the Sierpiński space. Every $T_0$ space is an equalizer of maps between powers of $S$, so a representing object for maps into $S$ constructs exponentials for every target by products and equalizers.

The exponential $B^A$ in the category of posets is the poset of monotone maps $A\to B$ ordered pointwise. Evaluation is monotone, and currying gives the cartesian-closed adjunction.

An object $A$ of a Cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.

The extra adjunction defining a tiny object should not be confused with the filtered-colimit condition for a compact object (mathematics).

Let $\mathcal{C}$ be a small Cauchy-complete category with initial object $0$. The terminal object of $[\mathcal{C},\mathbf{Set}]$ is $h_0$, so $\mathrm{Nat}(P,F)\cong (F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a left adjoint and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\mathrm{Nat}(P,-)$ preserves coproducts and epimorphisms, since every epimorphism in a set-valued functor category is the coequalizer of its kernel pair. Thus $P$ is an irreducible projective in a set-valued functor category, hence representable.

For a small category $\mathcal{C}$ with binary coproducts, exponentiation by $h_A=\mathcal{C}(A,-)$ in $[\mathcal{C},\mathbf{Set}]$ is naturally $F\mapsto F(-⨿A)$: the exponential at $B$ is $\mathrm{Nat}(h_B\times h_A,F)\cong \mathrm{Nat}(h_{B⨿A},F)\cong F(B⨿A)$ by the Yoneda lemma. This precomposition functor has a right adjoint given by Right Kan extension, so $h_A$ is a tiny object.

In a Cartesian closed category, exponentiation satisfies $(-{)}^{A\times B}\cong ((-{)}^B{)}^A$. If the two exponential functors have right adjoints, their composite has the composite of those right adjoints in reverse order. Therefore a product of tiny objects is tiny. The terminal object is tiny since exponentiation by it is the identity, covering the empty product.

If a small finite-product category $\mathcal{C}$ splits idempotents, the tiny objects of its presheaf topos are exactly the representable presheaves. Exponentiation by $yA$ is precomposition with $-\times A$ and therefore has a right adjoint given by Right Kan extension. Conversely, if $P$ is tiny, then $\mathrm{Nat}(P,-)$ preserves all colimits; idempotent completeness makes every such presheaf representable.