Exponential object ($B^A$)

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.