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Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$

Statement

For every set $A$ , the product functor $- \times A : S e t \to S e t$ is left adjoint to the function-set functor $(-)^{A}$ . Naturally in $X$ and $Y$ ,

$S e t (X \times A, Y) ≅ S e t (X, Y^{A}) .$

The bijection sends $h$ to $\hat{h} (x) (a) = h (x, a)$ and sends $k$ to $\overset{ˇ}{k} (x, a) = k (x) (a)$ .

Facts & Assumptions

Given: Sets $A, X, Y$ .

[F1]

The functions $A \to Y$ form the set $Y^{A}$ (The set $B^{A}$ of all functions $A \to B$ ).

[F2]

The function collection between two sets is a set (For sets $A$ and $B$ the collection of all functions $A \to B$ is a set, being a subset of $P (A \times B)$ ).

[F3]

The cartesian product $X \times A$ consists of ordered pairs $(x, a)$ with $x \in X$ , $a \in A$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ).

[F4]

Sets and functions form the locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[L1]

Natural hom-set bijections determine an adjunction (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1F1F3construct

For $h : X \times A \to Y$ , define $\hat{h} : X \to Y^{A}$ by $\hat{h} (x) (a) = h (x, a)$ ; [F1] and [F3] make this a well-defined function.

1.2F1F3construct

For $k : X \to Y^{A}$ , define $\overset{ˇ}{k} : X \times A \to Y$ by $\overset{ˇ}{k} (x, a) = k (x) (a)$ .

2.1step 1.1step 1.2F2

For all $(x, a)$ , $\overset{ˇ}{\hat{h}} (x, a) = h (x, a)$ , so function extensionality gives $\overset{ˇ}{\hat{h}} = h$ ; similarly $\hat{\overset{ˇ}{k}} (x) (a) = k (x) (a)$ gives $\hat{\overset{ˇ}{k}} = k$ .

2.2step 1.1step 1.2algebra

Precomposition in $X$ and postcomposition in $Y$ commute with evaluation at $(x, a)$ , so the bijection is natural in both variables.

3.1step 2.1step 2.2F4L1∎

Since $S e t$ is locally small by [F4], [L1] applies and gives $- \times A ⊣ (-)^{A}$ . The formulas also cover $A = \emptyset$ without exception.

Depends on

Used by

An adjunction read as a duality of endofunctors Example
Evaluation of functions is dinatural in its argument set Example
The state monad threads a fixed state set through a computation Example
A presheaf category on a small category is cartesian closed Theorem
Set is cartesian closed Theorem
Set is locally cartesian closed Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.4.9 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Example 2.1.6 (standard reference, not scraped)

Currying gives the adjunction −×A⊣(−)A−×A⊣(−)A in SetSet