Exponential law for sets

2020 Mathematics Subject Classification: Primary: 03E20 [MSN][ZBL]

The correspondence between the sets $A^{B \times C}$ and $(A^B)^C$, where $X^Y$ denotes the set of all maps from the set $Y$ to the set $X$. Given $f \in A^{B \times C}$, that is, a pairing $f : B \times C \rightarrow A$, and given $c \in C$, let $f_c$ denote the map $f_c : B \rightarrow A$ by $f_c : b \mapsto f(b,c)$. Then $c \mapsto f_c$ defines a map from $A^{B \times C} \rightarrow (A^B)^C$. In the opposite direction, let $G \in (A^B)^C$. Given $b \in B$ and $c \in C$, define $g(b,c)$ to be $G(c)$ applied to $b$. Then $G \mapsto g$ defines a map from $(A^B)^C \rightarrow A^{B \times C}$. These two correspondences are mutually inverse.

In computer science, this operation is known as "Currying" after Haskell Curry (1900-1982).

In category-theoretic terms, the exponential law makes the category of sets a Cartesian-closed category.

In cardinal arithmetic, this corresponds to the formula $$ \mathfrak{a}^{\mathfrak{b}\mathfrak{c}} = (\mathfrak{a}^{\mathfrak{b}})^{\mathfrak{c}} \ . $$

There are further correspondences relating the sets $A^{B \coprod C}$ and $A^B \times A^C$ and the sets $(A \times B)^C$ and $A^C \times B^C$, corresponding to the formulae in cardinal arithmetic $\mathfrak{a}^{\mathfrak{b} + \mathfrak{c}} = \mathfrak{a}^{\mathfrak{b}}\mathfrak{a}^{\mathfrak{c}}$ and $(\mathfrak{a} \mathfrak{b})^{\mathfrak{c}} = \mathfrak{a}^{\mathfrak{c}} \mathfrak{b}^{\mathfrak{c}}$.

References

• Benjamin C. Pierce, Basic Category Theory for Computer Scientists, MIT Press (1991) ISBN 0262660717
• Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press (1999) ISBN 0-521-63107-6