Exponential object

In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. An exponential object may also be called a power object or map object (but note that the term "power object" means something different in topos theory, analogous to "power set"; see power set for a simplified explanation).

Definition

Let C be a category with binary products and let Y and Z be objects of C. The exponential object Z^Y can be defined as a universal morphism from the functor – × Y to Z. (The functor – × Y from C to C maps objects X to X × Y and morphisms φ to φ × id_Y.)

Explicitly, the definition is as follows. An object Z^Y, together with a morphism

$\mathrm{eval}\colon (Z^Y \times Y) \rightarrow Z$

is an exponential object if for any object X and morphism g : (X × Y) → Z there is a unique morphism

$\lambda g\colon X\to Z^Y$

such that the following diagram commutes:

If the exponential object Z^Y exists for all objects Z in C, then the functor that sends Z to Z^Y is a right adjoint to the functor – × Y. In this case we have a natural bijection between the hom-sets

$\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y).$

(Note: In functional programming languages, the morphism eval is often called apply, and the syntax $\lambda g$ is often written curry(g). The morphism eval here must not to be confused with the eval function in some programming languages, which evaluates quoted expressions.)

The morphisms $g$ and $\lambda g$ are sometimes said to be exponential adjoints of one another.^[1]

It should be noted that for $A,B$ in the category of sets, $A^B = \mathrm{Hom}(B,A)$.^[2]

Examples

In the category of sets, the exponential object $Z^Y$ is the set of all functions from $Y$ to $Z$. The map $\mathrm{eval}\colon (Z^Y \times Y) \to Z$ is just the evaluation map, which sends the pair (f, y) to f(y). For any map $g\colon (X \times Y) \rightarrow Z$ the map $\lambda g\colon X\to Z^Y$ is the curried form of $g$:

$\lambda g(x)(y) = g(x,y).\,$

In the category of topological spaces, the exponential object Z^Y exists provided that Y is a locally compact Hausdorff space. In that case, the space Z^Y is the set of all continuous functions from Y to Z together with the compact-open topology. The evaluation map is the same as in the category of sets. If Y is not locally compact Hausdorff, the exponential object may not exist (the space Z^Y still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to be cartesian closed. However, the category of locally compact topological spaces is not cartesian closed either, since Z^Y need not be locally compact for locally compact spaces Z and Y. A cartesian closed category of spaces is for example given by the full subcategory spanned by the compactly generated Hausdorff spaces.

References

• Adámek, Jiří; Horst Herrlich; George Strecker (2006) [1990]. Abstract and Concrete Categories (The Joy of Cats). John Wiley & Sons.

1. Goldblatt, Robert (1984). "Chapter 3: Arrows instead of epsilon". Topoi : the categorial analysis of logic. Studies in Logic and the Foundations of Mathematics #98 (Revised ed.). North-Holland. p. 72. ISBN 978-0-444-86711-7.
2. Mac Lane, Saunders (1978). "Chapter 4: Adjoints". categories for the working mathematician. graduate texts in mathematics (2nd ed.). Springer-Verlag. p. 98. ISBN 978-0387984032.

External links

• Interactive Web page which generates examples of exponential objects and other categorical constructions. Written by Jocelyn Paine.