Exponential Object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law. Definition Let \mathbf be a category, let Z and Y be objects of \mathbf, and let \mathbf have all binary products with Y. An object Z^Y together with a morphism \mathrm\colon (Z^Y \times Y) \to Z is an ''exponential object'' if for any object X and morphism g \colon X\times Y \to Z there is a unique morphism \lambda g\colon X\to Z^Y (called the ''transpose'' of g) such that the following diagram commutes: This assignment of a unique \lambda g to each g establishes an isomorphism (bijection) of hom-sets, \mathrm(X\times Y,Z) \cong \mathrm(X,Z^Y). If Z^Yexists for all objects Z, Y in \mathbf, then the functor (-)^Y \colon \m ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no gener