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.

Let ${\displaystyle \mathbf {C} }$ be a category with binary products and let ${\displaystyle Z}$ and ${\displaystyle Y}$ be objects of ${\displaystyle \mathbf {C} }$. An object ${\textstyle Z^{Y}}$ together with a morphism ${\textstyle \mathrm {eval} \colon (Z^{Y}\times Y)\rightarrow Z}$ is an exponential object if for any object ${\displaystyle X}$ and morphism ${\textstyle g\colon X\times Y\to Z}$ there is a unique morphism ${\textstyle \lambda g\colon X\to Z^{Y}}$ (called the transpose of ${\displaystyle g}$) such that the following diagram commutes:

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