Editing Map (mathematics) (section)

==As morphisms==
{{Main article|Morphism}}

In category theory, "map" is often used as a synonym for "[[morphism]]" or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does.<ref>{{cite book |title=An Introduction to Category Theory |first=H. |last=Simmons |publisher=Cambridge University Press |year=2011 |isbn=978-1-139-50332-7 |page=2 |url=https://books.google.com/books?id=VOCQUC_uiWgC&pg=PA2 }}</ref> For example, a morphism <math>f:\, X \to Y</math> in a [[concrete category]] (i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source <math>X</math> of the morphism) and its codomain (the target <math>Y</math>). In the widely used definition of a function <math>f:X\to Y</math>, <math>f</math> is a subset of <math>X\times Y</math> consisting of all the pairs <math>(x,f(x))</math> for <math>x\in X</math>. In this sense, the function does not capture the set <math>Y</math> that is used as the codomain; only the range <math>f(X)</math> is determined by the function.