nLab morphism




The notion of morphism in category theory is an abstraction of the notion of homomorphism.

In a general category, a morphism is an arrow between two objects. The word map is also sometimes used as a synonym.


Given two objects in a (locally small) category, say xx and yy, there is a set hom(x,y)hom(x,y), called a hom-set, whose elements are morphisms from xx to yy. Given a morphism ff in this hom-set, we write f:xyf:x \to y to indicate that it goes from xx to yy.

More generally, a morphism is what goes between objects in any n-category.


The most familiar example is the category Set, where the objects are sets and the morphisms are functions. Here if xx and yy are sets, a morphism f:xyf: x \to y is a function from xx to yy.

Last revised on November 16, 2022 at 11:52:01. See the history of this page for a list of all contributions to it.