nLab
morphism

Contents

Idea

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.

Definition

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.

Examples

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.

Revised on November 18, 2014 10:37:29 by Urs Schreiber (217.155.201.6)