# Functor

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In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories.

Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, [1]who used the term in a linguistic context.[2]

## Contents

### Definition

Let C and D be categories. A functor F from C to D is a mapping that[3]

• associates to each object $X \in C$ an object $F(X) \in D$,
• associates to each morphism $f:X\rightarrow Y \in C$ a morphism $F(f):F(X) \rightarrow F(Y) \in D$ such that the following two conditions hold:
• $F(\mathrm{id}_{X}) = \mathrm{id}_{F(X)}\,\!$ for every object $X \in C$
• $F(g \circ f) = F(g) \circ F(f)$ for all morphisms $f:X \rightarrow Y\,\!$ and $g:Y\rightarrow Z.\,\!$

That is, functors must preserve identity morphisms and composition of morphisms.