Concept

# Functor

Summary
In mathematics, specifically , a functor is a mapping between . Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context; see function word. Definition Let C and D be . A functor F from C to D is a mapping that
• associates each object X in C to an object F(X) in D,
• associates each morphism f \colon X \to Y in C to a morphism F(f) \colon F(X) \to F(Y) in D such that the following two conditio
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