related topics
{math, number, function}
{group, member, jewish}
{theory, work, human}

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]



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.

Full article ▸

related documents
Direct sum of modules
Integration by parts
Newton's method
Boolean satisfiability problem
Riemann zeta function
Series (mathematics)
List of trigonometric identities
Principal components analysis
Stone–Čech compactification
Johnston diagram
Forcing (mathematics)
Pascal's triangle
Russell's paradox
Cauchy sequence
Complete lattice
Ruby (programming language)
Denotational semantics
Bra-ket notation
Non-standard analysis
Mathematical induction
Numerical analysis
Kernel (algebra)
Cardinal number
Sequence alignment
Gaussian elimination
Entropy (information theory)
Huffman coding