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functor (Definition)

Given two categories $ \mathcal{C}$ and $ \mathcal{D}$, a covariant functor $ T:\mathcal{C}\to\mathcal{D}$ consists of an assignment for each object $ X$ of $ \mathcal{C}$ an object $ T(X)$ of $ \mathcal{D}$ (i.e. a “function” $ T:{\rm Ob}(\mathcal{C})\to{\rm Ob}(\mathcal{D})$) together with an assignment for every morphism $ f\in{\rm Hom}_{\mathcal{C}}(A,B)$, to a morphism $ T(f)\in{\rm Hom}_{\mathcal{D}}(T(A),T(B))$, such that:

  • $ T(1_A) = 1_{T(A)}$ where $ 1_X$ denotes the identity morphism on the object $ X$ (in the respective category).
  • $ T(g \circ f) = T(g)\circ T(f)$, whenever the composition $ g\circ f$ is defined.

A contravariant functor $ T :\mathcal{C}\to\mathcal{D}$ is just a covariant functor $ T:\mathcal{C}^{\rm op}\to\mathcal{D}$ from the opposite category. In other words, the assignment reverses the direction of maps. If $ f\in{\rm Hom}_{\mathcal{C}}(A,B)$, then $ T(f)\in{\rm Hom}_{\mathcal{D}}(T(B),T(A))$ and $ T(g\circ f) = T(f)\circ T(g)$ whenever the composition is defined (the domain of $ g$ is the same as the codomain of $ f$).

Given a category $ \mathcal{C}$ and an object $ X$ we always have the functor $ T : \mathcal{C}\to{\bf Sets}$ to the category of sets defined on objects by $ T(A) = {\rm Hom}(X, A)$. If $ f : A \to B$ is a morphism of $ \mathcal{C}$, then we define $ T(f) : {\rm Hom}(X,A)\to {\rm Hom}(X,B)$ by $ g\mapsto f\circ g$. This is a covariant functor, denoted by $ {\rm Hom}(X,-)$.

Similarly, one can define a contravariant functor $ {\rm Hom}(-,X) :\mathcal{C}\to{\bf Sets}$.



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See Also: endofunctor, monad

Other names:  covariant functor, contravariant functor
Keywords:  category

Attachments:
identity functor (Definition) by CWoo
diagonal functor (Definition) by CWoo
constant functor (Definition) by CWoo
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Cross-references: category of sets, codomain, domain, maps, opposite category, composition, identity, morphism, object, categories
There are 108 references to this entry.

This is version 3 of functor, born on 2001-12-12, modified 2002-02-03.
Object id is 1093, canonical name is Functor.
Accessed 15400 times total.

Classification:
AMS MSC18-00 (Category theory; homological algebra :: General reference works )

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