Functors  

In General > s.a. category theory.
* Idea: Something that takes a space (a bunch of spaces) into a new space, and morphisms into morphisms.
> Online resources: see Wikipedia page.

Covariant
$ Of one variable: A map C: AB and Mor(A) → Mor(B) between two categories, such that for all X, YA, and all f ∈ Hom(X, Y), f*:= C(f) ∈ Hom(C(X), C(Y)), and composition and identity are preserved, C(gf) = C(g) C(f) and C(idX) = idC(X).
* Examples: C: Top → Top defined by C(X):= X × X, C(f):= f × f ; HomR(A, · ): Rmod → AbelGr; T : Man → Man, defined by T(M):= TM, the tangent bundle, T(f):= f*.

Contravariant
$ Of one variable: A map C : AB between two categories, such that for all X, YA, and all f ∈ Hom(X,Y), f *:= C(f) ∈ Hom(C(Y), C(X)), and composition and id are preserved, C(gf) = C(f) C(g) and C(idX) = idC(X).
* Examples: HomR( · , B): Rmod → AbelGr; T*: Man → Man, defined by T*(M):= T*M, the cotangent bundle, T*(f):= f *.
* Properties: Functors take equivalences into equivalences (easy to show).
* Composition: Contravariant functors can be composed, but their composition is a covariant functor, etc.

Special Types of Functors
* Duality: A contravariant functor with an inverse; Every category is the domain (and the range) of some duality.
* Forgetful: A functor from a category to another whose structure is less rich.

And Physics > see canonical quantum theory; category theory in physics.


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