Functors| Functors |
In General > s.a. category theory.
* Idea: Something that takes a space
(or a bunch of spaces) into a new one, and morphisms into morphisms.
> Online resources: see Wikipedia
page.
Covariant
$ Of one variable: A map C:
A → B and Mor(A) → Mor(B) between
two categories, such that for all X, Y ∈ A, and all
f ∈ Hom(X, Y), f*:=
C(f) ∈ Hom(C(X), C(Y)), and
composition and the 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 : A → B between two categories, such
that for all X, Y ∈ A, and all f
∈ Hom(X,Y), f *:= C(f)
∈ Hom(C(Y), C(X)), and composition
and the identity 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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send feedback and suggestions to bombelli at olemiss.edu – modified 14 apr 2018