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*(id_{X})
= id_{C(X)}.

* __Examples__: *C*:
Top → Top defined by *C*(*X*):= *X* × *X*,
*C*(*f*):= *f* × *f* ;
Hom_{R}(*A*, · ):
*R*mod → AbelGr; *T* : Man → Man,
defined by *T*(*M*):= T*M*, 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*(id_{X})
= id_{C(X)}.

* __Examples__:
Hom_{R}( · , *B*): *R*mod
→ 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