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*:
*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 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 id 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.

main page – abbreviations – journals – comments – other
sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 11
jul 2014