ONT Re: Category Theory
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CAT. Note 23
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| 1.4. Natural Transformations
|
| Given two functors S, T : C -> B, a 'natural transformation'
| !t! : S -> T is a function which assigns to each object c of C
| an arrow !t!_c = !t!c : Sc -> Tc of B in such a way that every
| arrow f : c -> c' in C yields a diagram:
|
| !t!c
| c o Sc o------------>o Tc
| | | |
| | | |
| f | Sf | | Tf (1)
| | | |
| v v v
| c' o Sc' o------------>o Tc'
| !t!c'
|
| which is commutative. When this holds, we also say that
| !t!_c : Sc -> Tc is 'natural' in c. If we think of the
| functor S as giving a picture in B of (all the objects
| and arrows of) C, then a natural transformation !t! is
| a set of arrows mapping (or, translating) the picture S
| to the picture T, with all squares (and parallegrams!)
| like that above commutative:
|
| a Sa !t!a Ta
| o o---------------------->o
| |\ |\ |\
| | \ f | \ Sf | \ Tf
| | \ | \ | \
| | v | v Sb Th | v
| h | o b Sh | o------------------|--->o Tb
| | / | / !t!b | /
| | / | / | /
| | / g | / Sg | / Tg
| vv vv vv
| o o---------------------->o
| c Sc !t!c Tc
|
| We call !t!a, !t!b, !t!c, ..., the 'components'
| of the natural transformation !t!.
|
| A natural transformation is often called a 'morphism of functors';
| a natural transformation !t! with every component !t!c invertible in
| B is called a 'natural equivalence' or better a 'natural isomorphism';
| in symbols, !t! : S ~=~ T. In this case, the inverses (!t!c)^(-1) in B
| are the components of a natural isomorphism !t!^(-1) : T -> S.
|
| Mac Lane, 'Cat Work Math', p. 16.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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