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natural transformation
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(Definition)
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Let
 and
 be categories, and let
 be functors. A natural transformation
 is a function that carries each object  of
 to a morphism
 of
 , and such that for any morphism  of
 , the diagram
is commutative. If every  happens to be an isomorphism, then  is called a natural isomorphism, a natural equivalence, or an isomorphism of functors.
Natural transformations arise frequently in mathematics. One example is the boundary map
 in a homology theory. By definition, every morphism of presheaves is a natural transformation. More prosaically, the determinant
 is natural.
If
 is a
 -category, then we can also define the functor category
 ; the objects of
 are the functors
 , and the morphisms are the natural transformations
 . The composition of two composable functions which are natural transformations is again a natural transformation, and so
 is a category.
- 1
- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
- 2
- S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer-Verlag, 1997.
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"natural transformation" is owned by mps. [ full author list (2) | owner history (1) ]
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Cross-references: composition, determinant, presheaves, homology, boundary map, frequently in, isomorphism, morphism, object, function, functors, categories
There are 37 references to this entry.
This is version 12 of natural transformation, born on 2002-01-23, modified 2006-09-15.
Object id is 1570, canonical name is NaturalTransformation.
Accessed 15875 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) | | | 18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{& S(A) \ar[d]^{Sf} \ar[r]^{\tau_A} & T(A) \ar[d]^{Tf} \ & S(B) \ar[r]^{\tau_{B}} & T(B) } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/1570/l2h/img11.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{& S(A) \ar[d]^{Sf} \ar[r]^{\tau_A} & T(A) \ar[d]^{Tf} \ & S(B) \ar[r]^{\tau_{B}} & T(B) } } \end{xy}$")