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Exact Functor

A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into short exact sequences.

A covariant functor is called left exact if it preserves the exactness of all sequences

0-->A-->B-->C,

and it is called right exact if it preserves the exactness of all sequences

A-->B-->C-->0.

("Left" and "right" are interchanged in the corresponding definitions for contravariant functors.)

A functor is exact iff it is both left and right exact.

Every tensor product functor is right exact. For every module M over a unit ring R, the covariant functor Hom_R(M,-) and the contravariant functor Hom_R(-,M) are left exact; the first is exact iff M is projective and the second iff M is injective.

See also

Flat module, Hom, Injective Module, Projective Module, Short Exact Sequence

This entry contributed by Margherita Barile

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References

Herrlich, H. and Strecker, G. E. Category Theory: An Introduction. Boston, MA: Allyn and Bacon, p. 202, 1973.

Referenced on Wolfram|Alpha

Exact Functor

Cite this as:

Barile, Margherita. "Exact Functor." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExactFunctor.html

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