PlanetMath: exact functor

(more info)

Math for the people, by the people.

Main Menu

exact functor

(Definition)

A covariant functor is said to be left exact if whenever

$\displaystyle 0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$

is an exact sequence, then

$\displaystyle 0 \to FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC$

is also an exact sequence.

A covariant functor is said to be right exact if whenever

$\displaystyle A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$

is an exact sequence, then

$\displaystyle FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC \to 0$

is also an exact sequence.

A contravariant functor is said to be left exact if whenever

$\displaystyle A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$

is an exact sequence, then

$\displaystyle 0 \to FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA$

is also an exact sequence.

A contravariant functor is said to be right exact if whenever

$\displaystyle 0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$

is an exact sequence, then

$\displaystyle FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA \to 0$

is also an exact sequence.

A (covariant or contravariant) functor is said to be exact if it is both left exact and right exact.

"exact functor" is owned by antizeus.

(view preamble)

Other names:

left exact functor, right exact functor

Log in to rate this entry.
(view current ratings)

Cross-references: right, exact sequence, covariant functor
There are 7 references to this entry.

This is version 3 of exact functor, born on 2002-01-05, modified 2003-09-20.
Object id is 1362, canonical name is ExactFunctor.
Accessed 5561 times total.

Classification:

18A22 (Category theory; homological algebra :: General theory of categories and functors :: Special properties of functors )

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

No messages.

Interact