Exact additive category

Exact additive category is a category which is both additive and exact.

1. An additive category definition can be found in the attached reference^[1]

2. Exact categories properties are related to those of abelian categories in the following way. assume $Ab$ to be an abelian category and let $E$ be any strictly full additive subcategory which is closed under taking extensions in the sense that given an exact sequence

0 \to M' \to M \to M'' \to 0\

in $Ab$ , and then if $M', M''$ are in $E$ , so is $M$ . We can take the class $E$ to be simply the sequences in $E$ which are exact in $Ab$ ; that is,

M' \to M \to M''\

is in $E$ iff

0 \to M' \to M \to M'' \to 0\

is exact in $Ab$ . Then $E$ is an exact category in the following sense.

3. An exact category, $E$ , is an additive category possessing a class Es of "short exact sequences", that is, triples of objects connected by arrows

M' \to M \to M''\

satisfying the following axioms that are related to the properties of short exact sequences of an abelian category:

- $E$ is closed under isomorphisms and contains the canonical ("split exact") sequences:

M' \rightarrow M' \oplus M''\rightarrow M'';

- Admissible epimorphisms (respectively, admissible monomorphisms) are stable under pullbacks (resp. pushouts): given an exact sequence of objects in $E$ ,

0 \to M' \xrightarrow{f} M \to M'' \to 0,\

and a map

N \to M''

with

N

in $E$ , one verifies that the following sequence is also exact; since $E$ is stable under extensions, this means that

M \times_{M''} N

is in $E$ :

0 \to M' \xrightarrow{(f,0)} M \times_{M''} N \to N \to 0.\

Every admissible monomorphism is the kernel of its corresponding admissible epimorphism, and vice-versa: this is true as morphisms in A, and E is a full subcategory.
If $M \to M''$ admits a kernel in E and if $N \to M$ is such that $N \to M \to M''$ is an admissible epimorphism, then so is $M \to M''$ : See Quillen (1972).

Note

Conversely, if $E$ is any exact category, we can take $Ab$ to be the category of left-exact functors from $E$ into the category of abelian groups, which is itself abelian and in which $E$ is a natural subcategory (via the Yoneda embedding, since Hom is left exact), stable under extensions, and in which a sequence is in Es if and only if it is exact in $Ab$ . $E$ is any exact category, we can take $Ab$ to be the category of left-exact functors from $E$ into the category of abelian groups, which is itself abelian and in which $E$ is a natural subcategory (via the Yoneda embedding, since the Hom functor is left exact), stable under extensions, and in which a sequence is in Es if and only if it is exact in $Ab$ .

References

↑ http://images.planetmath.org/cache/objects/7922/pdf/AdditiveCategory.pdf Additive Category definition.

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