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

• 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 $Ab$.