Exact Category - Motivation

Motivation

Exact categories come from abelian categories in the following way. Suppose A is abelian and let E be any strictly full additive subcategory which is closed under taking extensions in the sense that given an exact sequence

in A, then if are in E, so is . We can take the class E to be simply the sequences in E which are exact in A; that is,

is in E iff

is exact in A. Then E is an exact category in the above sense. We verify the axioms:

• E is closed under isomorphisms and contains the split exact sequences: these are true by definition, since in an abelian category, any sequence isomorphic to an exact one is also exact, and since the split sequences are always exact in A.
• Admissible epimorphisms (respectively, admissible monomorphisms) are stable under pullbacks (resp. pushouts): given an exact sequence of objects in E,

and a map with in E, one verifies that the following sequence is also exact; since E is stable under extensions, this means that is in E:

• 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 admits a kernel in E and if is such that is an admissible epimorphism, then so is : See Quillen (1972).

Conversely, if E is any exact category, we can take A 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 E if and only if it is exact in A.