An exact category E is an additive category possessing a classE of "short exact sequences": triples of objects connected by arrows satisfying the following axioms inspired by the properties of short exact sequences in an abelian category:
E is closed under isomorphisms and contains the canonical sequences:
Suppose occurs as the second arrow of a sequence in E and is any arrow in E. Then their pullback exists and its projection to is also an admissible epimorphism. Dually, if occurs as the first arrow of a sequence in E and is any arrow, then their pushout exists and its coprojection from is also an admissible monomorphism. ;
Admissiblemonomorphisms are kernels of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible ;
Suppose is a map in E which admits a kernel in E, and suppose is any map such that the composition is an admissible epimorphism. Then so is Dually, if admits a cokernel and is such that is an admissible monomorphism, then so is
Admissible monomorphisms are generally denoted and admissible epimorphisms are denoted These axioms are not minimal; in fact, the last one has been shown by to be redundant. One can speak of an exact functor between exact categories exactly as in the case of exact functors of abelian categories: an exact functor from an exact category D to another one E is an additive functor such that if is exact in D, then is exact in E. If D is a subcategory of E, it is an exact subcategory if the inclusion functor is fully faithful and exact.
Motivation
Exact categories come from abelian categories in the following way. Suppose A is abelian and letE be any strictly fulladditive 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 classE 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 are stable under pullbacks : given an exact sequence of objects 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.
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, stable under extensions, and in which a sequence is in Eif and only if it is exact in A.
Examples
Any abelian category is exact in the obvious way, according to the construction of #Motivation.
A less trivial example is the category Abtf of torsion-free abelian groups, which is a strictly full subcategory of the category Ab of all abelian groups. It is closed under extensions: if
The following example is in some sense complementary to the above. Let Abt be the category of abelian groups with torsion. This is additive and a strictly full subcategory of Ab again. It is even easier to see that it is stable under extensions: if
