Exact Category - Examples

Examples

• Any abelian category is exact in the obvious way, according to the construction of #Motivation.
• A less trivial example is the category Ab_tf of torsion-free abelian groups, which is a strictly full subcategory of the (abelian) category Ab of all abelian groups. It is closed under extensions: if

is a short exact sequence of abelian groups in which are torsion-free, then is seen to be torsion-free by the following argument: if is a torsion element, then its image in is zero, since is torsion-free. Thus lies in the kernel of the map to, which is, but that is also torsion-free, so . By the construction of #Motivation, Ab_tf is an exact category; some examples of exact sequences in it are:

where the last example is inspired by de Rham cohomology ( and are the closed and exact differential forms on the circle group); in particular, it is known that the cohomology group is isomorphic to the real numbers. This category is not abelian.

• The following example is in some sense complementary to the above. Let Ab_t be the category of abelian groups with torsion (and also the zero group). This is additive and a strictly full subcategory of Ab again. It is even easier to see that it is stable under extensions: if

is an exact sequence in which have torsion, then naturally has all the torsion elements of . Thus it is an exact category; some examples of its exact sequences are

where in the second example, the means inclusion as the first summand, and in the last example, the means projection onto the second summand. One interesting feature of this category is that it illustrates that the notion of cohomology does not make sense in general exact categories: for consider the "complex"

which is obtained by pasting the marked arrows in the last two examples above. The second arrow is an admissible epimorphism, and its kernel is (from the last example), . Since the two arrows compose to zero, the first arrow factors through this kernel, and in fact the factorization is the inclusion as the first summand. Thus the quotient, if it were to exist, would have to be, which is not actually in Ab_t. That is, the cohomology of this complex is undefined.