Exact category

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

${\displaystyle 0\to M^{\prime }\to M\to M^{″}\to 0}$

in A, then if ${\displaystyle M^{\prime },M^{″}}$ are in E, so is ${\displaystyle M}$. We can take the class E to be simply the sequences in E which are exact in A; that is,

${\displaystyle M^{\prime }\to M\to M^{″}}$

is in E iff

${\displaystyle 0\to M^{\prime }\to M\to M^{″}\to 0}$

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,

${\displaystyle 0\to M^{\prime }\stackrel{f}{\to }M\to M^{″}\to 0,}$

and a map ${\displaystyle N\to M^{″}}$ with ${\displaystyle N}$ in E, one verifies that the following sequence is also exact; since E is stable under extensions, this means that ${\displaystyle M{\times }_{M^{″}}N}$ is in E:

${\displaystyle 0\to M^{\prime }\stackrel{(f,0)}{\to }M{\times }_{M^{″}}N\to N\to 0.}$

• 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 ${\displaystyle M\to M^{″}}$ admits a kernel in E and if ${\displaystyle N\to M}$ is such that ${\displaystyle N\to M\to M^{″}}$ is an admissible epimorphism, then so is ${\displaystyle M\to M^{″}}$: 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.

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

${\displaystyle 0\to A\to B\to C\to 0}$

is a short exact sequence of abelian groups in which ${\displaystyle A,C}$ are torsion-free, then ${\displaystyle B}$ is seen to be torsion-free by the following argument: if ${\displaystyle b}$ is a torsion element, then its image in ${\displaystyle C}$ is zero, since ${\displaystyle C}$ is torsion-free. Thus ${\displaystyle b}$ lies in the kernel of the map to ${\displaystyle C}$, which is ${\displaystyle A}$, but that is also torsion-free, so ${\displaystyle b=0}$. By the construction of #Motivation, Ab_tf is an exact category; some examples of exact sequences in it are:

${\displaystyle 0\to \mathbb{Z}\stackrel{\left(\begin{array}{c}1\\ 2\end{array}\right)}{\to }{\mathbb{Z}}^2\stackrel{(-2,1)}{\to }\mathbb{Z}\to 0,}$

${\displaystyle 0\to d{\mathrm{\Omega }}^0(S^1)\to {\mathrm{\Omega }}_c^1(S^1)\to H_{\text{dR}}^1(S^1)\to 0,}$

where the last example is inspired by de Rham cohomology (${\displaystyle {\mathrm{\Omega }}_c^1(S^1)}$ and ${\displaystyle d{\mathrm{\Omega }}^0(S^1)}$ 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

${\displaystyle 0\to A\to B\to C\to 0}$

is an exact sequence in which ${\displaystyle A,C}$ have torsion, then ${\displaystyle B}$ naturally has all the torsion elements of ${\displaystyle A}$. Thus it is an exact category.