5
$\begingroup$

Let $\operatorname{Coim}f$ be the cokernel of the kernel pair of an arrow. Let $\operatorname{Im}f$ be the kernel of the cokernel pair.

An interesting property of a category is the isomorphy of the canonical arrow $\operatorname{Coim}\to \operatorname{Im}$. For a regular category this is equivalent to being balanced, so e.g holds for groups but fails for commutative rings.

Another interesting property of a pointed category is the isomorphy of the canonical arrow $\operatorname{Coker}\operatorname{ker}\to \operatorname{Ker}\operatorname{coker}$. This property has the important consequence that monomorphy is measured by the kernel.

I would like to have nice names for these properties but cannot really think of any good ones. At first I thought of calling the former one "the first isomorphism theorem", but then realized the first isomorphism is really about $\operatorname{Coker}\operatorname{ker}\cong \operatorname{Coim}$.

I did think about calling $\operatorname{Coker}\operatorname{ker}$ and $\operatorname{Ker}\operatorname{coker}$ respectively the linear regular coimage and linear regular image. Is this a good idea? Any suggestions in general?

$\endgroup$
2
  • 1
    $\begingroup$ The property $\operatorname{Coim} \overset{\sim}{\to} \operatorname{Im}$ is AB2 in Grothendieck's Tôhoku paper. $\endgroup$ Dec 9, 2016 at 15:46
  • 1
    $\begingroup$ You can find some terminology of this sort in A. Deitmar, Belian categories, Far East J. Math. Sci. 70 (2012), 1-46. $\endgroup$ Feb 6, 2017 at 20:05

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.