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Let $A$ be an additive category with kernels and cokernels. A morphism $f$ is called strict if the natural morphism from the coimage to the image is an isomorphism. In Schneiders: Quasi-abelian categories and sheaves one finds a proof that the composition of strict morphisms is strict, so the strict morphisms form a subcategory, which we denote by $A_{\mathrm{st}}$.

Is $A_{\mathrm{st}}$ abelian? Where can I find a proof or counterexample?

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  • $\begingroup$ No, all morphisms are strict in that sense in the category of groups, which is not Abelian. Not all monomorphisms are the kernel of their cokernel, there are not biproducts, etc. $\endgroup$
    – Fernando Muro
    Commented Apr 28, 2023 at 8:09
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    $\begingroup$ @FernandoMuro : the category of groups is not additive $\endgroup$
    – Maxime Ramzi
    Commented Apr 28, 2023 at 8:10
  • $\begingroup$ Ooops, I missed that requirement. $\endgroup$
    – Fernando Muro
    Commented Apr 28, 2023 at 8:10
  • $\begingroup$ Jochen Wengenroth has pointed out an "abstract counterexample" but it may be worth noting a more "pedestrian" counterexample based on the fact that Schneiders's QACS monograph explicitly mentions that Ban is an examble of a quasi-abelian category, and in Ban it is well known that following an isometric embedding (=strict mono) with a quotient map (=strict epi) you can get something which does not have closed range (hence not strict) $\endgroup$
    – Yemon Choi
    Commented Apr 30, 2023 at 18:35

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I do not believe that the composition of strict morphisms is strict, proposition 1.1.7 of Schneider's Quasi-Abelian Categories and Sheaves claims this only for strict epimorphisms as well as for strict monomorphisms.

I think that rather the opposite of the OP's claim is true:

If all compostions of strict morphisms in an additive category with kernels and cokernels are again strict, then the category is abelian.

To prove this we write an arbitrary morphism $f:X\to Y$ as a composition of a strict epimorphism and a strict monomorphism $$f=\pi_Y\circ [id_X,f]: X\to X\times Y\to Y.$$

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