Geodesic
On a strange homotopy category
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 31, Tome 455 (2017), pp. 33-41 
A. I. Generalov. On a strange homotopy category. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 31, Tome 455 (2017), pp. 33-41. http://geodesic.mathdoc.fr/item/ZNSL_2017_455_a3/
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     title = {On a~strange homotopy category},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

For an additive category $\mathcal C$ in which each morphism has a kernel, it is proved that the homotopy category of the category of complexes over $\mathcal C$ which are concentrated in degrees 2,1,0 and are exact in degrees 2 and 1 is abelian. Under assumption that a category $\mathcal C$ is abelian, earlier this result was obtained by considering the heart of a suitable $t$-structure on the homotopy category of $\mathcal C$.

[1] E. Backelin, O. Jaramillo, “Auslander–Reiten sequences and $t$-structures on the homotopy category of an abelian category”, J. Algebra, 339:1 (2011), 80–96 | DOI | MR | Zbl

[2] A. A. Beilinson, J. Bernstein, J. Deligne, Faisceaux pervers, Astérisque, 100, 1982 | MR | Zbl

[3] S. I. Gelfand, Yu. I. Manin, Metody gomologicheskoi algebry, v. I, Nauka, M., 1988 | MR

[4] T. H. Fay, K. A. Hardie, P. J. Hilton, “The two-square lemma”, Publ. Mat., 33:1 (1989), 133–137 | DOI | MR | Zbl