Geodesic
The Two-Square Lemma and the connecting morphism in a~preabelian category
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 3, pp. 316-325.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994. We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.
Keywords: strict morphism, preabelian category, pullback, pushout, Snake Lemma, connecting morphism.
Mots-clés : semi-stable (co)kernel 
@article{JSFU_2012_5_3_a2,
     author = {Yaroslav A. Kopylov},
     title = {The {Two-Square} {Lemma} and the connecting morphism in a~preabelian category},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {316--325},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a2/}
}
TY  - JOUR
AU  - Yaroslav A. Kopylov
TI  - The Two-Square Lemma and the connecting morphism in a~preabelian category
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2012
SP  - 316
EP  - 325
VL  - 5
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a2/
LA  - en
ID  - JSFU_2012_5_3_a2
ER  - 
%0 Journal Article
%A Yaroslav A. Kopylov
%T The Two-Square Lemma and the connecting morphism in a~preabelian category
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2012
%P 316-325
%V 5
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a2/
%G en
%F JSFU_2012_5_3_a2
Yaroslav A. Kopylov. The Two-Square Lemma and the connecting morphism in a~preabelian category. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 3, pp. 316-325. http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a2/

[1] T. Bühler, “Exact categories”, Expo. Math., 28:1 (2010), 1–69 | DOI | MR

[2] A. I. Generalov, “The Ker-Coker sequence for pre-abelian categories”, Abelian Grouus and Modules, 11–12, Tomsk. Gos. Univ., Tomsk, 1994, 78–89 (Russian) | MR

[3] Sib. Math. J., 43:1 (2002), 28–35 | DOI | MR | Zbl

[4] Sib. Math. J., 41:3 (2000), 509–517 | DOI | MR | Zbl

[5] Sib. Math. J., 50:1 (2009), 86–95 | DOI | DOI | MR | Zbl

[6] F. Borceux, D. Bourn, Mal'cev, Protomodular, Homological and Semi-abelian Categories, Mathematics and its Applications, 566, Kluwer Academic Publishers, Dordrecht, 2004 | MR | Zbl

[7] M. Grandis, “On the categorical foundations of homological and homotopical algebra”, Cah. Topol. Géom. Différ. Catég., 33:2 (1992), 135–175 | MR | Zbl

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

[9] G. M. Kelly, “Monomorphisms, epimorphisms and pullbacks”, J. Austral. Math. Soc., 9 (1969), 124–142 | DOI | MR | Zbl

[10] F. Richman, E. A. Walker, “Ext in pre-Abelian categories”, Pacific J. Math., 71:2 (1977), 521–535 | MR | Zbl

[11] I. Bucur, A. Deleanu, Introduction to the Theory of Categories and Functors, Pure and Applied Mathematics, 19, Interscience Publication John Wiley Sons, Ltd., London–New York–Sydney, 1968 | MR

[12] J. Bonet, S. Dierolf, “The pullback for bornological and ultrabornological spaces”, Note Mat., 25:1 (2006), 63–67 | MR | Zbl

[13] F. Prosmans, “Derived categories for functional analysis”, Publ. Res. Inst. Math. Sci., 36:1 (2000), 19–83 | DOI | MR | Zbl

[14] W. Rump, “A counterexample to Raikov's conjecture”, Bull. London Math. Soc., 40:6 (2008), 985–994 | DOI | MR | Zbl

[15] W. Rump, “Analysis of a problem of Raikov with applications to barreled and bornological spaces”, J. of Pure Appl. Algebra, 215:1 (2011), 44–52 | DOI | MR | Zbl

[16] J.-P. Schneiders, Quasi-Abelian Categories and Sheaves, Mém. Soc. Math. Fr. (N.S.), 76, 1999 | MR

[17] St. Petersburg Math. Jour., 4:1 (1993), 93–113 | MR | Zbl

[18] Russ. Math. Surv., 26:1 (1971), 1–64 | DOI | MR | Zbl

[19] Sib. Math. J., 13:6 (1972), 895–902 | DOI | MR

[20] Ya. A. Kopylov, S.-A. Wegner, “On the notion of a semi-abelian category in the sense of Palamodov”, Appl. Categor. Struct., Published online 25 March 2011 | DOI

[21] J. Soviet Mathematics, 19:1 (1982), 1060–1067 | DOI | MR | Zbl | Zbl

[22] Sib. Math. J., 50:5 (2009), 867–873 | DOI | MR | Zbl

[23] R. Beyl, “The connecting morphism in the kernel-cokernel sequence”, Arch. Math., 32 (1979), 305–308 | DOI | MR | Zbl

[24] S. Maclane, Categories for the Working Mathematician, Graduate Texts in Math., 5, New York–Heidelberg–Berlin, 1972 | MR