Lambek invariants in a~p-semi-abelian category
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 332-341.

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We consider the well-known invariants $\mathrm{Ker}$ and $\mathrm{Img}$ for commutative squares in P-semi-abelian categories. These invariants were introduced by Lambek for groups and then studied in a more general context by Hilton and Nomura. In this paper, P-semi-abelian analogs are proved for Lambek's isomorphism and acyclic sequences that include these invariants are found.
Keywords: P-semi-abelian category, commutative square
Mots-clés : Lambek invariants.
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Ya. A. Kopylov. Lambek invariants in a~p-semi-abelian category. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 332-341. http://geodesic.mathdoc.fr/item/SEMR_2022_19_1_a13/

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

[2] I. Bucur, A. Deleanu, Introduction to the theory of categories and functors, Pure and Applied Mathematics, XIX, John, Wiley Sons, London etc, 1968 | MR | Zbl

[3] B. Eckmann, P.J. Hilton, “Exact couples in an Abelian category”, J. Algebra, 3 (1966), 38–87 | DOI | MR | Zbl

[4] S.I. Gelfand, Yu.I. Manin, Methods of homological algebra, Springer, Berlin, 2003 | MR | Zbl

[5] A.I. Generalov, “Derived categories of an additive category”, St. Petersbg. Math. J., 4:5 (1993), 909–919 | MR | Zbl

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

[7] R. Henrard, S. Kvamme, A.-C. van Roosmalen, S.-A. Wegner, The left heart and exact hull of an additive regular category, 2021, arXiv: 2105.11483 [math.CT] | Zbl

[8] P.J. Hilton, “On systems of interlocking exact sequences”, Fundam. Math., 61 (1967), 111–119 | DOI | MR | Zbl

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

[10] Ya.A. Kopylov, “On the Lambek invariants of commutative squares in a quasi-abelian category”, Sci. Ser. A Math. Sci. (N.S.), 11 (2005), 57–67 | MR | Zbl

[11] Ya.A. Kopylov, “Homology in P-semi-abelian categories”, Sci. Ser. A Math. Sci. (N.S.), 17 (2009), 105–114 | MR | Zbl

[12] Ya.A. Kopylov, “On the homology sequence in a $P$-semi-abelian category”, Sib. Èlektron. Mat. Izv., 9 (2012), 190–200 | MR | Zbl

[13] Ya.A. Kopylov, V.I. Kuz'minov, “Exactness of the cohomology sequence for a short exact sequence of complexes in a semiabelian category”, Sib. Adv. Math., 13:3 (2003), 72–80 | MR | Zbl

[14] Ya.A. Kopylov, V.I. Kuz'minov, “The Ker-Coker-sequence and its generalization in some classes of additive categories”, Sib. Math. J., 50:1 (2009), 86–95 | DOI | MR | Zbl

[15] Ya.A. Kopylov, S.-A. Wegner, “On the notion of a semi-abelian category in the sense of Palamodov”, Appl. Categ. Struct., 20:5 (2012), 531–541 | DOI | MR | Zbl

[16] Ya.A. Kopylov., S.-A. Wegner, “Exact couples in semiabelian categories revisited”, J. Algebra, 414 (2014), 264–270 | DOI | MR | Zbl

[17] V.I. Kuz'minov, A.Yu. Cherevikin, “Semiabelian categories”, Sib. Math. J., 13:6 (1972), 895–902 | DOI | MR | Zbl

[18] J. Lambek, “Goursat's theorem and homological algebra”, Can. Math. Bull., 7 (1964), 597–608 | DOI | MR | Zbl

[19] J.B. Leicht, “Axiomatic proof of J. Lambek's homological theorem”, Can. Math. Bull., 7 (1964), 609–613 | DOI | MR | Zbl

[20] B.M. Makarov, “Some pathological properties of inductive limits of B-spaces”, Usp. Mat. Nauk, 18:3(111) (1963), 171–178 | MR | Zbl

[21] A. Neeman, “The derived category of an exact category”, J. Algebra, 135:2 (1990), 388–394 | DOI | MR | Zbl

[22] Y. Nomura, “An exact sequence generalizing a theorem of Lambek”, Arch. Math., 22 (1971), 467–478 | DOI | MR | Zbl

[23] Y. Nomura, “Induced morphisms for Lambek invariants of commutative squares”, Manuscr. Math., 4 (1971), 263–275 | DOI | MR | Zbl

[24] V.P. Palamodov, “Homological methods in the theory of locally convex spaces”, Russ. Math. Surv., 26:1 (1972), 1–64 | DOI | MR | Zbl

[25] V.P. Palamodov, “On a Stein manifold the Dolbeault complex splits in positive dimensions”, Math. USSR, Sb., 17:1972 (1973), 289–316 | MR | Zbl

[26] W. Rump, “Almost abelian categories”, Cah. Topol. Géom. Différ. Catég., 42:3 (2001), 163–225 | MR | Zbl

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

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

[29] D. Sieg, S.-A. Wegner, “Maximal exact structures on additive categories”, Math. Nachr., 284:16 (2011), 2093–2100 | DOI | MR | Zbl

[30] L. Ubeda Bescansa, Teorema homologico de J. Lambek en una categoria Hofmaniana, Alxebra, 7, Dept. Algebra y Fundamentos, Univ. Santiago de Compostela, 1971 | MR | Zbl

[31] L. Ubeda Bescansa, “Invariantes de Lambek en categorías Hofmanianas”, Alxebra, 14 (1974), 1–34 | MR | Zbl

[32] J. Wengenroth, “The Raikov conjecture fails for simple analytical reasons”, J. Pure Appl. Algebra, 216:7 (2012), 1700–1703 | DOI | MR | Zbl

[33] A.V. Yakovlev, “Homological algebra in pre-Abelian categories”, J. Sov. Math., 19:1 (1982), 1060–1067 | DOI | Zbl