×

Duality in non-abelian algebra. IV. Duality for groups and a universal isomorphism theorem. (English) Zbl 07060775

Summary: Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a self-dual context which allows one to establish the same theorems in the case of non-abelian group-like structures; the question of whether such a context can be found has been left open for seventy years. We also formulate and prove in our context a universal isomorphism theorem from which all other isomorphism theorems can be deduced.

MSC:

20A05 Axiomatics and elementary properties of groups
20J15 Category of groups
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
08A30 Subalgebras, congruence relations
06A15 Galois correspondences, closure operators (in relation to ordered sets)
18D30 Fibered categories
18E10 Abelian categories, Grothendieck categories
18G50 Nonabelian homological algebra (category-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Barr, M., Exact Categories, Lecture Notes in Mathematics, vol. 236, 1-120, (1971), Springer: Springer Berlin
[2] Beutler, E.; Kaiser, H.; Matthiessen, G.; Timm, J., Biduale algebren, (1979), Universität Bremen, Mathematik Arbeitspapiere Nr. 21 · Zbl 0493.08001
[3] Borceux, F.; Bourn, D., Mal’cev, Protomodular, Homological and Semi-Abelian Categories, Math. Appl., vol. 566, (2004), Kluwer · Zbl 1061.18001
[4] Borceux, F.; Grandis, M., Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra, 208, 665-689, (2007) · Zbl 1109.18006
[5] Bourn, D., Normalization Equivalence, Kernel Equivalence, and Affine Categories, Lecture Notes in Mathematics, vol. 1488, 43-62, (1991), Springer · Zbl 0756.18007
[6] Brümmer, G., Topological categories, Topology Appl., 18, 27-41, (1994) · Zbl 0551.18003
[7] Buchsbaum, D. A., Exact categories and duality, Trans. Amer. Math. Soc., 80, 1, 1-34, (1955) · Zbl 0065.25502
[8] Chao, K.; Jacobs, B.; Westerbaan, B.; Westerbaan, B., Quotient-comprehension chains, EPTCS, 195, 136-147, (2015)
[9] Dedekind, R., Über Zerlegungen von Zahlen Durch Ihre Grössten Gemeinsamen Theiler, (Fest-Schrift der Herzoglichen Technischen Hochschule Carolo-Wilhelmina, (1897)), 1-40
[10] Grandis, M., Transfer functors and projective spaces, Math. Nachr., 118, 147-165, (1984) · Zbl 0556.18005
[11] Grandis, M., On the categorical foundations of homological and homotopical algebra, Cah. Topol. Géom. Différ. Catég., 33, 135-175, (1992) · Zbl 0814.18006
[12] Grandis, M., Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups, (2012), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore · Zbl 1280.18001
[13] Grandis, M., Homological Algebra in Strongly Non-Abelian Settings, (2013), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore · Zbl 1280.18002
[14] Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku Math. J., 9, 119-221, (1957) · Zbl 0118.26104
[15] Grothendieck, A., Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, (Séminaire N. Bourbaki 5, (1959)), 299-327, Exposé 190
[16] Jacobson, N., Basic Algebra I, (1985), W. H. Freeman and Company: W. H. Freeman and Company New York · Zbl 0557.16001
[17] Janelidze, G.; Márki, L., Kurosh-Amitsur radicals via a weakened Galois connection, Comm. Algebra, 31, 241-258, (2003) · Zbl 1025.17003
[18] Janelidze, G.; Márki, L., A simplicial approach to factorization systems and Kurosh-Amitsur radicals, J. Pure Appl. Algebra, 213, 2229-2237, (2009) · Zbl 1182.18013
[19] Janelidze, G.; Márki, L.; Tholen, W., Semi-abelian categories, J. Pure Appl. Algebra, 168, 367-386, (2002) · Zbl 0993.18008
[20] Janelidze, Z., On the form of subgroups in semi-abelian and regular protomodular categories, Appl. Categ. Structures, 22, 755-766, (2014) · Zbl 1309.18008
[21] Janelidze, Z.; Weighill, T., Duality in non-abelian algebra I. From cover relations to Grandis ex2-categories, Theory Appl. Categ., 29, 11, 315-331, (2014) · Zbl 1305.18028
[22] Janelidze, Z.; Weighill, T., Duality in non-abelian algebra II. From Isbell bicategories to Grandis exact categories, J. Homotopy Relat. Struct., 11, 553-570, (2016) · Zbl 1359.18004
[23] Janelidze, Z.; Weighill, T., Duality in non-abelian algebra III. Normal categories and 0-regular varieties, Algebra Universalis, 77, 1-28, (2017) · Zbl 1359.18002
[24] Mac Lane, S., Groups, categories and duality, Proc. Natl. Acad. Sci. USA, 34, 263-267, (1948)
[25] Mac Lane, S., Duality for groups, Bull. Amer. Math. Soc., 56, 485-516, (1950)
[26] Mac Lane, S., Homology, Die Grundlehren der mathematischen Wissenschaften, vol. 114, (1963), Academic Press, Inc., Publishers/Springer-Verlag: Academic Press, Inc., Publishers/Springer-Verlag New York/Berlin-Göttingen-Heidelberg · Zbl 0818.18001
[27] Mac Lane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0906.18001
[28] Mac Lane, S.; Birkhoff, G. D., Algebra, (1999), AMS Chelsea Publishing
[29] Mitchell, B., Theory of Categories, Pure and Applied Mathematics, vol. 17, (1965), Academic Press
[30] Noether, E., Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Math. Ann., 96, 26-61, (1927) · JFM 52.0130.01
[31] Ore, O., On the foundation of abstract algebra I, Ann. of Math., 36, 406-437, (1935) · JFM 61.0111.09
[32] Ore, O., On the foundation of abstract algebra II, Ann. of Math., 37, 265-292, (1936) · JFM 62.1099.08
[33] Ore, O., Galois connexions, Trans. Amer. Math. Soc., 55, 493-513, (1944) · Zbl 0060.06204
[34] Puppe, D., Korrespondenzen in abelschen Kategorien, Math. Ann., 148, 1, 1-30, (1962) · Zbl 0109.25201
[35] Schmidt, R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics, vol. 14, (1994) · Zbl 0843.20003
[36] Schreier, O., Über den Jordan-Hölderschen Satz, Abh. Math. Semin. Univ. Hambg., 6, 300-302, (1928) · JFM 54.0147.05
[37] Street, R., Fibrations in bicategories, Cah. Topol. Géom. Différ. Catég., 21, 111-160, (1980) · Zbl 0436.18005
[38] Tsalenko, M. S., Correspondences over a quasi exact category, Mat. Sb. (N.S.), 73, 115, 564-584, (1967) · Zbl 0164.01401
[39] Ursini, A., Osservazioni sulle variet a BIT, Boll. Unione Mat. Ital., 7, 205-211, (1973) · Zbl 0274.08005
[40] van Niekerk, F. K., Contributions to Projective Group Theory, (2017), Stellenbosch University, MSc Thesis
[41] Weighill, T., Bifibrational Duality in Non-Abelian Algebra and the Theory of Databases, (2014), Stellenbosch University, MSc Thesis
[42] Wyler, O., Weakly exact categories, Arch. Math., XVII, 9-19, (1966) · Zbl 0163.01503
[43] Wyler, O., The Zassenhaus lemma for categories, Arch. Math., XVII, 561-569, (1971) · Zbl 0254.18004
[44] Zassenhaus, H., Zum Satz von Jordan-Hölder-Schreier, Abh. Math. Semin. Univ. Hambg., 10, 106-108, (1934) · JFM 60.0081.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.