×

Central reflections and nilpotency in exact Mal’tsev categories. (English) Zbl 1397.18021

Summary: We study nilpotency in the context of exact Mal’tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie’s functor calculus. We show in particular that the reflection into the subcategory of \(n\)-nilpotent objects is the universal endofunctor of degree \(n\) if and only if every \(n\)-nilpotent object is \(n\)-folded. In the special context of a semi-abelian category, an object is \(n\)-folded precisely when its Higgins commutator of length \(n+1\) vanishes.

MSC:

18D35 Structured objects in a category (MSC2010)
17B30 Solvable, nilpotent (super)algebras
18E10 Abelian categories, Grothendieck categories
18G50 Nonabelian homological algebra (category-theoretic aspects)
20F18 Nilpotent groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baues, H.-J., Pirashvili, T.: Quadratic endofunctors of the category of groups. Adv. Math. 141, 167-206 (1999) · Zbl 0923.20042 · doi:10.1006/aima.1998.1784
[2] Berstein, I., Ganea, T.: Homotopical nilpotency. Ill. J. Math. 5, 99-130 (1961) · Zbl 0096.17602
[3] Biedermann, G., Dwyer, B.: Homotopy nilpotent groups. Algebr. Geom. Topol. 10, 33-61 (2010) · Zbl 1329.55008 · doi:10.2140/agt.2010.10.33
[4] Borceux, F., Bourn, D.: Mal’cev, Protomodular, Homological and Semi-abelian Categories, vol. 566. Kluwer Academic Publisher, Dordrecht (2004) · Zbl 1061.18001 · doi:10.1007/978-1-4020-1962-3
[5] Bourn, D.: Normalization Equivalence, Kernel Equivalence and Affine Categories. Lect. Notes Math., vol. 1488, pp. 43-62. Springer Verlag, Berlin (1991) · Zbl 0756.18007
[6] Bourn, D.: Mal’tsev categories and fibration of pointed objects. Appl. Categ. Struct. 4, 307-327 (1996) · Zbl 0856.18004 · doi:10.1007/BF00122259
[7] Bourn, D.: The denormalized \[3\times 33\]×3 lemma. J. Pure Appl. Algebra 177, 113-129 (2003) · Zbl 1032.18007 · doi:10.1016/S0022-4049(02)00143-3
[8] Bourn, D.: Commutator theory in regular Mal’tsev categories. AMS Fields Inst. Commun. 43, 61-75 (2004) · Zbl 1067.18002
[9] Bourn, D.: Commutator theory in strongly protomodular categories. Theory Appl. Categ. 13, 27-40 (2004) · Zbl 1068.18006
[10] Bourn, D.: On the monad of internal groupoids. Theory Appl. Categ. 28, 150-165 (2013) · Zbl 1273.18007
[11] Bourn, D., Gran, M.: Central extensions in semi-abelian categories. J. Pure Appl. Algebra 175, 31-44 (2002) · Zbl 1023.18013 · doi:10.1016/S0022-4049(02)00127-5
[12] Bourn, D., Gran, M.: Centrality and connectors in Maltsev categories. Algebra Universalis 48, 309-331 (2002) · Zbl 1061.18006 · doi:10.1007/s000120200003
[13] Bourn, D., Gray, J.R.A.: Aspects of algebraic exponentiation. Bull. Belg. Math. Soc. 19, 823-846 (2012) · Zbl 1264.18003
[14] Bourn, D., Rodelo, D.: Comprehensive factorization and \[II\] -central extensions. J. Pure Appl. Algebra 216, 598-617 (2012) · Zbl 1266.18001 · doi:10.1016/j.jpaa.2011.07.013
[15] Bruck, R.H.: A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Verlag, Berlin (1958) · Zbl 0081.01704
[16] Carboni, A., Janelidze, G.: Smash product of pointed objects in lextensive categories. J. Pure Appl. Algebra 183, 27-43 (2003) · Zbl 1029.18002 · doi:10.1016/S0022-4049(03)00034-3
[17] Carboni, A., Kelly, G.M., Pedicchio, M.C.: Some remarks on Mal’tsev and Goursat categories. Appl. Categ. Struct. 1, 385-421 (1993) · Zbl 0799.18002 · doi:10.1007/BF00872942
[18] Carboni, A., Lack, S., Walters, R.F.C.: Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84, 145-158 (1993) · Zbl 0784.18001 · doi:10.1016/0022-4049(93)90035-R
[19] Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Malcev categories. J. Pure Appl. Algebra 69, 271-284 (1991) · Zbl 0722.18005 · doi:10.1016/0022-4049(91)90022-T
[20] Cigoli, A., Gray, J.R.A., Van der Linden, T.: Algebraically coherent categories. Theory Appl. Categ. 30, 1864-1905 (2015) · Zbl 1366.18005
[21] Costoya, C., Scherer, J., Viruel, A.: A torus theorem for homotopy nilpotent groups. arXiv:1504.06100 · Zbl 1331.55007
[22] Doro, S.: Simple Moufang loops. Math. Proc. Camb. Philos. Soc. 83, 377-392 (1978) · Zbl 0381.20054 · doi:10.1017/S0305004100054669
[23] Eilenberg, S., Mac Lane, S.: On the groups \[H(\pi\] π,n). II. Methods of computation. Ann. Math. (2) 60, 49-139 (1954) · Zbl 0055.41704 · doi:10.2307/1969702
[24] Eldred, R.: Goodwillie calculus via adjunction and LS cocategory. arXiv:1209.2384 · Zbl 1360.55013
[25] Everaert, T., Van der Linden, T.: Baer invariants in semi-abelian categories I: general theory. Theory Appl. Categ. 12, 1-33 (2004) · Zbl 1065.18011 · doi:10.1023/B:APCS.0000013953.15330.92
[26] Everaert, T., Van der Linden, T.: A note on double central extensions in exact Mal’tsev categories. Cah. Topol. Géom. Differ. Catég. 51, 143-153 (2010) · Zbl 1215.18013
[27] Freese, R.S., McKenzie, R.N.: Commutator Theory for Congruence Modular Varieties. London Math. Soc. Lect. Note Series. Cambridge Univ. Press, Cambridge (1987) · Zbl 0636.08001
[28] Glauberman, G.: On loops of odd order II. J. Algebra 8, 383-414 (1968) · Zbl 0155.03901
[29] Goodwillie, T.G.: Calculus III. Taylor series. Geom. Topol. 7, 645-711 (2003) · Zbl 1067.55006 · doi:10.2140/gt.2003.7.645
[30] Gran, M.: Central extensions and internal groupoids in Maltsev categories. J. Pure Appl. Alg. 155, 139-166 (2001) · Zbl 0974.18007 · doi:10.1016/S0022-4049(99)00092-4
[31] Gran, M.: Applications of categorical Galois theory in universal algebra. AMS Fields Inst. Commun. 43, 243-280 (2004) · Zbl 1067.18011
[32] Gran, M., Rodelo, D.: Beck-Chevalley condition and Goursat categories. arXiv:1512.04066 · Zbl 1397.18024
[33] Gran, M., Linden, T.: On the second cohomology group in semi-abelian categories. J. Pure Appl. Algebra 212, 636-651 (2008) · Zbl 1136.18003 · doi:10.1016/j.jpaa.2007.06.009
[34] Grishkov, A.N., Zavarnitsine, A.V.: Groups with triality. J. Algebra Appl. 5, 441-463 (2006) · Zbl 1110.20023 · doi:10.1142/S021949880600182X
[35] Gray, J.R.A.: Algebraic exponentiation in general categories. Appl. Categ. Struct. 20, 543-567 (2012) · Zbl 1276.18002 · doi:10.1007/s10485-011-9251-6
[36] Gray, J.R.A.: Algebraic exponentiation for categories of Lie algebras. J. Pure Appl. Algebra 216, 1964-1967 (2012) · Zbl 1275.18021 · doi:10.1016/j.jpaa.2012.02.034
[37] Hall, J.I.: Central automorphisms, \[Z^*\] Z∗-theorems, and loop structures. Quasigroup Relat. Syst. 19, 69-108 (2011) · Zbl 1250.20060
[38] Hartl, M., Loiseau, B.: On actions and strict actions in homological categories. Theory Appl. Categ. 27, 347-392 (2013) · Zbl 1275.18012
[39] Hartl, M., Linden, T.: The ternary commutator obstruction for internal crossed modules. Adv. Math. 232, 571-607 (2013) · Zbl 1258.18007 · doi:10.1016/j.aim.2012.09.024
[40] Hartl, M., Vespa, C.: Quadratic functors on pointed categories. Adv. Math. 226, 3927-4010 (2011) · Zbl 1235.18002 · doi:10.1016/j.aim.2010.11.008
[41] Higgins, P.J.: Groups with multiple operators. Proc. Lond. Math. Soc. 6, 366-416 (1956) · Zbl 0073.01704 · doi:10.1112/plms/s3-6.3.366
[42] Hovey, M.: Lusternik-Schnirelmann cocategory. Ill. J. Math. 37, 224-239 (1993) · Zbl 0802.55003
[43] Huq, S.A.: Commutator, nilpotency and solvability in categories. Quart. J. Math. Oxford 19, 363-389 (1968) · Zbl 0165.03301 · doi:10.1093/qmath/19.1.363
[44] Janelidze, G., Kelly, G.M.: Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97, 135-161 (1994) · Zbl 0813.18001 · doi:10.1016/0022-4049(94)90057-4
[45] Janelidze, G., Kelly, G.M.: Central extensions in universal algebra: a unification of three notions. Algebra Universalis 44, 123-128 (2000) · Zbl 1013.08009 · doi:10.1007/s000120050174
[46] Janelidze, G., Kelly, G.M.: Central extensions in Mal’tsev varieties. Theory Appl. Categ. 7, 219-226 (2000) · Zbl 0956.08002
[47] Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367-386 (2002) · Zbl 0993.18008 · doi:10.1016/S0022-4049(01)00103-7
[48] Janelidze, G., Sobral, M., Tholen, W.: Beyond Barr exactness: effective descent morphisms. Camb. Univ. Press Encycl. Math. Appl. 97, 359-405 (2004) · Zbl 1047.18010
[49] Jibladze, M., Pirashvili, T.: Linear extensions and nilpotence of Maltsev theories. Contrib. Algebra Geom. 46, 71-102 (2005) · Zbl 1081.18008
[50] Johnson, B., McCarthy, R.: A classification of degree \[n\] n functors I/II. Cah. Topol. Géom. Différ. Catég. 44, 2-38, 163-216 (2003) · Zbl 1025.18009
[51] Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. E. N. S. 71, 101-190 (1954) · Zbl 0055.25103
[52] Liebeck, M.W.: The classification of finite simple Moufang loops. Math. Proc. Camb. Philos. Soc. 102, 33-47 (1987) · Zbl 0622.20061 · doi:10.1017/S0305004100067025
[53] Mal’cev, A.I.: On the general theory of algebraic systems. Mat. Sbornik N. S. 35, 3-20 (1954) · Zbl 0057.02403
[54] Mantovani, S., Metere, G.: Normalities and commutators. J. Algebra 324, 2568-2588 (2010) · Zbl 1218.18001 · doi:10.1016/j.jalgebra.2010.07.043
[55] Mostovoy, J.: Nilpotency and dimension series for loops. Comm. Algebra 36, 1565-1579 (2008) · Zbl 1151.20052 · doi:10.1080/00927870701864189
[56] Moufang, R.: Zur Struktur von Alternativkörpern. Math. Ann. 110, 416-430 (1935) · JFM 60.0093.02 · doi:10.1007/BF01448037
[57] Pedicchio, M.C.: A categorical approach to commutator theory. J. Algebra 177, 647-657 (1995) · Zbl 0843.08004 · doi:10.1006/jabr.1995.1321
[58] Penon, J.: Sur les quasi-topos. Cah. Topol. Géom. Diff. 18, 181-218 (1977) · Zbl 0401.18002
[59] Quillen, D.: Homotopical Algebra. Lect. Notes Math., vol. 43. Springer Verlag, Berlin (1967) · Zbl 0168.20903
[60] Smith, J.D.H.: Mal’cev Varieties. Lect. Notes Math., vol. 554. Springer Verlag, Berlin (1976) · Zbl 0344.08002 · doi:10.1007/BFb0095447
[61] Stanovsky, D., Vojtěchovskỳ, P.: Commutator theory for loops. J. Algebra 399, 290-322 (2014) · Zbl 1319.20056 · doi:10.1016/j.jalgebra.2013.08.045
[62] Van der Linden, T.: Simplicial homotopy in semi-abelian categories. J. K-theory 4, 379-390 (2009) · Zbl 1190.18007 · doi:10.1017/is008008022jkt070
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.