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Keywords:
unitary magma; split extension; firm split extension; semidirect product
unitary magma; split extension; firm split extension; semidirect product
Summary:
We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) actions of unitary magmas on unitary magmas. The class of split extensions is pullback stable but not closed under composition. We introduce two subclasses of it that have both of these properties.
References:
[1] Borceux F., Bourn D.: Mal'cev, Protomodular, Homological and Semi-abelian Categories. Mathematics and Its Applications, 566, Kluwer Academic Publishers, Dordrecht, 2004. MR 2044291
[2] Borceux F., Janelidze G., Kelly G. M.: Internal object actions. Comment. Math. Univ. Carolin. 46 (2005), no. 2, 235–255. MR 2176890 | Zbl 1121.18004
[3] Bourn D.: Normalization equivalence, kernel equivalence and affine categories. Category Theory, Como, 1990, Lecture Notes in Math., 1488, Springer, Berlin, 1991, pages 43–62. DOI 10.1007/BFb0084212 | MR 1173004
[4] Bourn D., Janelidze G.: Protomodularity, descent, and semidirect products. Theory Appl. Categ. 4 (1998), no. 2, 37–46. MR 1615341
[5] Bourn D., Martins-Ferreira N., Montoli A., Sobral M.: Schreier Split Epimorphisms in Monoids and in Semirings. Textos de Matemática, Série B, 45, Universidade de Coimbra, Departamento de Matemática, Coimbra, 2013. MR 3157484
[6] Bourn D., Martins-Ferreira N., Montoli A., Sobral M.: Monoids and pointed S-protomodular categories. Homology Homotopy Appl. 18 (2016), no. 1, 151–172. MR 3485342
[7] Cigoli A., Mantovani S., Metere G.: A push forward construction and the comprehensive factorization for internal crossed modules. Appl. Categ. Structures 22 (2014), no. 5–6, 931–960. DOI 10.1007/s10485-013-9348-1 | MR 3275283
[8] Janelidze G., Márki L., Tholen W.: Semi-abelian categories. J. Pure Appl. Algebra 168 (2002), no. 2–3, 367–386. DOI 10.1016/S0022-4049(01)00103-7 | MR 1887164 | Zbl 0993.18008
[9] Mac Lane S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, 5, Springer, New York, 1998. MR 1712872 | Zbl 0906.18001
[10] Martins-Ferreira N., Montoli A.: On the “Smith is Huq" condition in S-protomodular categories. Appl. Categ. Structures 25 (2017), no. 1, 59–75. DOI 10.1007/s10485-015-9411-1 | MR 3606494
[11] Martins-Ferreira N., Montoli A., Sobral M.: Semidirect products and crossed modules in monoids with operations. J. Pure Appl. Algebra 217 (2013), no. 2, 334–347. DOI 10.1016/j.jpaa.2012.06.022 | MR 2969256
[12] Martins-Ferreira N., Montoli A., Sobral M.: The nine lemma and the push forward construction for special Schreier extensions of monoids with operations. Semigroup Forum 97 (2018), no. 2, 325–352. DOI 10.1007/s00233-018-9962-1 | MR 3852777
[13] Orzech G.: Obstruction theory in algebraic categories I. J. Pure Appl. Algebra 2 (1972), 287–314. DOI 10.1016/0022-4049(72)90008-4 | MR 0323859
[14] Patchkoria A.: Crossed semimodules and Schreier internal categories in the category of monoids. Georgian Math. J. 5 (1998), no. 6, 575–581. DOI 10.1023/B:GEOR.0000008133.94825.60 | MR 1654760
[15] Porter T.: Extensions, crossed modules and internal categories in categories of groups with operations. Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 3, 373–381. MR 0908444
[16] Yoneda N.: On $ Ext$ and exact sequences. J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507–576. MR 0225854
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