Split (Pre)crossed Modules over $R$-Algebroids

Abstract

In this paper, we introduce (d-)split and (d-)split$^{+}$ epimorphisms and (d-)split and d-split$^{\ast}$ (pre)crossed modules in the context of algebroids. Moreover, we examine their categorical properties, and in particular, we give a necessary and sufficient condition for a morphism of pre-$R$-algebroids to be a d-split precrossed module and a necessary and sufficient condition for a d-split$^{\ast}$ precrossed module to be a crossed module. In addition, we examine the hierarchical relations between the categories obtained and look over some results for split (pre)crossed modules over associative $R$-algebras, as a reduced case.

Keywords

• Algebroids
• crossed modules
• split epimorphisms.

References

1. Akça, İ.İ., Avcıoğlu, O., Equivalence between (pre)cat1-R-algebroids and (pre)crossed modules of R-algebroids, Bull. Math. Soc. Sci. Math. Roumanie, Tome 65(113)(2022), no. 3, 267–288.
2. Alp, M., Pullback crossed modules of algebroids, Iran. J. Sci. Technol., 32(A1)(2008), 1–5.
3. Alp, M., Pushout crossed modules of algebroids, Iran. J. Sci. Technol., 32(A3)(2008), 175–181.
4. Amgott, S.M., Separable categories, J. Pure Appl. Algebra, 40(1986), 1–14.
5. Arvasi, Z., Porter, T., Simplicial and crossed resolutions of commutative algebras, Journal of Algebra, 181(2)(1996), 426–448.
6. Arvasi, Z., Crossed modules of algebras, Math. Comput. Appl., 9(2)(2004), 173–182.
7. Avcıoğlu, O., Homotopies of crossed modules of R-algebroids, Applied Categorical Structures, 29(2021), 827–847.
8. Avcıoğlu, O., Akça, İ.İ., On generators of Peiffer ideal of a pre-R-algebroid in a precrossed module and applications, New Trends in Mathematical Sciences, 5(4)(2017), 148–155.

9. Avcıoğlu, O., Akça, İ.İ., Free modules and crossed modules of R-algebroids, Turk. J. Math., 42(6)(2018), 2863–2875.
10. Brown, R., Higgins, P.J., Sivera, R., Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, EMS Tracts in Mathematics, 15, European Mathematical Society (EMS), Z¨urich, 2011.
11. Brown, R., İçen, İ., Homotopies and automorphisms of crossed modules of groupoids, Applied Categorical Structures, 11(2003), 185–206.
12. Brown, R., Spencer, C.B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagationes Mathematicae (Proceedings), 79(4)(1976), 296–302.
13. B¨ohm, G., Crossed modules of monoids II: Relative crossed modules, Applied Categorical Structures, 28(2020), 601–653.
14. Carrasco, P., Porter, T., Coproduct of 2-crossed modules: applications to a definition of a tensor product for 2-crossed complexes, Collect. Math., 67(2016), 485–517.
15. Dedecker, P., Lue, A.S.-T., A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72(1966), 1044–1050.
16. Ege Arslan, U., Akça, İ.İ., Onarlı Irmak, G., Avcıoğlu, O., Fibrations of 2-crossed modules, Math. Meth. Appl. Sci., 42(2019), 5293–5304.
17. Gerstenhaber, M., On the deformation of rings and algebras: II, Annals of Mathematics, 84(1)(1966), 1–19.
18. Janalidze, G., Internal Crossed Modules, Georgian Mathematical Journal, 10(1)(2003), 99–114.
19. Lavendhomme, R., Roisin, J.R., Cohomologie non ab´elienne de structures alg´ebriques, Journal of Algebra, 67(2)(1980), 385–414.
20. Lichtenbaum, S., Schlessinger, M., The cotangent complex of a morphism, Trans. Amer. Math. Soc., 128(1967), 41–70.
21. Lue, A.S.-T., Non-abelian cohomology of associative algebras, The Quarterly Journal of Mathematics, 19(1)(1968), 159–180.
22. Mitchell, B., Rings with several objects, Advances in Mathematics, 8(1)(1972), 1–161.
23. Mitchell, B., Some applications of module theory to functor categories, Bull. Amer. Math. Soc., 84(1978), 867–885.
24. Mitchell, B., Separable algebroids, Mem. Amer. Math. Soc., 57, no. 333, iv+96pp., 1985.
25. Mosa, G.H., Higher dimensional algebroids and crossed complexes, Ph.D. Thesis, University of Wales, Bangor, 1986.
26. Peiffer, R., U¨ ber Identita¨ten zwischen Relationen, Math. Ann., 121(1949), 67–99.
27. Reidemeister, K., U¨ ber Identita¨ten von Relationen, Abh. Math. Semin. Univ. Hambg., 16(1949), 114–118.
28. Porter, T., Homology of commutative algebras and an invariant of simis and vasconcelos, Journal of Algebra, 99(2)(1986), 458–465.
29. Porter, T., Some categorical results in the theory of crossed modules in commutative algebras, Journal of Algebra, 109(2)(1987), 415–429.
30. Porter, T., The Crossed Manegarie: an introduction to crossed gadgetry and cohomology in algebra and topology. 17 April 2018. https://ncatlab.org/nlab/files/menagerie12a.pdf
31. Shammu, N.M., Algebraic and categorical structure of categories of crossed modules of algebras, Ph.D. Thesis, University of Wales, Bangor, 1992.
32. Temel, S., S¸ ahan, T., Mucuk, O., Crossed modules, double group-groupoids and crossed squares, Filomat, 34(6)(2020), 1755–1769.
33. Whitehead, J.H.C., On adding relations to homotopy groups, Annals of Mathematics, 42(2)(1941), 409–428.
34. Whitehead, J.H.C., Note on a previous paper entitled ”On adding relations to homotopy groups.”, Annals of Mathematics, 47(4)(1946), 806–810.