Homotopies of 2-Algebra Morphisms

Year 2022, Volume: 5 Issue: 4, 170 - 179, 30.12.2022

İbrahim Akça Ummahan Ege Arslan

https://doi.org/10.33434/cams.1180773

Abstract

In [1] it is defined the notion of 2-algebra as a categorification of algebras, and shown that the category of strict 2-algebras is equivalent to the category of crossed modules in commutative algebras. In this paper we define the notion of homotopy for 2-algebras and we explore the relations of crossed module homotopy and 2-algebra homotopy.

Keywords

2-categories, Crossed modules, Homotopy.

References

• [1] ˙I. Akc¸a, U. E. Arslan, Categorification of algebras:2-algebras, Ikonion J. Math., Submitted.
• [2] ˙I. Akc¸a, K. Emir , F. M. Martins, Pointed homotopy of maps between 2-crossed modules of commutative algebras, Homol. Homotopy Appl., 18(1)(2016), 99-128.
• [3] Z. Arvasi , U. Ege , Annihilators, multipliers and crossed modules, Appl. Categ. Struct., 11 (2003), 487-506.
• [4] J. C. Baez , A. S. Crans , Higher dimensional algebra VI: Lie 2-Algebras, Theory Appl. Categ., 12(15) (2004), 492-538.
• [5] H. J. Baues , Combinatorial Homotopy and 4-Dimensional Complexes, Berlin etc.: Walter de Gruyter, 1991.
• [6] R. Brown , M. Golasinski, A model structure for the homotopy theory of crossed complexes, Cah. Topologie Geom. Diff´er. Cat´egoriques, 30(1) (1989),61-82.
• [7] R. Brown , C. Spencer , G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad.v. Wet, 79 (1976), 296-302.
• [8] F. Borceux , Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge, Cambridge U. Press, 1994.
• [9] J. G. Cabello, A. R. Garz´on, Closed model structures for algebraic models of n-types, J. Pure Appl. Algebra, 103(3) (1995), 287-302.
• [10] C. Ehresmann ,Categories structures, Ann. Ec. Normale Sup., 80 (1963).
• [11] C. Elvira-Donazar, L. J. Hernandez-Paricio, Closed model categories for the n-type of spaces and simplicial sets, Math. Proc. Camb. Philos. Soc., 118(7) (1995), 93-103.
• [12] J. W. Gray, Formal Category Theory Adjointness for 2-Categories, Lecture Notes in Math 391, Springer-Verlag, 1974.
• [13] ˙I.˙Ic¸en , The equivalence of 2-groupoids and crossed modules, Commun. Fac. Sci, Univ. Ank. Series A1, 49 (2000), 39-48.
• [14] E. Khmaladze, On associative and Lie 2-algebras, Proc. A. Razmadze Math. Inst., 159 (2012), 57-64.
• [15] J. L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Alg., 24 (1982), 179-202.
• [16] A. S. T. Lue, Semi-complete crossed modules and holomorphs of groups, Bull. London Math. Soc., 11 (1979), 8-16.
• [17] S. Mac Lane, Extension and obstructures for rings, Illinois J. Math., 121 (1958), 316-345.
• [18] K. J. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (1990), 129-146.
• [19] T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, J. Algebra, 109 (1987), 415-429.
• [20] T. Porter , The Crossed Menagerie: An Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, http://ncatlab.org/timporter/files/menagerie10.pdf