Home Homotopies of crossed complex morphisms of associative R-algebras
Article
Licensed
Unlicensed Requires Authentication

Homotopies of crossed complex morphisms of associative R-algebras

  • İbrahim İlker Akça EMAIL logo and Osman Avcıoğlu
Published/Copyright: December 4, 2019
Become an author with De Gruyter Brill

Abstract

In this study, given two crossed complexes 𝒞 and 𝒟 of associative R-algebras and a crossed complex morphism f:𝒞𝒟, we construct a homotopy as a pair (H,f), where H=(Hn) is a sequence of R-linear maps Hn:CnDn+1. Then we show that for a fixed pair 𝒞 and 𝒟 of crossed complexes of associative R-algebras, the family of all homotopies between crossed complex morphisms from 𝒞 to 𝒟 has a groupoid structure with crossed complex morphisms as objects and homotopies as morphisms.

References

[1] İ. İ. Akça, K. Emir and J. Faria Martins, Pointed homotopy of maps between 2-crossed modules of commutative algebras, Homology Homotopy Appl. 18 (2016), no. 1, 99–128. 10.4310/HHA.2016.v18.n1.a6Search in Google Scholar

[2] Z. Arvasi, Crossed modules of algebras, Math. Comput. Appl. 9 (2004), no. 2, 173–182. 10.3390/mca9020173Search in Google Scholar

[3] Z. Arvasi and M. Koçak, Crossed N-cubes and n-crossed complexes of commutative algebras, Turkish J. Math. 22 (1998), no. 2, 127–143. Search in Google Scholar

[4] Z. Arvasi and T. Porter, Simplicial and crossed resolutions of commutative algebras, J. Algebra 181 (1996), no. 2, 426–448. 10.1006/jabr.1996.0128Search in Google Scholar

[5] H. J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, De Gruyter Exp. Math. 2, Walter de Gruyter, Berlin, 1991. 10.1515/9783110854480Search in Google Scholar

[6] A. L. Blakers, Some relations between homology and homotopy groups, Ann. of Math. 49 (1948), no. 2, 428–461. 10.2307/1969290Search in Google Scholar

[7] R. Brown and M. Golasiński, A model structure for the homotopy theory of crossed complexes, Cah. Topol. Géom. Différ. Catég. 30 (1989), no. 1, 61–82. Search in Google Scholar

[8] R. Brown and P. J. Higgins, Crossed complexes and non-abelian extensions, Category Theory Proceedings (Gummersbach 1981), Lecture Notes in Math. 962, Springer, Berlin (1982), 39–50. 10.1007/BFb0066884Search in Google Scholar

[9] R. Brown and P. J. Higgins, Tensor products and homotopies for ω-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), no. 1, 1–33. 10.1016/0022-4049(87)90099-5Search in Google Scholar

[10] R. Brown and P. J. Higgins, Crossed complexes and chain complexes with operators, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 1, 33–57. 10.1017/S0305004100068353Search in Google Scholar

[11] R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology, EMS Tracts Math. 15, European Mathematical Society (EMS), Zürich, 2011. 10.4171/083Search in Google Scholar

[12] R. Brown and İ. İçen, Homotopies and automorphisms of crossed modules of groupoids, Appl. Categ. Structures 11 (2003), no. 2, 185–206. 10.1023/A:1023544303612Search in Google Scholar

[13] D. Conduché, Modules croisés généralisés de longueur 2, J. Pure Appl. Algebra 34 (1984), no. 2–3, 155–178. 10.1016/0022-4049(84)90034-3Search in Google Scholar

[14] G. J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra 52 (1988), no. 3, 277–282. 10.1016/0022-4049(88)90095-3Search in Google Scholar

[15] C. Elvira-Donazar and L.-J. Hernandez-Paricio, Closed model categories for the n-type of spaces and simplicial sets, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 93–103. 10.1017/S0305004100073485Search in Google Scholar

[16] M. Gerstenhaber, On the deformation of rings and algebras: II, Ann. of Math. 84 (1966), no. 1, 1–19. 10.2307/1970528Search in Google Scholar

[17] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302–313. 10.1007/BF02566688Search in Google Scholar

[18] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967), 41–70. 10.1090/S0002-9947-1967-0209339-1Search in Google Scholar

[19] J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982), no. 2, 179–202. 10.1016/0022-4049(82)90014-7Search in Google Scholar

[20] A. S.-T. Lue, Non-abelian cohomology of associative algebras, Quart. J. Math. 19 (1968), no. 1, 159–180. 10.1093/qmath/19.1.159Search in Google Scholar

[21] G. H. Mosa, Higher dimensional algebroids and crossed complexes, Ph.D. Thesis, University of Wales, Bangor, 1986. Search in Google Scholar

[22] T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconcelos, J. Algebra 99 (1986), no. 2, 458–465. 10.1016/0021-8693(86)90038-4Search in Google Scholar

[23] T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, J. Algebra 109 (1987), no. 2, 415–429. 10.1016/0021-8693(87)90147-5Search in Google Scholar

[24] T. Porter, The Crossed Manegarie: an introduction to crossed gadgetry and cohomology in algebra and topology (Notes initially prepared for the XVI Encuentro Rioplasente de Álgebra y GeometrÄ́±a Algebraica, in Buenos Aires, December 12-15, 2006), 2018, https://ncatlab.org/nlab/files/menagerie12a.pdf. Search in Google Scholar

[25] N. M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Ph.D. Thesis, University of Wales, Bangor, 1992. Search in Google Scholar

[26] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. 42 (1941), no. 2, 409–428. 10.2307/1968907Search in Google Scholar

[27] J. H. C. Whitehead, Note on a previous paper entitled “On adding relations to homotopy groups”, Ann. of Math. 47 (1946), no. 4, 806–810. 10.2307/1969237Search in Google Scholar

[28] J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949), no. 5, 453–496. 10.1090/S0002-9904-1949-09213-3Search in Google Scholar

Received: 2017-03-13
Revised: 2018-04-17
Accepted: 2018-04-26
Published Online: 2019-12-04
Published in Print: 2021-04-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 1.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2019-2065/html?lang=en
Scroll to top button