Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

Pointed homotopy of maps between 2-crossed modules of commutative algebras

Pages: 99 – 128

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a6

Authors

İ. İlker Akça (Department of Mathematics and Computer Science, Faculty of Science and Letters, Eskişehir Osmangazi University, Eskişehir, Turkey)

Kadir Emir (Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal; and Department of Mathematics and Computer Science, Faculty of Science and Letters, Eskişehir Osmangazi University, Eskişehir, Turkey)

João Faria Martins (Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal)

Abstract

We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.

Keywords

simplicial commutative algebra, crossed module of commutative algebras, 2-crossed module of commutative algebras, quadratic derivation

2010 Mathematics Subject Classification

Primary 55U10. Secondary 18D05, 18D20, 55Q15.

Full Text (PDF format)

Published 31 May 2016