Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras

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 ve...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2015-10
Main Authors: İ. lker Akça, adir Emir, oão Faria Martins
Format: Article
Language:English
Subjects:
Algebra
Equivalence
Homotopy theory
Modules
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1411.6931