2-quasi crossed modules of commutative algebras
type: Item
instance of: scholarly article
title: 2-quasi crossed modules of commutative algebras
zbMATH Open document ID: 1502.18031
published in: Ukrainian Mathematical Journal
publication date: 2022-11-08T00:00:00Z
review text: The aim of this paper is to construct the coproduct in the category of \(2\)-crossed modules of commutative algebras over a fixed tail. To do this, the author introduces the weaker notion of a \(2\)-quasi crossed module of commutative algebras, inspired by [\textit{P. Carrasco} and \textit{T. Porter}, Collect. Math. 67, No. 3, 485--517 (2016; Zbl 1348.18008)], and first constructs the analogous coproduct in this category. A \(2\)-crossed module of (non-unital) commutative algebras is a chain complex of \(R\)-algebras and morphisms \[ L\stackrel{\partial_2} {\rightarrow} E \stackrel{\partial _1}{\rightarrow} R, \] equipped with a Pfeiffer lifting \(\{ , \} : E \otimes _R E \rightarrow L\) that satisfies certain axioms; for instance, the restriction along \(\partial_2 \otimes \partial_2 : L \otimes _R L \rightarrow E\otimes _R E\) gives the product of \(L\). The category of \(2\)-crossed modules is denoted \(\mathrm{X_2 Mod}\) and the full subcategory of objects with given `tail' \(E \stackrel{\partial _1}{\rightarrow} R\) is denoted \( \mathrm{X_2 Mod}/ (E\rightarrow R)\). The author defines \(2\)-quasi crossed modules of commutative algebras similarly, replacing two of the axioms by a single, weaker axiom. These form a category \(\mathrm{Q X_2 Mod}\) and, for a fixed tail, one has the full subcategory \( \mathrm{Q X_2 Mod}/ (E\rightarrow R)\). The author shows that any \(2\)-crossed module is a \(2\)-quasi crossed module, so that there is an inclusion \( \mathrm{ X_2 Mod} \hookrightarrow \mathrm{Q X_2 Mod}\). He proves that this admits a left adjoint; this is given by replacing \(L\) in a \(2\)-quasi crossed module by the quotient \(L / [L,L]\) for the appropriate \(R\)-invariant ideal \([L,L]\). The category of \(2\)-crossed modules is equivalent to the category of simplicial commutative algebras with Moore complex of length two [\textit{Z. Arvasi} and \textit{T. Porter}, Theory Appl. Categ. 3, 1--23 (1997; Zbl 0874.18011)]; the author indicates an analogous result…
Mathematics Subject Classification ID: 18G45
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 7613707
zbMATH Keywords: crossed module
zbMATH Keywords: 2-crossed module
zbMATH Keywords: 2-quasi crossed module
zbMATH Keywords: coproduct
zbMATH Keywords: commutative algebras
zbMATH Keywords: Moore complex
author: Kadir Emir
reviewed by: Geoffrey Powell
MaRDI profile type: Publication
full work available at URL: s11253-022-02068-4
OpenAlex ID: W4307818380
cites work: Q4693062
cites work: Q4771408
cites work: Q4342435
cites work: Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups
cites work: Coproduct of 2-crossed modules: applications to a definition of a tensor product for 2-crossed complexes
cites work: Modules croisés généralisés de longueur 2
cites work: On the homology of commutative algebras
cites work: Q3744265
cites work: Simplicial methods for operads and algebraic geometry. Edited by Carles Casacuberta and Joachim Kock
cites work: Q4219215
cites work: Homology of commutative algebras and an invariant of Simis and Vasconcelos
cites work: On adding relations to homotopy groups
DOI: 10.1007/S11253-022-02068-4
description: scientific article; zbMATH DE number 7613707