2-quasi crossed modules of commutative algebras
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 for \(2\)-quasi crossed modules. He constructs the categorical coproduct in \( \mathrm{Q X_2 Mod}/ (E\rightarrow R)\) by generalizing the construction of the coproduct of crossed modules of commutative algebras. He then obtains the coproduct in \( \mathrm{X_2 Mod}/ (E\rightarrow R)\) using the above adjunction.
scientific article; zbMATH DE number 7613707