In this work, we define the quadratic modules for commutative algebras and give relations among 2-crossed modules, crossed squares, quadratic modules and simplicial commutative algebras with Moore complex of length 2.
We show how to construct a free or a totally free quadratic module on suitable construction data by using free simplicial groups. Similar freeness results are also explored for quadratic chain complexes.
We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras. M i 6 6 J J J J J J J J J J
In this work, we explain the relations among braided regular crossed modules, simplicial groups, 2-crossed modules, quadratic modules and crossed squares, and the role of hypercrossed complex pairings in these structures.
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