The non-abelian tensor product of groups and related constructions

Abstract

The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [ 5, 6 ]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [ 10 ] and R. K. Dennis [ 7 ]. The tensor product of crossed complexes was introduced by R. Brown and the second author [ 3 ] in connection with the fundamental crossed complex π( X ) of a filtered space X , which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π( X ⊗ Y ) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.