caa a22 4500 386371911 CHVBK 20180307112001.0 cr unu---uuuuu 161130e198901 xx s 000 0 eng 10.1017/S0017089500007515 doi S0017089500007515 pii (NATIONALLICENCE)cambridge-10.1017/S0017089500007515 The non-abelian tensor product of groups and related constructions [Elektronische Daten] 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. Copyright © Glasgow Mathematical Journal Trust 1989 Gilbert N. D. Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd LL57 2UW Higgins P. J. Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE Glasgow Mathematical Journal Cambridge University Press 31/1(1989-01), 17-29 0017-0895 31:1<17 1989 31 GMJ https://doi.org/10.1017/S0017089500007515 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0017089500007515 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gilbert N. D. Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd LL57 2UW NATIONALLICENCE 700 1- Higgins P. J. Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE NATIONALLICENCE 773 0- Glasgow Mathematical Journal Cambridge University Press 31/1(1989-01), 17-29 0017-0895 31:1<17 1989 31 GMJ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge