In his 1967 thesis (but actually essentially completed by 1964) Beck defined an A-module, for an object A of a category _A_ to be an abelian group object in the slice _A_/A. This turned out to mean 2-sided A-modules, left A-modules, left A-modules and left A-modules in the categories of associative algebras, commutative (associative) algebras, groups and Lie algebras, resp., that is in all cases the desired coefficients for cohomology. This is all widely known among categorists. Has a full exposition of this ever been published and, if so, where? --Michael +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I too would like such an exposition. Hopefully it will also explain why modules may have tensor products. Also what about the dual formulation for say distributive categories, in particular explaining why the infinitesimal neighborhoods of the diagonal " can't be defined" for arbitrary objects in the gros Zariski topos. Bill +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I have been using modules of universal algebras in my work, and plan to continue doing so. The following papers of mine use the concept: Enveloping ringoids of universal algebras. Dissertation, University Microfilms International, 1992. The modules of a universal algebra A are the same as the left modules of the enveloping ringoid Z[A] of the universal algebra. An appendix treats modules in detail, including restriction and induction functors, the centralizer of a module, and some important examples. Enveloping ringoids. To appear in Algebra Universalis in Day conference special issue. Summary of thesis. Maximal subalgebras of CM algebras. In review. Classifies maximal subalgebras of CM algebras (e.g. groups, rings, Lie algebras, lattices) into 7 types. I was disappointed recently when the concept, which I feel is very important, was never mentioned at the July UACT conference at MSRI. (Perhaps if Dr. Barr had attended...) William H. Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Jon Beck told me that his thesis was never published and that he wrote quite a bit about the "modules" that never appeared even in the thesis. I guess if someone wanted to publish it as a technical report of some description, Beck might be persuaded to give them the manuscript. But this is off the top of my head, one would have to contact him and ask. Unfortunately I couldn't even try do this in Cambridge, as only the Computer Laboratory produces technical reports... Valeria de Paiva ------------------------------------------------------------------------------ Valeria de Paiva, | University of Cambridge | Phone: +44 (0)223 334418 Computer Laboratory | Fax: +44 (0)223 334678 New Museums Site, Pembroke Street | JANET: Valeria.Paiva@uk.ac.cam.cl Cambridge CB2 3QG, England, UK | Internet: Valeria.Paiva@cl.cam.ac.uk ------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (4)
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barr＠triples.Math.McGill.CA -
MTHFWL＠ubvms.cc.buffalo.edu -
rowan＠garnet.berkeley.edu -
Valeria de Paiva