JOURNAL OF ALGEBRA 179, 936]963Ž. 1996 ARTICLE NO. 0046 On Modules and Crossed Modules R. Lavendhomme and Th. Lucas Uni¨ersite Catholique de Lou¨ain, B-1348 Lou¨ain-la-Neu¨e, Belgium Communicated by Walter Feit View metadata, citation and similar papersReceived at core.ac.uk December 21, 1994 brought to you by CORE provided by Elsevier - Publisher Connector INTRODUCTION This is a paper without any result, it is just a meditation on the concept of module. Besides the aesthetic appeal of such questions, we find some justification in that our concept of a crossed module covers classical examples, gives a unified proof that the semi-direct product is an internal category, allows computations of non-abelian cocycles, and gives general meaning to algebras of derivations. Let G be a familiar equational algebraic theory, such as the theory of groups or the theory of rings. The theory G is described as a set of equations written in a one-sorted language L. An interpretation of L is given by a non-empty set A and interpretations in A of the operation symbols of A. A model of G is an interpretation of L which moreover satisfies the axioms of G. On the other hand, the well-known example of modules is often de- scribed as a model of a theory written in a 2-sorted language. We prefer here to consider them as 2-sorted interpretations of the 1-sorted theory of rings. This allows us to present in a unified setting basic notions such as modules, crossed modules, and derivation algebras. One of the intuitions we want to convey to the reader is that the definition of a crossed module is almost forced by the decision to pass from a 1-sorted interpretation to a 2-sorted interpretation of the same 1-sorted language. Section 1 will present our notions of a modulated model of a theory G and of a crossed G-module. Section 2 is devoted to examples. We show in particular that our notion of a crossed G-module covers the classical examples of Lie crossed mod- ules and crossed modules in groups. Section 3 defines semi-direct products under general conditions, gives examples, and proves that semi-direct products are internal categories. 936 0021-8693r96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. MODULES AND CROSSED MODULES 937 In Section 4, we comment on the notion of two-cocycle and check that with our presentation of crossed modules in groups, the general notion of two-cocycle and the classical notion coincide. In Section 5, we show that Lie R-algebra of derivations, the group of automorphisms of a group, MacLane's R-algebra of bimultiplications, etc., are all particular instances of the same fundamental concept. The notion of a crossed module originates in algebraic topology and more particularly in homotopy theory: see for example Mac Lane and Whitehead's basic paperwx 7 . Since then, crossed modules have been extensively used in the cohomology of groups, for example by Brown, Dedecker, and Duskin. A short historical note on the question has been written by Maclane as an appendix of Holt's paperwx 2 . The notion of a crossed module has also been extended to categories of algebras. See for example Luewx 5 and Lavendhomme and Roisinwx 3 , further developed in wx4 . The most interesting algebraic situations have been highlighted by Orzech inwx 8 under the name ``categories of interest''. 1. MODULATED G-STRUCTURES AND CROSSED G-MODULES We start with a usual language L having only one sort of variable and for each n a set Opn of n-ary operation symbols. Terms of L are defined as usual. To introduce the idea of a two-sorted interpretation, we consider two symbols a and b Žto be interpreted later by elements of the algebra and scalars respectively. An n-ary modulation is a sequence m s Ž.jj j a b j 1,..., nn; q1of n q 1 symbols or . We say that nq1is the target Ž. of m. If all ji 1 F i F n q 1 are a, we denote the corresponding modulation by ma and call it internal to a; similar conventions hold for b. An L-modulation M is fixed by giving for each n and each F g Opn a set MF of n-ary modulations such that mb g MF , Ž. if m s j 1,...,jnF;b gM then m s mb, Ž. if m s j 1,...,jnFi;a gM then j s a for some i,1FiFn. The elements of MF are called modulations admissible for F. From now on, we consider M as being part of the definition of L. To interpret L, we first associate with a a non-empty set <<a s A Žthe set underlying the algebra. and with b a non-empty set <<b s B Žthe set of scalars.Ž. A modulated interpretation F of L o¨er A, B , to be denoted by 938 LAVENDHOMME AND LUCAS Ž. A,B,F, is fixed by giving for each n, each F g Opn, and each m s Ž.jj j 1,..., nn; q1gMFa mapping << <j <??? <j < <j < Fm : 1 = = nnª q1, called Ž.m- modulated interpretation of F. We extend the notions of admissible modulation and of modulated interpretation to terms of the language L, more exactly to pairs txŽ.of terms t and sequences x of distinct variables containing those of t. Ž. m Žj j j . aIfxis the ith variable of the sequence x, s 1,..., nn; q1is Ž. j j admissible for xx if and only if nq1 s i and the corresponding modu- lated interpretation is the ith projection xxŽ.mspi:<<j1=??? = <<jniª <<j . Ž. Ž. Ž. Ž. bIftxsFt1 x ??? txkkwith F g Op , and if for each i,1FiFk, Ž. Ž.Ž. mi sj1,...,jni;h is admissible for tx i and n s h1,...,hkF;z gM , Ž. Ž. then m s j 1,...,jn;z is admissible for tx and the corresponding modulated interpretation <txŽ.<m is given by composition: <<h ??? <<h 1 =6 = k ²< Ž.< : txi mi <<Fn <Ž.< 6 tx m 6 <<j1=??? = <<jzn <<. Consider now a theory G in L, i.e., a set of equations of the form t12s t Ž.we consider only equational theories . We say that the modulated inter- Ž. pretation A, B, F satisfies or is a model of t12s t iff for every x and every modulation m admissible for tx12Ž.and tx Ž., tx12Ž.mstx Ž.m. ŽComposition with projections shows that this does not really depend on x.Ž.Ž. We say that A, B, F is a model of G or a modulated G-model . if Ž.A, B, F is a model of every element of G. According to our view, the notion of crossed module arises when putting coherence conditions on different modulations admissible for the same operation. Here are the details. Introduce a partial order on the set of n-ary modulations by letting a F b and extending it by j j j jX jX jX Ž.1,..., nn; q11FŽ.,..., nn; q1 iff j j X j jX j j X 11F ,..., nnF and nq1F nq1. MODULES AND CROSSED MODULES 939 For that order, there is a smallest n-ary modulation, namely ma , and a greatest n-ary modulation, mb . Recall that ma is admissible for every F. DEFINITION 1. A crossed G-module Ž.A, B, F, D is determined by a modulated interpretation Ž.A, B, F of L which is a modulated model of G and a mapping D : A ª B X such that for every n, every F g OpnF , and every m, m g M satisfying mŽ.jj j mXŽjXXXj j . s1,..., nn; q11F s ,..., nn; q1the following diagram commutes: << Fm 6 <<j ??? <<jj < < 1= = nnq1 DÄÄ1=??? =Dn DÄnq1 6 6 << Fm9 6 <<jXX??? <<j <<jX 1 = = n nq1 . Ž In the diagram, the convention is that for every i,1FiFnq1, DÄi is the XX. identity if jiis j and DÄ is D if j is a and j is b . Note that if ma g MF , one of the diagrams of the definition expresses Ž X . that D : A ª B is homomorphic for F take m s ma and m s mb .A reason why we do not include ma in every MF is that we want to cover some trivial examples as we will show in the next section. The notion of morphism of modulated interpretations makes no difficul- ties; it is an ordered pair X X X Ž.Žf , g : A, B, F .Žª A , B , F . such that f is a mapping from A to AX and g is a mapping from B to BX making the following diagram commute for every n, for every F g Opn , for every m g MF : << Fm 6 <<j ??? <<jj < < 1= = nnq1 w w1=??? =wn nq1 6 6 << Fm9 6 <<j9 ??? <<j 9 <<j 9 1 = = n nq1 . ŽIn the diagram, the convention is that<< is associated with Ž.A, B, F , <<X Ž XXX.. is associated with A , B , F , wiis f if j s a and w is g if j is b . 940 LAVENDHOMME AND LUCAS For crossed G-modules Ž.ŽA, B, F, D and AXXXX, B , F , D .we add the commutativity of the square D 6 AA9 f g 6 6 D96 A9B9. Those definitions clearly determine the category of modulated interpreta- tions and the category of crossed G-modules.