Cubical abelian groups with connections are equivalent to chain complexes Ronald Brown*, Philip J. Higginsy, Mathematics Division Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. December 11, 2002 University of Wales, Bangor, Maths Preprint 02.24 Abstract The theorem of the title is deduced from the equivalence between crosse* *d complexes and cubical !-groupoids with connections proved by the authors in 1981. In fact we pro* *ve the equivalence of five categories defined internally to an additive category with kernels. Introduction The theorem of the title is shown to be a consequence of the equivalence betwee* *n crossed complexes and cubical !-groupoids with connections proved by us in [4]. We assume the definit* *ions given in [4]. Thus this paper is a companion to others, for example [7], which show that a deficit* * of the traditional theory of cubical sets and cubical groups has been the lack of attention paid to the ö* *c nnections", defined in [4]. Indeed the traditional degeneracies of cubical theory identify certain * *opposite faces of a cube, unlike the degeneracies of simplicial theory which identify adjacent faces. The* * connections allow for a fuller analogy with the methods available for simplicial theory by giving forms* * of `degeneracies' which identify adjacent faces of cubes. They are used in [4] and [1] to give a defini* *tion of a `commutative cube'. Part of the interest of these results is that the family of categories equi* *valent to that of crossed complexes can be regarded as a foundation for a non-abelian approach to algebra* *ic topology and the cohomology of groups. These results show that a form of abelianisation of these* * categories leads to well-known structures. *_____________________________________ email: r.brown@bangor.ac.uk yemail: p.j.higgins@durham.ac.uk 1 Crossed complexes internal to an additive category with kernels The basic elements of what we say next are well known, but are given for comple* *teness. Suppose we are given an action of a group P on the right of a group M such * *that the action OE : M x P ! M is a morphism of groups. Then, as is well known, the action is t* *rivial. The proof is easy: let m 2 M, p 2 P. Then mp = OE(m, p) = OE(m, 1)OE(1, p) = m11p = m. It fo* *llows that a crossed module internal to the category of groups is just a morphism of abelian groups. We need to consider below the more general case of crossed modules over gro* *upoids. Internally to the category of groups, these are more complicated; but internally to the categ* *ory of abelian groups they are again equivalent to morphisms of abelian groups. This result is essent* *ially in [5]. Theorem Let A be an additive category with kernels. The following categories, d* *efined internally to A, are equivalent. B1 : The category of chain complexes. B2 : The category of crossed complexes B3 : The category of cubical sets with connections. B4 : The category of cubical !-groupoids with connections. B5 : The category of globular !-groupoids. Proof: By working on the morphism sets, we can as usual assume that we are work* *ing in the category of abelian groups. Note that the theorem of the title follows from the equivale* *nce B3 ' B1. B1 ' B2 : By a chain complex we shall always mean a sequence of objects and* * morphisms ffi : An ! An-1, n > 1, such that ffiffi = 0. Let C be a crossed complex internal to * *A. The associated chain complex ffC will be defined by (ffC)0= C0, (ffC)1= Ker(ffi0 : C1 ! C0), (ffC)n= Cn(0), n > 2. The crossed complex fiA associated to a chain complex A will be defined by (fiA)0= A0, (fiA)1= A0x A1, (fiA)n= A0x An, n > 2. The groupoid structure on fiA in dimension 1 is defined as usual by ffi0 = pr1,* * ffi1 = pr1+ (@ O pr2), and with composition (a, b) + (a + @b, c) = (a, b + c). The structure on (fiA)n* * for n > 2 is that the only addition is (a, b)+(a, c) = (a, b+c). The operation of (fiA)1 on (fiA)n, n > 2,* * is (a, b)(a,c)= (a+@c, b). This gives our first equivalence, between chain complexes and crossed complexes. B2 ' B3 : An equivalence between crossed complexes and cubical !-groupoids * *with connections internally to the category of sets is established in [4]. Although choices are * *involved in this, the end 2 result is a natural equivalence. It follows that this can be applied internally* * to a category A, simply by applying it to the morphism sets A(X, A) for all objects X of A. This yields ou* *r equivalence between crossed complexes and cubical !-groupoids with connections internal to A. B2 ' B5 : This follows, in a similar way, from the equivalence between cros* *sed complexes and globular !-groupoids proved in [3]. (Reference [2] is relevant to the equivalen* *ce B1 ' B5. B3 ' B4 : Let K be a cubical abelian group with connections, in the sense o* *f [4]. Lemma If G is an abelian group, and if s, t : G ! G are endomorphism of G such * *that st = s, ts = t, then we can define a groupoid structure on G with source and target maps s, t by g O h = g - tg + h, for g, h 2 G with tg = sh, and this defines on G the structure of groupoid inte* *rnal to abelian groups. This result comes from [5], and is also a special case of a non-abelian res* *ult on cat1-groups [6], where the condition [Ker s, Ker t] = 1 is required, and is here trivially satis* *fied. This result can be applied to Kn, n > 1, and for each i = 1, . .,.n, with si= ffli@0i, ti= ffli@1i* *, giving n compositions and so a cubical complex with compositions and connections in the sense of [1, 4]. * *The interchange law is easily verified, and there remains essentially only the transport law for th* *e connections, which is again simple, showing that K is now a cubical !-groupoid with connections. It i* *s easy to see that the functor thus defined is adjoint to the forgetful functor B4 ! B3. * * 2 References [1]Al-Agl, F.A., Brown, R. and Steiner, R., `Multiple categories: the equivale* *nce between a globular and cubical approach', Advances in Mathematics 170 (2002) 71-118. [2]Bourn, D., `Another denormalization theorem for the abelian chain complexes* *', J. Pure Appl. Algebra 66 (1990) 229-249. [3]Brown, R. and Higgins, P.J. , `The equivalence of 1-groupoids and crossed c* *omplexes', Cah. Top. G'eom. Diff. 22 (1981) 371-386. [4]Brown, R. and Higgins, P.J., `The algebra of cubes', J. Pure Appl. Algebra * *21 (1981) 233-260. [5]Grothendieck, A., `Cat'egories cofibr'ees additives et complexe cotangent r* *elatif', Springer Lec- ture Notes in Math. 79 (1968) Springer-Verlag, Berlin, 167pp. [6]Loday, J.-L., `Spaces with finitely many non-trivial homotopy groups', J. P* *ure Appl. Algebra 24 (1982) 179-202. [7]Tonks, A.P., `Cubical groups which are Kan', J. Pure Appl. Algebra 81 (1992* *) 83-87. 3