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In this paper we give a categorical treatment of matrix Toda brackets, both in the preand post-compositional versions. Explicitly the setting in which we work is, a la Gabriel-Zisman, a 2-category with zeros. The development parallels that in the topological setting but with homotopy groups replaced by nullity groups of invertible 2-morphisms. A central notion is that of conjugation of 2-morphisms. Our treatment of matrix Toda brackets is carried forward to the point of establishing appropriate indeterminacies. 0. INTRODUCTION When P. Gabriel and M. Zisman wrote their book [3] they presented their subject (homotopy theory of simplicial sets via categories of fractions) from the most general point of view available going "back to the beginning of the theory ... in the hope of providing the reader with ... proofs which are easier or more conceptual than those already published." Their study of homotopy having begun in Chapter IV they set out in Chapter V "to give a standard and self-dual proof of various exact sequences occurring in Algebraic Topology ... the main idea is to construct an exact sequence in the category of pointed groupoids and then to reduce the study of a large class of 2-categories to the preceding one." They soon specialise to 2-categories ?1 which satisfy conditions A and B below: A. If x and y are two objects of 2', Hom sw(x, y) is a groupoid. B. There is an object o of 2' such that, for any object x of XP, Homr(o, x) and Homw(x, o) are isomorphic to the zero category Continuing in the spirit of Gabriel and Zisman the present paper is a contribution to an abstract theory of homotopy that assumes the existence of a 2-category &' satisfying only condition B. Indeed it can be regarded as sequel to a paper [8] by the third author which, in the setting of an arbitrary 2-category, studies the notion of homotopy equivalence and characterises the homotopy equivalences in various lax categories. Here the development parallels that in the topological setting but with track groups of the form 7r(EX, Y) or 7r(X, Q2Y) replaced by nullity groups of invertible 2-morphisms. A central notion is that of conjugation of 2-morphisms. This notion is available in any 2-category but does not seem to have been used previously. In fact when Received by the editors June 9, 1993. 1991 Mathematics Subject Classification. Primary 18D05, 55Q05, 55Q35. ?1995 American Mathematical Society

In this note we present a variant of bridging theorems in abstract homotopy theory which applies to the category of semisimplicial sets yielding bridging theorems for semisimplicial sets which might prove useful in semi-algebraic topology.

In this paper we present comparison theorems in semicubical homotopy theory. Topological versions are more or less well known and have been obtained amongst others by A. DOLD ([6], [7]), I. M. JAMES ([11]), P. R. HEATH ([9]), S. P. LAM ([15]), and the first author ([2], [3]). A partially abstract approach has been given by the first author in [3]. We use operations on homotopy sets and the notion of CHEP (covering homotopy extension property).

A Kervaire diagram of group homomorphisms and pointed maps arising from a standard homotopy pullback Z of groupoid morphisms with common codomain is studied. The diagram yields two exact Mayer-Vietoris sequences that specialize in various ways. A classification theorem is obtained for the set of components of z. Applications are given to elementary group theory and to homotopy pair theory.