M. Bullejos

On the Geometry of 2-Categories and their Classifying Spaces

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillens Theorem A.

On catn-groups and homotopy types☆

Abstract We give an algebraic proof of Lodays ‘Classification theorem’ for truncated homotopy types. In particular we give a precise construction of the homotopy cat n -group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusies ‘total Dec’.

Higher dimensional obstruction theory in algebraic categories

In this paper we generalize Duskins low dimensional obstruction theory, established for the Barr-Becks cotriple cohomogy HG2, to higher dimensions by giving a new interpretation of HGn+1 in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to HGn+1(S,A) in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over HGn(S,A).

Proportionally modular diophantine inequalities and the Stern-Brocot tree

Given positive integers a, b and c to compute a generating system for the numerical semigroup whose elements are all positive integer solutions of the inequality ax mod b < cx is equivalent to computing a Bezout sequence connecting two reduced fractions. We prove that a proper Bezout sequence is completely determined by its ends and we give an algorithm to compute the unique proper Bezout sequence connecting two reduced fractions. We also relate Bezout sequences with paths in the Stern-Brocot tree and use this tree to compute the minimal positive integer solution of the above inequality.

A FULL AND FAITHFUL NERVE FOR 2-CATEGORIES

The notion of geometric nerve of a 2-category (Street, J. Pure Appl. Algebra49 (1987), 283–335) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding simplicial maps. These facts allow us to prove a representation theorem of the general non-abelian cohomology of groupoids (classifying non-abelian extensions of groupoids) by means of homotopy classes of simplicial maps.

On the equivariant 2-type of a G-space☆

Abstract A classical theorem of Mac Lane and Whitehead states that the homotopy type of a topological space with trivial homotopy at dimensions 3 and greater can be reconstructed from its π1 and π2, and a cohomology class k3 /gE H3(π1, π2). More recently, Moerdijk and Svensson suggested the possibility of using Bredon cohomology to extend this result to the equivariant case, that is, for spaces X equipped with an action by a fixed group G. In this paper we carry out this suggestion and prove an analogue of the classical result in the equivariant case.

Cohomology and Higher Dimensional Baer Invariants

In this paper we work in a categorical setting which includes the most usual algebraic categories, for example, groups, rings (associative, com- mutative), or algebras (associative, commutative, Jordan, Lie, etc.). Such a setting is supplied by the theory of Q-groups in the sense of Higgins [S]. Let 9 be a variety of Q-groups and g E 9 a subvariety, with associated quotient functor U: 9 + 99. Several authors have shown that the first Baer invariant of an object A in 9, g,(A) = L, U(A), plays an important role in the study of the relationship between extensions of U(A) in the subvariety &J and general extensions of