# Algebraic Topology

### Freeness conditions for crossed squares and squared complexes

#### Abstract:

Following Ellis, we investigate the notion of totally free crossed squares and related square complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, in terms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result of Brown and Loday.

#### Published in:

K-Theory 20 (2000) 345-368.

#### ftp access:

xxx-archive : http://uk.arXiv.org/abs/math.KT/9904041

### A homotopy 2-groupoid from a fibration

#### Abstract:

In this paper we give an elementary derivation of a 2-groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2-groupoids and cat^1-groupoids.

HHA: n2.dvi, .ps, .dvi.gz, .ps.gz files.

#### Published in:

Homology, Homotopy and Applications 2 (1999) 79-93.

### Double categories, 2-categories, thin structures and connections

#### Abstract:

The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2-categories is equivalent to the category of edge symmetric double categories with thin structure.

#### Published in:

Theory and Applications of Categories 5 No.7 (1999) 163-175.

### Double algebroids and crossed modules

#### Abstract:

We define and show an equivalence between certain crossed modules of algebroids and certain double algebroids with thin structure, or connection.

#### 99.21 : BROWN, R.

Some problems in non-abelian homotopical and homological algebra
(Latex, edited  version of 89.07)

#### Abstract:

This is an edited Latex version of the paper Some problems in non-Abelian homotopical and homological algebra'', {\em Homotopy theory and related topics, Proceedings Kinosaki, 1988}, ed. M. Mimura, Springer Lecture Notes in Math., 1418 (1990) 105-129.

Part of the motivation of the paper was to give a kind of survey of the area in terms of what had not been done, thus allowing a lighter touch on what had been done.

Many of the problems came from grant proposals unsupported as either irrelevant to mainstream algebraic topology', or because calculating integral homotopy types is not a central problem of homotopy theory'. It is hoped now to be able to see how or if the mainstream is catching up.

Full details of papers published since 1989 are given in the References. Comments on progress on some of the problems since 1989 will be given as an Appendix, with an additional Bibliography.

### Global actions, groupoid atlases, and related topics

revised as 06.02.

### Computing Resolutions Over Finite p-Groups

#### Abstract:

A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any finite p-group is detailed. The resolutions we construct are, as vector spaces, as small as the minimal resolution of Fp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented.