### U.W. Bangor - School of Informatics - Mathematics Preprints 2002

# Algebraic Topology

#### 02.02 : BROWN, R. & PORTER, T.

### The intuitions of higher dimensional algebra for the study of structured space

#### Summary:

Higher dimensional algebra frees mathematics from the restriction to a purely linear notation, in order to improve the modelling of geometry and so obtain more understanding and more modes of computation. It gives new tools for non-commutative, higher dimensional, local to global problems, through the notion of*algebraic inverse to subdivision*. We explain the way these ideas arose for the writers, in extending first the classical notion of abstract group to abstract groupoid, in which composition is only partially defined, as in composing journeys, and which brings a spatial component to the usual group theory. An example from knot theory is is used to explain how such algebra can be used to describe some structure of a space. The extension to dimension 2 uses compositions of squares in two directions, and the richness of the resulting algebra is shown by some 2-dimensional calculations. The difficulty of the jump from dimension 1 to dimension 2 is also illustrated by the comparison of the commutative square with the commutative cube - discussion of the latter requires new ideas. The importance of category theory is explained, and a range of current and potential applications of higher dimensional algebra is indicated.

#### Published in:

*Revue de Synthèse*124 (2003) 173-203.

#### Download:

- pdf file: paris-cogn8.pdf
- gzipped postscript file: paris-cogn8.ps.gz

#### 02.03 : MUTLU, A. & PORTER, T.

### Crossed squares and 2-crossed modules

#### Summary:

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gzipped pdf file: 02_03.pdf.gz#### 02.09 : BROWN, R., MOORE, E.J., PORTER, T. & WENSLEY, C.D.

### Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups

#### Summary:

The category of crossed complexes gives an algebraic model of*CW*-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases.

#### Published in:

*Georgian Math. J.*9 (2002) 623-644.

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gzipped postscript file: 02_09.ps.gz#### 02.21 : BROWN, R.

### Multiple groupoids as a non commutative tool for higher dimensional local-to-global problems

#### Talk given at:

Categorical Structures for Descent and Galois Theory, Fields Institute, September 23-28, 2002#### Download:

gzipped pdf file: 02_21.pdf.gz#### 02.22 : BROWN, R. & GLAZEBROOK, J.F.

### Connections, local subgroupoids, and a holonomy Lie groupoid of a line bundle gerbe

#### Abstract:

Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2-holonomy on line bundles may be formulated using the notion of a connection pair on a double category, due to Brown-Spencer, but now formulated in terms of double groupoids using the thin fundamental groupoids introduced by Caetano-Mackaay-Picken. To obtain a locally Lie groupoid to which globalisation applies, we use methods of local subgroupoids as developed by Brown-Icen-Mucuk.#### Published in:

*Univ. Iagellonicae Acta Math.*41 (2003) 283-296.

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gzipped pdf file: 02_22.pdf.gz#### 02.23 : ARVASI, Z. & PORTER, T.

### Freeness conditions for crossed squares of commutative algebras

#### Abstract:

We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras.#### Published in:

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gzipped pdf file: 02_23.pdf.gz#### 02.24 : BROWN, R. & HIGGINS, P.J.

### Cubical abelian groups with connections are equivalent to chain complexes

#### Abstract:

The theorem of the title is shown to be a consequence of the equivalence between crossed complexes and cubical*omega*-groupoids with connections proved by us in [BH3]. We assume the definitions given in [BH3]. Thus this paper is a companion to others, for example [T1], which show that a deficit of the traditional theory of cubical sets and cubical groups has been the lack of attention paid to the ``connections'', defined in [BH3]. 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 [BH3] and [ABS] to give a definition of a `commutative cube'.

Part of the interest of these results is that the family of categories equivalent to that of crossed complexes can be regarded as a foundation for a non-abelian approach to algebraic topology and the cohomology of groups. These results show that a form of abelianisation of these categories leads to well-known structures.

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pdf file: 02_24.pdf#### 02.25 : BROWN, R. & HIGGINS, P.J.

### The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid

#### Abstract:

The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space is the orbit groupoid of the fundamental groupoid of the space. This result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology [Brown:1988]. Since the book is out of print, and the result seems not well known, we now advertise it here. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space.This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections of [Brown:1988]. It is also hoped that this publication will allow wider views of this result, for example in topos theory and descent theory.

Because of its provenance, this should be read as a graduate text rather than an article. This explains also the inclusion of exercises. It is expected that this material will be part of a new edition of the book.

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pdf file: 02_25.pdf#### 02.26 (revised as 08.07) : BROWN, R.

### Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

#### Abstract:

(This is an extended account of a lecture given at the meeting on `Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories', Fields Institute, September 23-28, 2002.)We outline the main features of the definitions and applications of crossed complexes and cubical omega-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain non commutative results and compute homotopy types.