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

# Category Theory & Homotopy Theory

#### 02.01 : KAMPS, K.H. & PORTER, T.

### 2-groupoid enrichments in homotopy theory and algebra

#### Abstract:

The use of groupoid enrichments in abatract homotopy theory is well known and classical. Recently enrichments by higher dimensional groupoids have been considered. Here we will describe enrichment by 2-groupoids with respect to the Gray tensor product and will examine several examples (2-groupoids, 2-crossed complexes, chain complexes, etc.) from an elementary viewpoint. The enrichment of the category of chain complexes is examined in detail and questions of the existence of analogues of classical constructions (categories over B, under A, etc.) are explored.#### Published in:

*K-Theory*25 (2002) 373-409.

#### 02.14 (revised as 04.19) : BROWN, R., MORRIS, I., SHRIMPTON, J. & WENSLEY, C.D.

### Graphs of morphisms of graphs

#### Abstract:

From the categorical viewpoint, the endomorphisms of a graph should not only be composable, giving a monoid structure, but should also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph.We extend Shrimpton's investigations on the morphism-digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal.

#### Download:

04_19.ps.gz#### Published in:

#### 02.15 : PORTER, T.

### Geometric aspects of multiagent systems

#### Abstract:

Recent advances in Multiagent Systems (MAS) and Epistemic Logic within Distributed Systems Theory, have used various combinatorial structures that model both the geometry of the systems and the Kripke model structure of models for the logic. Examining one of the simpler versions of these models, interpreted systems, and the related Kripke semantics of the logic*S5_n*(an epistemic logic with

*n*-agents), the similarities with the geometry/homotopy theoretic structure of groupoid atlases is striking. These latter objects arise in problems within algebraic K-theory, an area of algebra linked to the study of decomposition and normal form theorems in linear algebra. They have a natural well structured notion of path and constructions of path objects, etc., that yield a rich homotopy theory.

In this paper, we examine what a geometric analysis of the model may tell us of the MAS. Also the analogous notion of path will be analysed for interpreted systems and

*S5_n*-Kripke models, and is compared to the notion of `run' as used with MASs. Further progress may need adaptions to handle

*S4_n*rather than

*S5_n*and to use directed homotopy rather than standard `reversible' homotopy.

#### Published in:

*Electronic Notes in Theoretical Computer Science*81 (2003) 1-26.

#### Download:

At Elsevier or locally as 02_15.pdf#### 02.16 : BOOTH, P.I.

### Mapping Spaces for Homotopy Theory

#### Abstract:

Let*q*be a map (= continuous function) from

*Y*to

*B*and let

*Z*be a space. We will use

*Y ! Z*to denote a set of partial maps from

*Y*to

*Z*, i.e. those such maps whose domains are individual fibres of

*q*. If

*B*is a

*T_1*-space, we will equip

*Y ! Z*with a suitable version of the compact-open toplogy, thereby defining the

*freerange mapping space*

*Y ! Z*. There is an obvious associated

*freerange functional projection*

*q ! Z*from

*Y ! Z*to

*B*. If

*B*is a Hausdorff space, we will show that maps from

*Y*to

*Z*determine sections (= right inverses) to

*q ! Z*. Further, if

*Y*is a compactly generated space, this correspondence will be bijective. If

*q*is a fibration, then

*q ! Z*may also be a fibration; in that case it may be considerably easier to derive information about sections to

*q ! Z*, and hence about maps from

*Y*to

*Z*.

This paper gives a simple and direct approach to the theory of freerange mapping spaces. So the topic achieves the status of a theory in its own right, i.e. one that is independent of other related theories. The sequel to this paper, 02.17, describes applications of freerange mapping spaces to the concepts referred to in its title.

#### Download:

gzipped postscript: 02_16.ps.gz#### Published in:

#### 02.17 : BOOTH, P.I.

### Cofibrations, Cohomology, Fibrations, Identifications, and Mapping Spaces

#### Abstract:

Let*q*be a map from

*Y*to

*B*and let

*Z*be a space. In 02.16 the author developed a theory of freerange mapping spaces. These spaces have underlying sets of partial maps from

*Y*to

*Z*, i.e. those partial maps whose domains are individual fibres of

*q*, and are equipped with a suitable form of the compact-open topology.

In this paper we present applications of freerange mapping spaces to the theories of cofibrations, the cohomology of fibrations, sectioned fibrations, identifications and Moore-Postnikov factorizations. We also relate our construction to those of fibrewise and pairwise mapping spaces. In particular we show that, in suitable circumstances, each of these three constructions can be used to define the two others.

#### Download:

gzipped postscript: 02_17.ps.gz#### Published in:

#### 02.18 : BROWN, R. & JANELIDZE, G.

### Galois theory and a new homotopy double groupoid of a map of spaces

#### Abstract:

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets.Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat1-group or crossed module.

An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.

#### Download:

pdf: 02_18.pdf#### Published in:

*Applied Categorical Structures*12 (2004) 63-80.