Bangor University - School of Computer Science
Mathematics Preprints 2007
Category Theory & Homotopy Theory
07.01 : PORTER, T.
Enriched categories and models for spaces of dipaths: a discussion document and overview of some techniques.
Summary:Partially ordered sets, causets, partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics. The order models `time' which is often not globally given. In this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system. Many physical problems rely on the analysis of models of the path space of a space-time manifold. Many problems in concurrent systems use `spaces' of paths in a system. We review some ideas from algebraic topology and discrete differential geometry that suggest how to model the dipath space of a pospace by an enriched category. Much of the earlier material is `well known', but, coming from different areas, is dispersed in the literature.
Published in:in: Computational Structures for Modelling Space, Time and Causality,
eds. R. Kopperman, P. Panangaden, M.B. Smyth, D. Spreen.
Schloss Dagstuhl, seminar 06341 (2007).
07.08 PORTER, T. & TURAEV, V.
Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras.
Abstract:Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds endowed with extra structure in the form of homotopy classes of maps into a given `target' space B. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2-type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2-types that we use are crossed modules, C, and we introduce a notion of formal C-map, which extends the usual lattice-type constructions to this setting. This leads to a classification of `formal' 2-dimensional HQFTs with target C in terms of crossed C-algebras.
Published in:Journal of Homotopy & Related Structures 3 (2008) 113-159.
07.09 : LEWIS, R.P.I.
Stacks and formal maps of crossed modules
Abstract:If X is a topological space then there is an equivalence between the category \pi_1(X)-Set, of actions of the fundamental group of X on sets, and the category of covering spaces on X. Moreover the latter is also equivalent to the category of locally constant sheaves on X.
Grothendieck has conjectured that this should be the 'n=1' case of a result which is true for all n, and it is the 'n=2' case we look at in this thesis.
The desired generalisation should replace actions of the group \pi_1(X) (which is an algebraic model for the 1-type of X) by actions of a crossed module (i.e., by an algebraic model for the 2-type) on groupoids; 'locally constant sheaves of sets' by 'locally constant stacks of groupoids'; and 'covering space' by a locally trivial object whose fibres are groupoids.
This last object we handle using the machinery of simplicial fibre bundles (twisted Cartesian products) and formal maps, building a simplicial object, Z(\lambda), where the fibre is now a (nerve of) a groupoid. To interpret Z(\lambda) as a stack, we show that just as sheaves on X are equivalent to etale spaces, we can define a notion of 2-etale space corresponding to stacks and show that from Z(\lambda) we can construct a locally constant stack on X.