Higher dimensional rewriting, concurrency and directed homotopy

Workshop as part of GEOCAL'06

Week 4: February 20 - 24

Marseille - Luminy - CIRM

scientific organizers : Eric Goubault & Yves Lafont



Monday February 20
History preserving bisimulation is a semantic equivalence relation on models of concurrency that tells when two systems have the same causal and branching structure. In this talk I show how to define this notion, which originally was proposed on event structures and related models, in the setting of higher dimensional automata. The definition is closely related to the concept of homotopy of paths in a higher dimensional automaton.
Studying the timely behaviour of concurrent processes by means of process graphs has been known since Dijkstra, yet the modern approach to the subject by algebro-topological means, intitiated by Fajstrup, Goubault and Raussen, has appeared only in the middle 1980's. The key observation was that a process graph is an ordered topological space and variants of classical algebro-topological tools were worked out in order to study invariants of such objects. There are basically two relevant variants of orders on topological spaces: global and local orders. The former are easier to handle and therefore better studied, with the drawback that no loops can be represented. The latter, called local po-spaces, do not exhibit this drawback, yet they are more complex artifacts. We present partial results about model structures for local po-spaces. The key observation here is a certain similarity with the situation in the category of (affine) schemes, leading to variants of methods introduced by Voevodsky when setting up the celebrated homotopy theory of schemes.
In this talk, we apply the polygraphic translation of term rewriting systems to SKS, a calculus of structures-type formalism describing the proofs of propositional calculus. We get a 3-polygraph which has three main advantages over the initial formulation: it allows an easy control over "structural bureaucracy"; it yields 3-dimensional representations for proofs; finally, local transformations of proofs have the rich algebraic structure of 4-dimensional objects.
Non abelian complexes are higher dimensional categories generalizing rewrite systems as well as CW-complexes. We introduce the fundamental notions for a homotopical theory of computation.
 
Tuesday February 21 
During this week, we will deal with the problem of customizing the construction of the components category to the case of models containing loops. We also try to go deeper in the understanding of the relations between M. Grandis spectra and our components categories, mostly when the underlying space is compact. Moreover, the fundamental category construction can be made in a purely categorical abstract setting that we will present. The purpose of this abstract approach is the comparison of the fundamental category functors over dTop, PoTop, LPoTop, Flow ... in other words the different known topological models for concurrency.
Wednesday February 22 
We prove that for any monoid M, the homology defined in terms of non abelian resolutions coincides with the homology defined in terms of resolutions by free ZM-modules.
Thursday February 23 
Pour une catégorie C complète et cocomplète, un foncteur nerf pour C apparaît comme un foncteur de la forme N : C -> Smpl et admettant un adjoint à gauche. Ce foncteur permet de ramener l'étude topologique d'un objet X de C à celle de l'ensemble simplicial N(X). La donnée de l'adjoint N' : Smpl -> C est le prolongement universel par limites inductives d'un objet cosimplicial de C. Dans tous les cas observés et familiers où un tel foncteur N est utilisé, on observe que l'objet cosimplicial est donné par l'application d'une monade T à l'objet initial (i.e. vide) de C. En particulier, ce schéma d'idée s'applique à la catégorie C des oméga-catégories strictes et l'objet cosimplic, al obtenu par ces méthodes fonctorielles (très différentes des méthodes combinatoires de Street) donne les orientaux de Street. Ces orientaux semblent être un outil de calcul adapté à l'homotopie dirigée. Nous donnerons également une description des algèbres de la monade T.
We study multinomial lattices, or lattices of paths in higher dimension, which arise from simple rewrite systems modeling concurrency. We propose a lattice theoretic notion of dimension, we point out relations to dihomotopy theory.
 
Friday February 24 
We define trivial fibrations and cofibrations in the category of strict n-categories, and show that the cofibrant objects are the free ones, that is those generated by polygraphs.



Lecture notes



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