a preprint: A symmetric cubical category associated to a directed space

a preprint: A symmetric cubical category associated to a directed space

[Marco Grandis]

The following preprint is available in pdf:

M. Grandis, A symmetric cubical category associated to a directed space.
Dip. Mat. Univ. Genova, Preprint 590 (2010). Available at:

       http://www.dima.unige.it/~grandis/Fnd.pdf

Abstract. The recent domain of directed algebraic topology studies
'directed spaces', where paths and homotopies cannot generally be
reversed. The general aim is modelling non reversible phenomena, but
the present applications are mostly concerned with the theories of
concurrent processes and rewrite systems. At the place of the
classical fundamental groupoid of a topological space, a directed
space has a fundamental category, whose applications to concurrency
have already been studied in many papers.
 	Here, we want to study an infinite dimensional version of the
fundamental category of a directed space, of a cubical type and more
precisely a symmetric cubical one, because transposition symmetries
occur naturally and simplify the coherence properties.
 	We introduce a 'Moore' strict symmetric cubical category of a
directed space  X,  with concatenation laws in the various directions
and transpositions (which permute variables).
 	On the other hand, standard cubes give a lax cubical structure,
where concatenations are associative up to invertible
reparametrisation but degeneracies are only lax-unital.

With best regards

Marco Grandis

PS. I have been informed by R. Brown of a recent preprint of his on
Moore n-paths for a topological space and their cubical structure,
see:   arXiv: 0909.2212v2
Sorry of missing that. I will add something about it, in my work.


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