Dipaths and dihomotopies in a cubical complex

Abstract

In the geometric realization of a cubical complex without degeneracies, a -set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipaths are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting -set. And that two combinatorial dipaths which are dihomotopic through a non-combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric -set. Moreover, we prove that in a geometric -set, the d-homotopy introduced in [M. Grandis, Directed homotopy theory, I, Cah. Topol. Géom. Différ. Catég. 44 (4) (2003) 281-316] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen, Algebraic topology and concurrency,

Simplicial models for trace spaces II: General higher dimensional automata

“…Properties of Higher Dimensional Automata (cf Section 1.1) are intimately related to the study of directed paths in a pre-cubical set, also called a -set; this term (cf [6]) is used in a similar way as a -set -as introduced in [20] -for a simplicial set without degeneracies. We use n as an abbreviation for the n-cube I n D OE0; 1 n with the product topology.…”

“…The subcomplex given by the carrier sequence corresponding to any directed path (see Fajstrup [6]), the sequence of cubes containing segments of that path, is obviously a subcomplex satisfying (NB).…”

Algebraic topology and concurrency

“…In a more abstract setting, see also our Section 7, this version of dihomotopy has been investigated by Grandis [46,45]. For nice enough lpo-spaces (like the cubical sets from Section 6, both dihomotopy concepts are shown to be equivalent by Fajstrup [19]). …”

A Category of Higher-Dimensional Automata

“…Employing recent developments by Fajstrup [8], we show that bisimilarity of HDA is equivalent to a certain dipath-lifting property, which can be attacked using (directed) homotopy techniques. This confirms a prediction from [13].…”

“…Note that the definition in [8] makes an extra assumption on X which, in fact, is not necessary. Figure 2 shows an example of a carrier sequence.…”