## A model category for the homotopy theory of concurrency.(English)Zbl 1069.55008

In this paper, the category of flows is introduced. An object $$X$$ in this category consists of a topological space $$\mathbb{P}X$$, a discrete space, $$X^0$$, a pair of continuous maps $$s,t:\mathbb{P}X\to X^0$$ and a concatenation map $$\star:\{(x,y)\in\mathbb{P}X\times \mathbb{P}X| s(y)=t(x)\}\to \mathbb{P}X$$ such that $$s(x\star y)=s(x)$$ and $$t(x\star y)=t(y)$$. A morphism $$f:X\to Y$$ is a map of the discrete spaces as sets and a continuous map of $$\mathbb{P}X\to\mathbb{P}Y$$ commuting with source and target maps and preserving concatenation.
The aim is a model for concurrent computations, and $$X^0$$ are then the states, $$\mathbb{P}X$$ the (non-constant) execution paths and $$s,t$$ the source and target of an execution. This category is complete and cocomplete – see section 4. A class of homotopy equivalences, $$S$$-homotopy, is defined in section 7, and the main part of the paper is devoted to constructing a model structure on the category of flows. This model structure is cofibrantly generated, any flow is fibrant, two cofibrant flows are homotopy equivalent in the model structure if and only if they are $$S$$-homotopy equivalent.

### MSC:

 55P99 Homotopy theory 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) 55U35 Abstract and axiomatic homotopy theory in algebraic topology

### Keywords:

concurrency; model category; higher dimensional automaton
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