Geodesic
Inverting weak dihomotopy equivalence using homotopy continuous flow
Theory and applications of categories, Tome 16 (2006), pp. 59-83.

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A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead's theorem for the full dihomotopy relation, and not only for S-homotopy as in previous works of the author. This fact is not the consequence of the existence of a model structure on the category of flows because it is known that there does not exist any model structure on it whose weak equivalences are exactly the weak dihomotopy equivalences. This fact is an application of a general result for the localization of a model category with respect to a weak factorization system.
Classification : 55U35, 55P99, 68Q85
Keywords: concurrency, homotopy, Whitehead theorem, directed homotopy, weak factorization system, model category, localization 
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     author = {Philippe Gaucher},
     title = {Inverting weak dihomotopy equivalence using homotopy continuous flow},
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     volume = {16},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2006_16_a2/}
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Philippe Gaucher. Inverting weak dihomotopy equivalence using homotopy continuous flow. Theory and applications of categories, Tome 16 (2006), pp. 59-83. http://geodesic.mathdoc.fr/item/TAC_2006_16_a2/