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Inverting weak dihomotopy equivalence using homotopy continuous flow

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Philippe Gaucher

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.

Keywords:
concurrency, homotopy, Whitehead theorem, directed homotopy, weak
factorization system, model category, localization

2000 MSC:
55U35, 55P99, 68Q85

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 3, pp 59-83.

http://www.tac.mta.ca/tac/volumes/16/3/16-03.dvi

http://www.tac.mta.ca/tac/volumes/16/3/16-03.ps

http://www.tac.mta.ca/tac/volumes/16/3/16-03.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/3/16-03.dvi

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