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Homotopical interpretation of globular complex by multipointed d-space

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Philippe Gaucher

Globular complexes were introduced by E. Goubault and the author to
model higher dimensional automata. Globular complexes are
topological spaces equipped with a globular decomposition which is
the directed analogue of the cellular decomposition of a CW-complex.
We prove that there exists a combinatorial model category such that
the cellular objects are exactly the globular complexes and such
that the homotopy category is equivalent to the homotopy category of
flows. The underlying category of this model category is a variant
of M. Grandis' notion of d-space over a topological space colimit
generated by simplices. This result enables us to understand the
relationship between the framework of flows and other works in
directed algebraic topology using d-spaces. It also enables us to
prove that the underlying homotopy type functor of flows can be
interpreted up to equivalences of categories as the total left
derived functor of a left Quillen adjoint.

Keywords:
homotopy, directed homotopy, combinatorial model category,
simplicial category, topological category, delta-generated
space, d-space, globular complex, time flow

2000 MSC:
55U35, 18G55, 55P99, 68Q85

*Theory and Applications of Categories,*
Vol. 22, 2009,
No. 22, pp 588-621.

http://www.tac.mta.ca/tac/volumes/22/22/22-22.dvi

http://www.tac.mta.ca/tac/volumes/22/22/22-22.ps

http://www.tac.mta.ca/tac/volumes/22/22/22-22.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/22/22-22.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/22/22-22.ps

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/22/22-22.pdf

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