#
The branching nerve of HDA and the Kan condition

##
Philippe Gaucher

One can associate to any strict globular $\omega$-category three
augmented simplicial nerves called the globular nerve, the
branching and the merging semi-cubical nerves. If this strict
globular $\omega$-category is freely generated by a precubical
set, then the corresponding homology theories contain different
informations about the geometry of the higher dimensional
automaton modeled by the precubical set. Adding inverses in this
$\omega$-category to any morphism of dimension greater than 2
and with respect to any composition laws of dimension greater
than 1 does not change these homology theories. In such a
framework, the globular nerve always satisfies the Kan condition.
On the other hand, both branching and merging nerves never
satisfy it, except in some very particular and uninteresting
situations. In this paper, we introduce two new nerves (the
branching and merging semi-globular nerves) satisfying the Kan
condition and having conjecturally the same simplicial homology as
the branching and merging semi-cubical nerves respectively in
such framework. The latter conjecture is related to the thin
elements conjecture already introduced in our previous papers.

Keywords:
cubical set, thin element, Kan complex, branching, higher
dimensional automata, concurrency, homology theory

2000 MSC:
55U10, 18G35, 68Q85

*Theory and Applications of Categories*
, Vol. 11, 2003,
No. 3, pp 75-106.

http://www.tac.mta.ca/tac/volumes/11/3/11-03.dvi

http://www.tac.mta.ca/tac/volumes/11/3/11-03.ps

http://www.tac.mta.ca/tac/volumes/11/3/11-03.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/3/11-03.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/3/11-03.ps

TAC Home