theorem 1.
Let
be a complete sequence of commuting quantum spin `foams' (QSFs) in an arbitrary quantum state space (QSS), and let
be the corresponding sequence of pair subspaces of QST. If
is a sequence of CW-complexes such that for any
,
, then there exists a sequence of
-connected models
of
and a sequence of induced isomorphisms
for
, together with a sequence of induced monomorphisms
for
.
There exist weak homotopy
equivalences between each
and
spaces
in such a sequence. Therefore, there exists a
-complex approximation of QSS
defined by the sequence
of CW-complexes with dimension
. This
-approximation is
unique up to regular homotopy equivalence.
Corollary 2.
The
-connected models
of
form the Model category of
Quantum Spin Foams
, whose morphisms
are maps
such that
, and also such that the following diagram
is commutative:
Furthermore, the maps
are unique up to the homotopy rel
, and also rel
.
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