PlanetPhysics/Theorem on CW Complex Approximation of Quantum State Spaces in QAT

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html} 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 \htmladdnormallink{morphisms {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are maps such that , and also such that the following diagram is commutative:} \\

</math> \begin{CD} Z_j @> f_j >> QSS_j \\ @V h_{jk} VV @VV g V \\ Z_k @ > f_k >> QSS_k \end{CD} Failed to parse (syntax error): {\displaystyle \\ ''Furthermore, the maps <math>h_{jk'' } are unique up to the homotopy rel , and also rel }.

{Theorem 1} complements other data presented in the parent entry on QAT.