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

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html} 1.}

Let $[QF_{j}]_{j=1,...,n}$ be a complete sequence of commuting quantum spin `foams' (QSFs) in an arbitrary quantum state space (QSS), and let $(QF_{j},QSS_{j})$ be the corresponding sequence of pair subspaces of QST. If $Z_{j}$ is a sequence of CW-complexes such that for any $j$ , $QF_{j}\subset Z_{j}$, then there exists a sequence of $n$-connected models $(QF_{j},Z_{j})$ of $(QF_{j},QSS_{j})$ and a sequence of induced isomorphisms ${f_{*}}^{j}:\pi _{i}(Z_{j})\rightarrow \pi _{i}(QSS_{j})$ for $i>n$, together with a sequence of induced monomorphisms for $i=n$.

There exist weak homotopy equivalences between each $Z_{j}$ and $QSS_{j}$ spaces in such a sequence. Therefore, there exists a $CW$--complex approximation of QSS defined by the sequence $[Z_{j}]_{j=1,...,n}$ of CW-complexes with dimension $n\geq 2$. This $CW$--approximation is unique up to regular homotopy equivalence.

Corollary 2.

The $n$-connected models $(QF_{j},Z_{j})$ of $(QF_{j},QSS_{j})$ form the Model category of Quantum Spin Foams $(QF_{j})$, whose \htmladdnormallink{morphisms {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are maps $h_{jk}:Z_{j}\rightarrow Z_{k}$ such that $h_{jk}\mid QF_{j}=g:(QSS_{j},QF_{j})\rightarrow (QSS_{k},QF_{k})$, 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 $QF_{j}$ , and also rel $QF_{k}$}.

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