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%%% Primary Title: theorem on CW--complex approximation of quantum state spaces in QAT
%%% Primary Category Code: 00.
%%% Filename: TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT.tex
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\textbf{\htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} 1.}
Let $[QF_j]_{j=1,...,n}$ be a complete sequence of commuting \htmladdnormallink{quantum spin `foams}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html}' (QSFs) in an arbitrary \htmladdnormallink{quantum state space (QSS)}{http://planetphysics.us/encyclopedia/QuantumSpaceTimes.html}, and let $(QF_j,QSS_j)$ be the corresponding sequence of pair subspaces of \htmladdnormallink{QST}{http://planetphysics.us/encyclopedia/SUSY2.html}. 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 \htmladdnormallink{isomorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} ${f_*}^j : \pi_i (Z_j)\rightarrow \pi_i (QSS_j)$
for $i>n$, together with a sequence of induced \htmladdnormallink{monomorphisms}{http://planetphysics.us/encyclopedia/InjectiveMap.html} for $i=n$.
\begin{remark}
There exist \emph{weak} \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} equivalences between each $Z_j$ and $QSS_j$ spaces
in such a sequence. Therefore, there exists a $CW$--complex approximation of \htmladdnormallink{QSS}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} defined by the sequence
$[Z_j]_{j=1,...,n}$ of CW-complexes with dimension $n \geq 2$. This $CW$--approximation is
unique up to \emph{\htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html}} homotopy equivalence.
\end{remark}
\textbf{Corollary 2.}
\emph{The $n$-connected models} $(QF_j,Z_j)$ of $(QF_j,QSS_j)$ form the \emph{Model \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html}} of
\htmladdnormallink{Quantum Spin Foams}{http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html} $(QF_j)$, \emph{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 \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is commutative:} \\
$
\begin{CD}
Z_j @> f_j >> QSS_j
\\ @V h_{jk} VV @VV g V
\\ Z_k @ > f_k >> QSS_k
\end{CD}
$
\\
\emph{Furthermore, the maps $h_{jk}$ are unique up to the homotopy rel $QF_j$ , and also rel $QF_k$}.
\begin{remark}
{Theorem 1} complements other data presented in the \htmladdnormallink{parent entry on QAT}{http://planetphysics.us/encyclopedia/QuantumAlgebraicTopology.html}.
\end{remark}
\end{document}
