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

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

Let $[Q F_{j}]_{j = 1, . . ., n}$ be a complete sequence of commuting [[../TriangulationMethodsForQuantizedSpacetimes2/|quantum spin `foams]]' (QSFs) in an arbitrary [[../QuantumSpaceTimes/|quantum state space (QSS)]], and let $(Q F_{j}, Q S S_{j})$ be the corresponding sequence of pair subspaces of [[../SUSY2/|QST]]. If $Z_{j}$ is a sequence of CW-complexes such that for any $j$ , $Q F_{j} \subset Z_{j}$ , then there exists a sequence of $n$ -connected models $(Q F_{j}, Z_{j})$ of $(Q F_{j}, Q S S_{j})$ and a sequence of induced [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphisms]] ${f_{*}}^{j} : π_{i} (Z_{j}) \to π_{i} (Q S S_{j})$ for $i > n$ , together with a sequence of induced [[../InjectiveMap/|monomorphisms]] for $i = n$ .

There exist weak [[../ThinEquivalence/|homotopy]] equivalences between each $Z_{j}$ and $Q S S_{j}$ spaces in such a sequence. Therefore, there exists a $C W$ --complex approximation of [[../QuantumSpinNetworkFunctor2/|QSS]] defined by the sequence $[Z_{j}]_{j = 1, . . ., n}$ of CW-complexes with dimension $n \geq 2$ . This $C W$ --approximation is unique up to [[../CoIntersections/|regular]] homotopy equivalence.

Corollary 2.

The $n$ -connected models $(Q F_{j}, Z_{j})$ of $(Q F_{j}, Q S S_{j})$ form the Model [[../Cod/|category]] of [[../SpinNetworksAndSpinFoams/|Quantum Spin Foams]] $(Q F_{j})$ , whose \htmladdnormallink{morphisms {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are maps $h_{j k} : Z_{j} \to Z_{k}$ such that $h_{j k} ∣ Q F_{j} = g : (Q S S_{j}, Q F_{j}) \to (Q S S_{k}, Q F_{k})$ , and also such that the following [[../TrivialGroupoid/|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 $Q F_{j}$ , and also rel $Q F_{k}$ }.

{Theorem 1} complements other data presented in the [[../QuantumAlgebraicTopology/|parent entry on QAT]].

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

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