$C W$ -complex approximation of quantum state spaces in QAT

Theorem 1.

Let $[QF_{j}]_{j=1,...,n}$ be a complete sequence of commuting quantum spin ‘foams’(QSFs) in an arbitrary quantum state space (QSS) (http://planetmath.org/QuantumSpaceTimes), 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 $i=n$ .

Remark 0.1.

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

Corollary 2.

The $n$ -connected models $(QF_{j},Z_{j})$ of $(QF_{j},QSS_{j})$ form the Model Category ofQuantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) $(QF_{j})$ , whose morphisms 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:

$\\begin{CD}Z_{j}@>{f_{j}}>{}>QSS_{j}\\\\@V{h_{jk}}V{}V@V{}V{g}V\\\\Z_{k}@ >f_{k}>>QSS_{k}\\end{CD}$
Furthermore, the maps $h_{jk}$ are unique up to the homotopy rel $QF_{j}$ , and also rel $QF_{k}$ .

Remark 0.2.

Theorem 1 complements other data presented in the parent entry on QAT (http://planetmath.org/QuantumAlgebraicTopology).

单词	CWcomplexApproximationOfQuantumStateSpacesInQAT
释义	$C W$ -complex approximation of quantum state spaces in QAT Theorem 1. Let $[QF_{j}]_{j=1,...,n}$ be a complete sequence of commuting quantum spin ‘foams’(QSFs) in an arbitrary quantum state space (QSS) (http://planetmath.org/QuantumSpaceTimes), 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 $i=n$ . Remark 0.1. There exist weak* homotopy equivalences between each $Z_{j}$ and $QSS_{j}$ spacesin such a sequence. Therefore, there exists a $C W$ –complex approximation of QSS defined by the sequence $[Z_{j}]_{j=1,...,n}$ of CW-complexes with dimension $n\\geq 2$ . This $C W$ –approximation isunique up to regular homotopy equivalence. Corollary 2. The $n$ -connected models $(QF_{j},Z_{j})$ of $(QF_{j},QSS_{j})$ form the Model Category ofQuantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) $(QF_{j})$ , whose morphisms 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: $\\begin{CD}Z_{j}@>{f_{j}}>{}>QSS_{j}\\\\@V{h_{jk}}V{}V@V{}V{g}V\\\\Z_{k}@ >f_{k}>>QSS_{k}\\end{CD}$ Furthermore, the maps $h_{jk}$ are unique up to the homotopy rel $QF_{j}$ , and also rel $QF_{k}$ . Remark 0.2. Theorem 1 complements other data presented in the parent entry on QAT (http://planetmath.org/QuantumAlgebraicTopology).
随便看	the top 10 most beautiful theorems TheTopologistsSineCurveHasTheFixedPointProperty the topologist’s sine curve has the fixed point property TheTorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve the torsion subgroup of an elliptic curve injects in the reduction of the curve The type of booleans TheUnionOfALocallyFiniteCollectionOfClosedSetsIsClosed the union of a locally finite collection of closed sets is closed The unit type The universal cover as an identity system The universal cover in type theory The van Kampen theorem The van Kampen theorem with a set of basepoints The Yoneda lemma ThinAlgebraicSet thin algebraic set ThinDoubleTrack thin double track ThirdIsomorphismTheorem third isomorphism theorem Thirteen thirteen This is my sample file ThisIsMySampleFile1 ThomassensTheoremOn3connectedGraphs

C⁢W-complex approximation of quantum state spaces in QAT

Remark 0.1.

Remark 0.2.

$C W$ -complex approximation of quantum state spaces in QAT