theorem on CW--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), 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$ 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 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} VV @VV 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.



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As of this snapshot date, this entry was owned by bci1.