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# $CW$-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:*

$\begin{matrix}Z_{j}&\cd@stack{\rightarrowfill@}{f_{j}}{}&QSS_{j}\\
{h_{{jk}}}{\Big\downarrow}&&{}{\Big\downarrow}{g}&&\\
Z_{k}@ >f_{k}>>QSS_{k}\end{matrix}$
*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.

## Mathematics Subject Classification

81T25*no label found*81T05

*no label found*81T10

*no label found*

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