# $CW$-complex approximation of quantum state spaces in QAT

Theorem 1.

Let ${[Q{F}_{j}]}_{j=1,\mathrm{\dots},n}$ be a complete sequence of commuting quantum spin ‘foams’
(QSFs) in an arbitrary quantum state space^{} (QSS) (http://planetmath.org/QuantumSpaceTimes), and let $(Q{F}_{j},QS{S}_{j})$ be the corresponding sequence of pair subspaces^{} of 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
$(Q{F}_{j},QS{S}_{j})$ and a sequence of induced isomorphisms^{} $f_{*}{}^{j}:{\pi}_{i}({Z}_{j})\to {\pi}_{i}(QS{S}_{j})$
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 $QS{S}_{j}$ spaces
in such a sequence. Therefore, there exists a $CW$–complex approximation of QSS defined by the sequence
${[{Z}_{j}]}_{j=1,\mathrm{\dots},n}$ of CW-complexes with dimension^{} $n\ge 2$. This $CW$–approximation is
unique up to *regular ^{}* homotopy equivalence.

Corollary 2.

*The $n$-connected models* $(Q{F}_{j},{Z}_{j})$ of $(Q{F}_{j},QS{S}_{j})$ form the *Model Category* of
Quantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) $(Q{F}_{j})$, *whose morphisms ^{} are maps ${h}_{j\mathit{}k}\mathrm{:}{Z}_{j}\mathrm{\to}{Z}_{k}$ such that ${h}_{j\mathit{}k}\mathrm{\mid}Q{F}_{j}\mathrm{=}g\mathrm{:}\mathrm{(}QS{S}_{j}\mathrm{,}Q{F}_{j}\mathrm{)}\mathrm{\to}\mathrm{(}QS{S}_{k}\mathrm{,}Q{F}_{k}\mathrm{)}$, and also such that the following diagram is commutative^{}:*

$\begin{array}{ccc}\hfill {Z}_{j}\hfill & \hfill \stackrel{{f}_{j}}{\to}\hfill & \hfill QS{S}_{j}\hfill \\ \hfill {h}_{jk}\downarrow \hfill & & \hfill \downarrow g\hfill & & \\ \hfill {Z}_{k}\text{@}>{f}_{k}\gg QS{S}_{k}\hfill \end{array}$
*Furthermore, the maps ${h}_{j\mathit{}k}$ are unique up to the homotopy ^{} rel $Q\mathit{}{F}_{j}$ , and also rel $Q\mathit{}{F}_{k}$*.

###### Remark 0.2.

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

Title | $CW$-complex approximation of quantum state spaces in QAT |
---|---|

Canonical name | CWcomplexApproximationOfQuantumStateSpacesInQAT |

Date of creation | 2013-03-22 18:14:37 |

Last modified on | 2013-03-22 18:14:37 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 29 |

Author | bci1 (20947) |

Entry type | Theorem |

Classification | msc 81T25 |

Classification | msc 81T05 |

Classification | msc 81T10 |

Synonym | quantum spin networks approximations by $CW$-complexes |

Related topic | ApproximationTheoremForAnArbitrarySpace |

Related topic | HomotopyEquivalence |

Related topic | QuantumAlgebraicTopology |

Related topic | ApproximationTheoremForAnArbitrarySpace |

Related topic | SpinNetworksAndSpinFoams |

Related topic | QuantumSpaceTimes |

Defines | $CW$-complex approximation of quantum state spaces in QAT |