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quantum fundamental groupoid∗ bci1† 2013-03-22 1:26:54 Definition 0.1. A quantum fundamental groupoid FQ is defined as a functor FQ : HB → QG , where HB is the category of Hilbert space bundles, and QG is the category of locally compact quantum groupoids and their homomorphisms. 0.1 Fundamental groupoid functors and functor categories The natural setting for the definition of a quantum fundamental groupoid FQ is in one of the functor categories– that of fundamental groupoid functors, FG , and their natural transformations defined in the context of quantum categories of quantum spaces Q represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces HB . Other related functor categories are those specified with the general definition of the fundamental groupoid functor, FG : Top → G2 , where Top is the category of topological spaces and G2 is the groupoid category. Example 0.1. A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specialized one-dimensional CW-complexes whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spaces (also called Frechét nuclear spaces). ∗ hQuantumFundamentalGroupoidi created: h2013-03-2i by: hbci1i version: h40865i Privacy setting: h1i hDefinitioni h55Q05i h55U40i h20L05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1
