Definition 0.1   A quantum fundamental groupoid $F_{\Q}$ is defined as a functor $F_{\Q}: \mathbf{H}_B \to {\Q}_G$ , where $\mathbf{H}_B$ is the category of Hilbert space bundles, and ${\Q}_G$ is the category of locally compact quantum groupoids and their homomorphisms.

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).