In the past decade, A. Kitaev proposed that the set of invertible gapped quantum spin systems would form an $\Omega$-spectrum. This conjecture has potentially significant application to the study of SPT phases. We give a mathematically rigorous realization of this proposal with the language of functional analysis and operator algebra. Furthermore, we formulate the idea of placing a quantum spin system on a general space beyond R^n, which gives rise to a model of the generalized homology theory associated with Kitaev's spectrum. As an application, we study the topological phase of invertible phases protected by a crystallographic symmetry. This talk is based on the preprint https://arxiv.org/abs/2503.12618.