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Manifestly unitary higher Hilbert spaces

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arxiv 2410.05120 v1 pith:NXIKYPI5 submitted 2024-10-07 math.QA math.CTmath.OA

Quan Chen , Giovanni Ferrer , Brett Hungar , David Penneys , Sean Sanford This is my paper · ORCID
classification math.QAmath.CTmath.OA
keywords spaceshilberthighercategoryconstructiondimensionalmanifestlyunitary
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abstract

Higher idempotent completion gives a formal inductive construction of the $n$-category of finite dimensional $n$-vector spaces starting with the complex numbers. We propose a manifestly unitary construction of low dimensional higher Hilbert spaces, formally constructing the $\mathrm{C}^*$-3-category of 3-Hilbert spaces from Baez's 2-Hilbert spaces, which itself forms a 3-Hilbert space. We prove that the forgetful functor from 3-Hilbert spaces to 3-vector spaces is fully faithful.

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  1. Topological defects in reflection positive topological field theories

    math-ph 2026-08 conditional novelty 5.0 of 10

    A reflection-positive two-dimensional defect TQFT gives its defect bicategory an O(2)-dagger structure, and with positivity a structure close to a 3-Hilbert space.

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