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Manifestly unitary higher Hilbert spaces
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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.
Forward citations
Cited by 3 Pith papers
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On the Higher Categorical Structure of Topological Defects in Quantum Field Theories
Categories of topological defects with arbitrary tangential structures are proposed to be structured higher dagger categories, proven for stable structures admitting direct sums under the stratified cobordism hypothesis.
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Orthonormal bases for higher Hilbert spaces
For finite-dimensional 3-Hilbert spaces, orthonormal bases exist uniquely up to contractible choice, and the Yoneda embedding into the presheaf 3-Hilbert space is an isometric equivalence.
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Topological defects in reflection positive topological field theories
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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