Hermitian forms on Hilbert spaces arise from the monoid structure of complex conjugation in Z/2-equivariant real linear types within LHoTT, requiring only a negative unit term.
K(n)-local duality for finite groups and groupoids
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abstract
We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the usual inner product on the representation ring when n=1, and to the Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure. In many places we find it convenient to use groupoids rather than groups, and to assist with this we include a detailed treatment of the homotopy theory of groupoids. We also explain some striking formal similarities between our duality and Atiyah-Poincare duality for manifolds.
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quant-ph 2years
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The pushout of entangled and parameterized quantum information in monoidal categories yields the external tensor product on flat K-theory bundles.
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Quantum and Reality
Hermitian forms on Hilbert spaces arise from the monoid structure of complex conjugation in Z/2-equivariant real linear types within LHoTT, requiring only a negative unit term.
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Entanglement of Sections: The pushout of entangled and parameterized quantum information
The pushout of entangled and parameterized quantum information in monoidal categories yields the external tensor product on flat K-theory bundles.