2015/01/28 by Henry, Simon
#03G30 #18B25 #46L05 #46L10 #Category Theory (math.CT) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1501.07045
** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete C*-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal *-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal C*-categories yet to be discovered.