Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Toward a non-commutative Gelfand duality: Boolean locally separated
  toposes and Monoidal monotone complete $C^{*}$-categories   ** 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.

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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn