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Every topological category is convenient for Gelfand duality

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Abstract

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.

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  1. Fachbereich Mathematik, Universität Bremen, 2800, Bremen 33, Federal Republic of Germany

    Hans -E. Porst & Manfred B. Wischnewsky

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  1. Hans -E. Porst
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  2. Manfred B. Wischnewsky
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Porst, H.E., Wischnewsky, M.B. Every topological category is convenient for Gelfand duality. Manuscripta Math 25, 169–204 (1978). https://doi.org/10.1007/BF01168608

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