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Journal Articles Bulletin of the Belgian Mathematical Society - Simon Stevin Year : 2012

Grothendieck quantaloids for allegories of enriched categories

Abstract

For any small involutive quantaloid Q we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that Rel(Q) is the category of relations in a topos if and only if Q is a modular, locally localic and weakly semi-simple quantaloid; in this case we call Q a Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever Q is a Grothendieck quantaloid. Any locale L is a Grothendieck quantale, and Sh(L) is the topos of sheaves on L. Any small quantaloid of closed cribles is a Grothendieck quantaloid, and if Q is the quantaloid of closed cribles in a Grothendieck site (C,J) then Sh(Q) is equivalent to the topos Sh(C,J). Any inverse quantal frame is a Grothendieck quantale, and if O(G) is the inverse quantal frame naturally associated with an \'etale groupoid G then Sh(O(G)) is the classifying topos of G.

Dates and versions

hal-03606336 , version 1 (11-03-2022)

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Cite

Hans Heymans, Isar Stubbe. Grothendieck quantaloids for allegories of enriched categories. Bulletin of the Belgian Mathematical Society - Simon Stevin, 2012, 19 (5), ⟨10.36045/bbms/1354031554⟩. ⟨hal-03606336⟩

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