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Title
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Grothendieck quantaloids for allegories of enriched categories
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Author
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Abstract
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| 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 kale groupoid G then Sh (O (G)) is the classifying topos of G. |
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Language
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English
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Source (journal)
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Bulletin of the Belgian Mathematical Society Simon Stevin. - Brussels, 1994, currens
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Publication
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Brussels
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2012
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ISSN
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1370-1444
[print]
2034-1970
[online]
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DOI
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10.36045/BBMS/1354031554
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Volume/pages
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19
:5
(2012)
, p. 859-888
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ISI
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000313931500005
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Full text (Publisher's DOI)
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Full text (publisher's version - intranet only)
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