Grothendieck quantaloids for allegories of enriched categories
Faculty of Sciences. Mathematics and Computer Science

article

2012
Brussels
, 2012

Mathematics

Bulletin of the Belgian Mathematical Society Simon Stevin. - Brussels, 1994, currens

19(2012)
:5
, p. 859-888

1370-1444

2034-1970

000313931500005

E

English (eng)

University of Antwerp

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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