Title 



Grothendieck quantaloids for allegories of enriched categories
 
Author 



 
Abstract 



For any small involutive quantaloid Q we define, in terms of symmetric quantaloidenriched categories, an involutive quantaloid Rel(Q) of Qsheaves and relations, and a category Sh (Q) of Qsheaves 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 semisimple 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.   
Language 



English
 
Source (journal) 



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



Brussels : 2012
 
ISSN 



13701444 [print]
20341970 [online]
 
Volume/pages 



19:5(2012), p. 859888
 
ISI 



000313931500005
 
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