Title
Grothendieck quantaloids for allegories of enriched categories
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Brussels ,
Subject
Mathematics
Source (journal)
Bulletin of the Belgian Mathematical Society Simon Stevin. - Brussels, 1994, currens
Volume/pages
19(2012) :5 , p. 859-888
ISSN
1370-1444
2034-1970
ISI
000313931500005
Carrier
E
Target language
English (eng)
Affiliation
University of Antwerp
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 kale groupoid G then Sh (O (G)) is the classifying topos of G.
E-info
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000313931500005&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000313931500005&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848
https://repository.uantwerpen.be/docman/iruaauth/5bc922/30610383.pdf
Handle