Publication
Title
Grothendieck quantaloids for allegories of enriched categories
Author
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.
Language
English
Source (journal)
Bulletin of the Belgian Mathematical Society Simon Stevin. - Brussels, 1994, currens
Publication
Brussels : 2012
ISSN
1370-1444 [print]
2034-1970 [online]
Volume/pages
19:5(2012), p. 859-888
ISI
000313931500005
Full text (publisher's version - intranet only)
UAntwerpen
Faculty/Department
Research group
Publication type
Subject
Affiliation
Publications with a UAntwerp address
External links
Web of Science
Record
Identification
Creation 20.09.2013
Last edited 11.08.2017
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