For computer scientists lurking on the list, or others who want a really simple minded approach, the following might be helpful @article(fun3, title = "Some Fundamental Algebraic Tools for the Semantics of Computation, Part 3: Indexed Categories", author = "Andrzej Tarlecki and Rod Burstall and Joseph Goguen", journal = "Theoretical Computer Science", year = 1991, volume = 91, pages = "239--264") though of course John Gray's "Fibred and Cofibred Categories" is the ur-text. == joseph ************************************************************************ Joseph Goguen, Dept. Computer Science & Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla CA 92093-0114 USA email: jgoguen@ucsd.edu www: http://www.cs.ucsd.edu/users/goguen/ phone: (858) 534-4197 [my office]; -1246 [dept office]; -7029 [dept fax]; (858) 822-0702 [secy] office: 3131 Applied Physics and Math Building J Consciousness Studies: http://www.imprint-academic.com/jcs/ ************************************************************************
Date: Fri, 26 Nov 2004 14:40:21 +0000 From: Claudio Hermida <chermida@math.ist.utl.pt> User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en Precedence: bulk X-Spam-Level: Level X-Spamscanner: mailbox8.ucsd.edu (v1.5 Aug 25 2004 09:28:35, -2.8/5.0 3.0.0) X-MailScanner: PASSED (v1.2.8 89601 iAQITWiC097817 mailbox8.ucsd.edu) X-Spam-Flag: Spam NO X-Scanned-By: milter-spamc/0.15.245 (fast.ucsd.edu [132.239.15.4]); pass=YES; Fri, 26 Nov 2004 10:29:37 -0800 X-Spam-Status: NO, hits=-4.50 required=5.00
Dear Cat Community,
I have the following question about preservation of properties of fiber categories in the total category:
Given a contravariant indexed category F : B^Op -> Cat, and the Grothendieck translation from the indexed category to fibrations Flat(F). Then, the first projection P : Flat(F) -> B forms a (split) fibration, called the flattening of F.
Now assume that the fiber categories are complete/cocomplete/adhesive. Would it be the case that the total category Flat(F) of the described (split) fibration is also complete/cocomplete/adhesive?
Any references would be appreciated. Thank you kindly.
Best regards, Alexey Cherchago
Given a fibration P: Flat(F) -> B, assuming B is complete then (E is complete and P continuous) iff P is fibred complete (complete fibres and continuous reindexing) .
Dually for cofbirations (coming from 'covariant' indexed categories) and cocompleteness.
I am clueless about 'adhesive categories' but Google points out to the following def: C adhesive means it has pullbacks, pushout along monos and these latter satisfy a certain exactness condition involving pullback stability (a so-called "VK square" which is actually a cube(?)).
Assuming P: Flat(F) -> B preserves monos, an arrow in Flat(F) is monic iff its image in B and its vertical factor are monic. Ergo, assume:
B adhesive P has direct images along monos (m^* have left adjoints)
Then, (P has fibrewise pb/po along monos) implies (Flat(F) has same and P preserves that).. The VK condition is a little obtruse, and at first sight requires the following additional assumption: the ' po along monos' in Flat(F) preserves cartesian morphisms and reindexing functors preserve po along monos. Then it seems that one gets (if one cares to do the relevant calculations)
P fibrewise adhesive iff Flat(P) adhesive (for the only if, given a stack of 2 cubes where the bottom one has all side faces pb, the total cube is a VK square iff the top one is such).
References: On the correspondence of limits/colimits between fibres and total Flat(F) see:
- Gray, J. W., Fibred and Cofibred categories, Proceedings of the Conference on Categorical Algebra,1966.
There are more recent references which treat this subject (fibred vs. global properties) from a more abstract point of view (and it a broader context), but they are probably not the first place to look if one is not acquainted with the basics.
28-Nov-2004 13:37:05 -0400,1549;000000000000-00000000