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

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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

The subject of adhesive categories came up in this forum, first raised by Alexey Cherchago and then googled by Claudio Hermida. I thought that I could clear a few things up. Adhesive categories were introduced by Steve Lack and myself in a paper at FoSSaCS '04. Recently an extended version has been accepted for publication in Theoretical Informatics and Applications. Both versions are available to download from my homepage. A van Kampen (VK) square is a certain pushout. First, such a pushout must be stable under pullback -- in the sense that if we start with a VK square plus one morphism to the "result" of the pushout and keep taking pullbacks, we obtain a cube where the bottom face is our original VK square -- "stable under pullback" means that the top face is now also a pushout. Now let us imagine that a cube is oriented in such a way that all the arrows of the top and bottom face point out of the page. The second property satisfied by VK squares is that starting with such an oriented cube over a VK square where the top face is a pushout and the back faces are pullbacks, then the front faces are required to be pullbacks. Thus, the defining property of "VK square" can be stated reasonably concisely (and less confusingly, if one has a picture to look at): given any cube over a VK square with back faces pullbacks, the top face is a pushout iff the front faces are pullbacks. Unfortunately, not every pushout in Set is VK - which leads to the definition of adhesive categories: a category is adhesive if it has pushouts along monos, pullbacks and pushouts along monos are VK squares. A weaker assumption, pushouts along regular monos being VK, defines quasiadhesive categories. It turns out that all monos are regular in adhesive categories. Set is adhesive, as well as any elementary topos. Moreover, adhesivity is stable under slice, coslice, product and functor category. The VK property is very similar to the property satisfied by coproducts in extensive categories. And, as in extensive categories, it can be characterised by a suitable "equivalence of categories" definition - perhaps a little less `abstruse' for some. Moreover, many of the cute little lemmas about coproducts in extensive categories translate to pushouts which led us to use the slogan "adhesive categories have well-behaved pushouts along monos" just as "extensive categories have well-behaved coproducts". Extensive and adhesive categories are also directly related: any adhesive category with a strict initial object is extensive. However, there are adhesive categories which are not extensive and extensive categories which are not adhesive. I'll also mention that the algebra of subobjects is quite nice in adhesive categories, subobject union is calculated as the pushout along their intersection and the resulting lattice is distributive. Finally, perhaps I should apologise for the apparent poor quality of the explanations in our papers. Somehow they got through 5 thorough reviews without any of the referees being confused by the notion of VK square - perhaps Claudio could help by suggesting how the explanation could be made clearer. On 26 Nov 2004, at 15:40, Claudio Hermida wrote:

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.

29-Nov-2004 16:05:09 -0400,1649;000000000001-00000000