* Name for a concept? @ 2013-04-25 3:14 David Yetter 2013-04-25 14:19 ` Aleks Kissinger ` (3 more replies) 0 siblings, 4 replies; 10+ messages in thread From: David Yetter @ 2013-04-25 3:14 UTC (permalink / raw) To: categories Is there an existing name in the literature for a category in which every cospan admits a completion to a commutative square? (Just that, no uniqueness, no universal properties required, just every cospan sits inside at least one commutative square). If so, what have such things been called? If not, does anyone have a poetic idea for a good name for such categories? Best Thoughts, David Yetter [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 3:14 Name for a concept? David Yetter @ 2013-04-25 14:19 ` Aleks Kissinger 2013-04-26 1:05 ` Eduardo J. Dubuc 2013-04-25 14:37 ` Prof. Peter Johnstone ` (2 subsequent siblings) 3 siblings, 1 reply; 10+ messages in thread From: Aleks Kissinger @ 2013-04-25 14:19 UTC (permalink / raw) To: David Yetter; +Cc: categories Oops, forgot to send to list. I think its actually a stronger property, but: perhaps cofiltered category? On 25 April 2013 04:14, David Yetter <dyetter@math.ksu.edu> wrote: > Is there an existing name in the literature for a category in which every cospan admits a completion to a commutative square? (Just that, no uniqueness, no universal > properties required, just every cospan sits inside at least one commutative square). If so, what have such things been called? If not, does anyone have a poetic idea for a good name for > such categories? > > Best Thoughts, > David Yetter > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 14:19 ` Aleks Kissinger @ 2013-04-26 1:05 ` Eduardo J. Dubuc 0 siblings, 0 replies; 10+ messages in thread From: Eduardo J. Dubuc @ 2013-04-26 1:05 UTC (permalink / raw) To: Aleks Kissinger; +Cc: David Yetter, categories On 25/04/13 11:19, Aleks Kissinger wrote: > Oops, forgot to send to list. > > I think its actually a stronger property, but: perhaps cofiltered category? > > On 25 April 2013 04:14, David Yetter<dyetter@math.ksu.edu> wrote: >> Is there an existing name in the literature for a category in which every cospan admits a completion to a commutative square? (Just that, no uniqueness, no universal >> properties required, just every cospan sits inside at least one commutative square). If so, what have such things been called? If not, does anyone have a poetic idea for a good name for >> such categories? >> >> Best Thoughts, >> David Yetter >> > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] yes !, a cofiltered category is just a connected such category, Verdier's formulation, see Mac Lane's book if you don't like the SGA4. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 3:14 Name for a concept? David Yetter 2013-04-25 14:19 ` Aleks Kissinger @ 2013-04-25 14:37 ` Prof. Peter Johnstone 2013-04-25 18:47 ` Eduardo Pareja-Tobes 2013-04-26 10:44 ` Paul Taylor 3 siblings, 0 replies; 10+ messages in thread From: Prof. Peter Johnstone @ 2013-04-25 14:37 UTC (permalink / raw) To: David Yetter; +Cc: categories I call this the `right Ore condition' (e.g. on page 79 of the Elephant) because that's what it reduces to in a monoid. Peter Johnstone On Wed, 24 Apr 2013, David Yetter wrote: > Is there an existing name in the literature for a category in which every cospan admits a completion to a commutative square? (Just that, no uniqueness, no universal > properties required, just every cospan sits inside at least one commutative square). If so, what have such things been called? If not, does anyone have a poetic idea for a good name for > such categories? > > Best Thoughts, > David Yetter [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 3:14 Name for a concept? David Yetter 2013-04-25 14:19 ` Aleks Kissinger 2013-04-25 14:37 ` Prof. Peter Johnstone @ 2013-04-25 18:47 ` Eduardo Pareja-Tobes 2013-04-26 13:03 ` M. Bjerrum ` (2 more replies) 2013-04-26 10:44 ` Paul Taylor 3 siblings, 3 replies; 10+ messages in thread From: Eduardo Pareja-Tobes @ 2013-04-25 18:47 UTC (permalink / raw) To: David Yetter; +Cc: categories There is something about this, yes; I read about this sort of things some time ago, so take what follows with a grain of salt. First, I will work with the opposite of your category, so what we have is a category for which every span can be completed to a commutative square. Let's call this notion "span-directed". Obviously, this looks like some sort of generalized filteredness notion; every filtered category is span-directed. As defined, span-directed does not require connectedness, so this look more like "pseudo-filtered", which according to Mac Lane CftWM was something introduced by Verdier [SGA4I, I.2.7]. A category is pseudo-filtered iff is a coproduct of filtered categories. Now, in Definition 53 of [Protolocalisations of homological categories - Borceux, Clementino, Gran, Sousa], they name a category protofiltered if it is span-directed and connected. So, according to all this, if one was to follow the same pattern, span-directed should be named "pseudo-protofiltered" :) I think it would be possible to have a conceptual characterization of span-directed/pseudo-protofiltered categories, in terms of distributivity of colimits in SET indexed by them over some natural class of limits. That is, as D-filtered categories for D a "doctrine of D-limits" in the terminology of [A classification of accessible categories - Adámek, Borceux, Lack, Rosický]. There, for a small class of categories D a category I is said to be D-filtered if colimits indexed by I distribute over D-limits (any diagram indexed by a category in D) in SET. Then, you get that 1. If D = finite categories, D-filtered = filtered 2. If D = finite connected categories, D-filtered = pseudo-filtered 3. If D = finite discrete categories, D-filtered = sifted Speculative content follows, possibly everything after this point is wrong: Now, I think that if you take D = equalizers what you get could be D-filtered = protofiltered, or at least something similar. Something like this is in p30 of [Sur la commutation des limites - Foltz] , which I got from this MathOverflow answer: http://mathoverflow.net/questions/93262/which-colimits-commute-with-which-limits-in-the-category-of-sets . So, maybe what we need to take for obtaining D-filtered = span-directed/pseudo-protofiltered is D = coreflexive equalizers. As finite products and coreflexive equalizers = finite limits in SET, this would mean that sifted + span-directed/pseudo-protofiltered => filtered, thus providing an affirmative answer to "It remains an open problem to determine whether a sifted protofiltered category is filtered": [Protolocalisations of homological categories] -- Eduardo Pareja-Tobes oh no sequences! On Thu, Apr 25, 2013 at 5:14 AM, David Yetter <dyetter@math.ksu.edu> wrote: > Is there an existing name in the literature for a category in which every > cospan admits a completion to a commutative square? (Just that, no > uniqueness, no universal > properties required, just every cospan sits inside at least one > commutative square). If so, what have such things been called? If not, > does anyone have a poetic idea for a good name for > such categories? > > Best Thoughts, > David Yetter > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 18:47 ` Eduardo Pareja-Tobes @ 2013-04-26 13:03 ` M. Bjerrum [not found] ` <Prayer.1.3.5.1304261403520.11018@hermes-2.csi.cam.ac.uk> 2013-04-26 14:07 ` M. Bjerrum 2 siblings, 0 replies; 10+ messages in thread From: M. Bjerrum @ 2013-04-26 13:03 UTC (permalink / raw) To: Eduardo Pareja-Tobes; +Cc: David Yetter, categories Hello, I suppose that a span is a diagram of finitely many arrows of same domain (or the op-situation). And the question concerns a name for categories with co-cones over all such diagrams. I don't have a very poetic name for this. At the moment I'm content with saying that such categories have V-cocones (or V-cones) depending on directions. I've seen it being called the "Amalgamation Property". But as to what concerns the connection with the question of mixed interchange of limits in Set, one needs to be very careful: If for some doctrine D one defines D-filtered categories to be categories J such that J-colimits commute with D-limits in Set, then this terminology will not do, since we then have. 1) If D is equalizers then D-filtered=pseudofiltered. 2) If D is pullbacks then D-filtered=pseudofiltered. 3) If D is pullbacks and terminal objects, then D-filtered=filtered (and not proto-pseudofilterd as one could hope for) So one need to distinguish between three things: 1) having cocones over certain diagrams. 2) the categories of cocones over certain diagrams are connected. 3) commuting in Set with limits over certain diagrams. What has been called "sound doctrines", are the doctrines such that 2) and 3) are equivalent. As a short answer to the open question: If J is a sifted + proto-pseudofiltered category, i.e sifted and span-directed then J is pseudofiltered and connected and thus filtered. (since pseudofiltered categories are categories with filtered connected components) This kind of reflexions and more, with proofs, will soon be available via my PhD thesis. Best wishes, Marie Bjerrum. On Apr 26 2013, Eduardo Pareja-Tobes wrote: >There is something about this, yes; I read about this sort of things some >time ago, so take what follows with a grain of salt. > >First, I will work with the opposite of your category, so what we have is a >category for which every span can be completed to a commutative square. >Let's call this notion "span-directed". Obviously, this looks like some >sort of generalized filteredness notion; every filtered category is >span-directed. > >As defined, span-directed does not require connectedness, so this look more >like "pseudo-filtered", which according to Mac Lane CftWM was something >introduced by Verdier [SGA4I, I.2.7]. A category is pseudo-filtered iff is >a coproduct of filtered categories. > >Now, in Definition 53 of [Protolocalisations of homological categories - >Borceux, Clementino, Gran, Sousa], they name a category protofiltered if it >is span-directed and connected. > >So, according to all this, if one was to follow the same pattern, >span-directed should be named "pseudo-protofiltered" :) > >I think it would be possible to have a conceptual characterization of >span-directed/pseudo-protofiltered categories, in terms of distributivity >of colimits in SET indexed by them over some natural class of limits. That >is, as D-filtered categories for D a "doctrine of D-limits" in the >terminology of [A classification of accessible categories - Adámek, >Borceux, Lack, Rosický]. > >There, for a small class of categories D a category I is said to be >D-filtered if colimits indexed by I distribute over D-limits (any diagram >indexed by a category in D) in SET. Then, you get that > >1. If D = finite categories, D-filtered = filtered >2. If D = finite connected categories, D-filtered = pseudo-filtered >3. If D = finite discrete categories, D-filtered = sifted > >Speculative content follows, possibly everything after this point is wrong: > > Now, I think that if you take D = equalizers what you get could be > D-filtered = protofiltered, or at least something similar. Something like > this is in p30 of [Sur la commutation des limites - Foltz] , which I got > from this MathOverflow answer: > http://mathoverflow.net/questions/93262/which-colimits-commute-with-which-limits-in-the-category-of-sets > . > >So, maybe what we need to take for obtaining D-filtered = >span-directed/pseudo-protofiltered is D = coreflexive equalizers. As finite >products and coreflexive equalizers = finite limits in SET, this would mean >that sifted + span-directed/pseudo-protofiltered => filtered, thus >providing an affirmative answer to "It remains an open problem to determine >whether a sifted protofiltered category >is filtered": [Protolocalisations of homological categories] > >-- >Eduardo Pareja-Tobes >oh no sequences! > >On Thu, Apr 25, 2013 at 5:14 AM, David Yetter <dyetter@math.ksu.edu> wrote: > >> Is there an existing name in the literature for a category in which every >> cospan admits a completion to a commutative square? (Just that, no >> uniqueness, no universal >> properties required, just every cospan sits inside at least one >> commutative square). If so, what have such things been called? If not, >> does anyone have a poetic idea for a good name for >> such categories? >> >> Best Thoughts, >> David Yetter >> > >[For admin and other information see: http://www.mta.ca/~cat-dist/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
[parent not found: <Prayer.1.3.5.1304261403520.11018@hermes-2.csi.cam.ac.uk>]
* Re: Name for a concept? [not found] ` <Prayer.1.3.5.1304261403520.11018@hermes-2.csi.cam.ac.uk> @ 2013-04-26 14:05 ` Eduardo Pareja-Tobes 0 siblings, 0 replies; 10+ messages in thread From: Eduardo Pareja-Tobes @ 2013-04-26 14:05 UTC (permalink / raw) To: M. Bjerrum; +Cc: David Yetter, categories Hello Marie, Nice! So, you think that none of protofiltered and pseudo-protofiltered can be characterized as D-filtered for D a sound doctrine? waiting for your PhD thesis :) Eduardo Pareja-Tobes Math & CS freak *oh no sequences!* <http://ohnosequences.com> On Fri, Apr 26, 2013 at 3:03 PM, M. Bjerrum <mb617@cam.ac.uk> wrote: > Hello, > > I suppose that a span is a diagram of finitely many arrows of same domain > (or the op-situation). And the question concerns a name for categories with > co-cones over all such diagrams. I don't have a very poetic name for this. > At the moment I'm content with saying that such categories have V-cocones > (or V-cones) depending on directions. I've seen it being called the > "Amalgamation Property". > > But as to what concerns the connection with the question of mixed > interchange of limits in Set, one needs to be very careful: > > If for some doctrine D one defines D-filtered categories to be categories > J such that J-colimits commute with D-limits in Set, then this terminology > will not do, since we then have. > > 1) If D is equalizers then D-filtered=pseudofiltered. 2) If D is pullbacks > then D-filtered=pseudofiltered. 3) If D is pullbacks and terminal objects, > then D-filtered=filtered (and not proto-pseudofilterd as one could hope for) > > > So one need to distinguish between three things: > 1) having cocones over certain diagrams. > 2) the categories of cocones over certain diagrams are connected. 3) > commuting in Set with limits over certain diagrams. > > What has been called "sound doctrines", are the doctrines such that 2) and > 3) are equivalent. > > As a short answer to the open question: If J is a sifted + > proto-pseudofiltered category, i.e sifted and span-directed then J is > pseudofiltered and connected and thus filtered. (since pseudofiltered > categories are categories with filtered connected components) > > This kind of reflexions and more, with proofs, will soon be available via > my PhD thesis. > > Best wishes, > > Marie Bjerrum. > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 18:47 ` Eduardo Pareja-Tobes 2013-04-26 13:03 ` M. Bjerrum [not found] ` <Prayer.1.3.5.1304261403520.11018@hermes-2.csi.cam.ac.uk> @ 2013-04-26 14:07 ` M. Bjerrum 2 siblings, 0 replies; 10+ messages in thread From: M. Bjerrum @ 2013-04-26 14:07 UTC (permalink / raw) To: Eduardo Pareja-Tobes; +Cc: David Yetter, categories Sorry, I seem to have confused the names pseudo-protofiltered and protofiltered in the previous. I took "pseudo-protofiltered" as a suggested name for connected and "span-directed", instead of just span-directed (having V-cocones). But if you just ignore the "pseudo", this should not disturb the content of my previous mail... i.e correction of previous, if D-filtered means "commuting with D-limits" then: 2) If D is pullbacks (V-limits) then D-filtered=pseudofiltered (and not just pseudo-protofiltered) 3) If D is pullbacks and terminal objects ({V,Ø}-limits) then D-filtered=filtered (and not just protofiltered) ..sorry for the confusion.. On Apr 26 2013, Eduardo Pareja-Tobes wrote: >There is something about this, yes; I read about this sort of things some >time ago, so take what follows with a grain of salt. > >First, I will work with the opposite of your category, so what we have is a >category for which every span can be completed to a commutative square. >Let's call this notion "span-directed". Obviously, this looks like some >sort of generalized filteredness notion; every filtered category is >span-directed. > >As defined, span-directed does not require connectedness, so this look more >like "pseudo-filtered", which according to Mac Lane CftWM was something >introduced by Verdier [SGA4I, I.2.7]. A category is pseudo-filtered iff is >a coproduct of filtered categories. > >Now, in Definition 53 of [Protolocalisations of homological categories - >Borceux, Clementino, Gran, Sousa], they name a category protofiltered if it >is span-directed and connected. > >So, according to all this, if one was to follow the same pattern, >span-directed should be named "pseudo-protofiltered" :) > >I think it would be possible to have a conceptual characterization of >span-directed/pseudo-protofiltered categories, in terms of distributivity >of colimits in SET indexed by them over some natural class of limits. That >is, as D-filtered categories for D a "doctrine of D-limits" in the >terminology of [A classification of accessible categories - Adámek, >Borceux, Lack, Rosický]. > >There, for a small class of categories D a category I is said to be >D-filtered if colimits indexed by I distribute over D-limits (any diagram >indexed by a category in D) in SET. Then, you get that > >1. If D = finite categories, D-filtered = filtered >2. If D = finite connected categories, D-filtered = pseudo-filtered >3. If D = finite discrete categories, D-filtered = sifted > >Speculative content follows, possibly everything after this point is wrong: > > Now, I think that if you take D = equalizers what you get could be > D-filtered = protofiltered, or at least something similar. Something like > this is in p30 of [Sur la commutation des limites - Foltz] , which I got > from this MathOverflow answer: > http://mathoverflow.net/questions/93262/which-colimits-commute-with-which-limits-in-the-category-of-sets > . > >So, maybe what we need to take for obtaining D-filtered = >span-directed/pseudo-protofiltered is D = coreflexive equalizers. As finite >products and coreflexive equalizers = finite limits in SET, this would mean >that sifted + span-directed/pseudo-protofiltered => filtered, thus >providing an affirmative answer to "It remains an open problem to determine >whether a sifted protofiltered category >is filtered": [Protolocalisations of homological categories] > >-- >Eduardo Pareja-Tobes >oh no sequences! [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? 2013-04-25 3:14 Name for a concept? David Yetter ` (2 preceding siblings ...) 2013-04-25 18:47 ` Eduardo Pareja-Tobes @ 2013-04-26 10:44 ` Paul Taylor 3 siblings, 0 replies; 10+ messages in thread From: Paul Taylor @ 2013-04-26 10:44 UTC (permalink / raw) To: David Yetter; +Cc: categories David Yetter asked, > Is there an existing name in the literature for a category in which > every cospan admits a completion to a commutative square? (Just > that, no uniqueness, no universal properties required, just every > cospan sits inside at least one commutative square). If so, what > have such things been called? If not, does anyone have a poetic > idea for a good name for such categories? Eduardo Pareja-Tobes replied with quite a long discussion of filtered categories, but I am not sure whether he made the following point clear. Any group (the two-element one will do) considered as a one-object category has the property that David mentioned. However, it is not "filtered" in the sense of Chapter IX, Section 1 of Mac Lane's "Categories for the Working Mathematician". You have to look carefully at the printed diagram to see this. Whilst any (co)span in a group may be completed to a commutative square, a parallel pair does not have a common composite in this sense: -------> X Y -------> Z -------> David's definitive language suggests that he knows this and does not require this property of his categories, but other people who want to generalise ideas from posets to categories might fall into this trap. Maybe "amalgamation property" would be suitable for David, although usage of that term in the literature might require the maps to be monos. Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Name for a concept? @ 2013-04-27 13:58 Michael Barr 0 siblings, 0 replies; 10+ messages in thread From: Michael Barr @ 2013-04-27 13:58 UTC (permalink / raw) To: Categories list It seems to me that I called it an Ore condition, as suggested by Peter Johnstone, in my Acyclic Models book and I almost surely got that name from Gabriel-Zisman. Since it has been used that way in at least two books and generalizes a well-known condition from rings (also monoids) I see no good reason not to go with that. Michael -- The modern conservative is engaged in one of man's oldest exercises in moral philosophy--the search for a superior moral justification for selfishness. --J.K. Galbraith [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
end of thread, other threads:[~2013-04-27 13:58 UTC | newest] Thread overview: 10+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2013-04-25 3:14 Name for a concept? David Yetter 2013-04-25 14:19 ` Aleks Kissinger 2013-04-26 1:05 ` Eduardo J. Dubuc 2013-04-25 14:37 ` Prof. Peter Johnstone 2013-04-25 18:47 ` Eduardo Pareja-Tobes 2013-04-26 13:03 ` M. Bjerrum [not found] ` <Prayer.1.3.5.1304261403520.11018@hermes-2.csi.cam.ac.uk> 2013-04-26 14:05 ` Eduardo Pareja-Tobes 2013-04-26 14:07 ` M. Bjerrum 2013-04-26 10:44 ` Paul Taylor 2013-04-27 13:58 Michael Barr
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