Name for a concept?

Name for a concept?

[David Yetter]

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

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Re: Name for a concept?

[Aleks Kissinger]

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
>


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Re: Name for a concept?

[Eduardo J. Dubuc]

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.


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Re: Name for a concept?

[Prof. Peter Johnstone]

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


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Re: Name for a concept?

[Eduardo Pareja-Tobes]

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
>

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Re: Name for a concept?

[M. Bjerrum]

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/ ]
>



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Re: Name for a concept?

[Eduardo Pareja-Tobes]

​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.
>
>
>

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Re: Name for a concept?

[M. Bjerrum]

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!​



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Re: Name for a concept?

[Paul Taylor]

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




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Re: Name for a concept?

[Michael Barr]

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


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