Re: Name for a concept?

["M. Bjerrum" <mb617@cam.ac.uk>]

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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