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-li... .
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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