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-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!​

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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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/ ]