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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