​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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