Re: Name for a concept?

[Eduardo Pareja-Tobes <eparejatobes@ohnosequences.com>]

​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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