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When discussing the alternative notion of distributivity for sound doctrines, this page says
See Appendix A in AR for a comparison of this definition to the above explicit definition.
But I haven’t been able to find such a comparison in the cited appendix. It seems to me that the appendix just mentions that their sound-doctrine notion of distributivity is distinct from the notion of commutativity. Am I missing something?
Rewrote a confusing paragraph:
See Appendix A in AR for a comparison of this definition to the above explicit definition in the special case of distributivity of filtered colimits over small limits in locally finitely presentable categories and distributivity of sifted colimits over small limits in varieties of algebras.
Re #2: Example A.4 in the cited paper does talk about the special case of the doctrine of filtered colimits in some detail:
Example A.4. In every lfp category filtered colimits distribute over products. This was already proved by Artin, Grothendieck and Verdier in [10]. To verify this, recall the following description of Ind K from [18]: objects are all small filtered diagrams D : D → K. Morphisms into another filtered diagram ¯D : ¯D → K are compatible families of equivalence classes [fd]d∈obj D of morphisms fd : Dd → ¯Dd′, d′ ∈ obj ¯D, under the smallest equivalence ∼ with fd ∼ D¯u · fd for every morphism ¯u : d′ → d′′ in ¯D. Compatibility means that for every morphism v : d1 → d2 in D we have fd1 ∼ fd2 · Dv. Now a product in Ind K is easy to describe: given objects Di : Di → K for i ∈ I, their product is the filtered diagram D : i∈I Di → K, D(di)i∈I =i∈I Didi And in every lfp category the colimit of D is canonically given by the product of colimits of Di, i ∈ Ij, see e.g. [8], proof of 2.1. Now, distributivity of filtered colimits in K over products means that K has both, and that given filtered diagrams Di (i ∈ i) we have colim D =i∈I colim Di
The original definition is due to Jan-Erik Roos, see Definition 1 in
Was the PDF of Roos’s paper meant to be uploaded to the nLab? I wasn’t able to find a copy elsewhere.
It was uploaded, but the link-to-uploaded-file-functionality does not work here on the nForum.
You can see the working link on the nLab page here, it’s at ncatlab.org/nlab/files/roos-distributivity.pdf
I managed to find the first two Roos papers on the CRAS archive (Gallica), but I couldn’t find the third one, despite having a page reference. If someone could find it, I’d be grateful if you could add the link to the page.
In the case of presentation of ind-objects and pro-objects as formal limits or colimits of (co)representables, a corollary is that there is a functor
$Ind(Pro(C))\to Pro(Ind(C))$which is in some interesting cases inclusion.
Added (perhaps a separate article might be needed eventually for the (∞,1)-case:
The condition that small (∞,1)-limits distribute over (∞,1)-filtered (∞,1)-colimits precisely characterizes compactly assembled (∞,1)-categories among all presentable (∞,1)-categories.
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