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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMar 19th 2020

Added a theorem with a reference.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeApr 26th 2021

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?

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 26th 2021

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.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 26th 2021

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

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 27th 2021

The original definition is due to Jan-Erik Roos, see Definition 1 in

• Jan-Erik Roos, Introduction à l’étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 259 (1964), 969–972. roos-distributivity.pdf:file.
• CommentRowNumber6.
• CommentAuthorvarkor
• CommentTimeApr 28th 2021

Was the PDF of Roos’s paper meant to be uploaded to the nLab? I wasn’t able to find a copy elsewhere.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 28th 2021