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Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies (where I should reiterate that $\Phi$ contains all finite diagrams). Is this more than a linear order? (edited)

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

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  • $\begingroup$ In answer to your first bullet point (but not the question in the title), yes, there are tons of other limit-colimit commutation classes. See Bjerrum, Johnstone, Leinster & Sawin, Notes on commutation of limits and colimits and Adámek, Borceux, Lack and Rosicky, A classification of accessible categories. But maybe you know all this and I've misunderstood what you're asking $\endgroup$
    – Tom Leinster
    Commented Feb 22 at 22:08
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    $\begingroup$ While it's true that there are infinitely many interesting classes contained in finite limits, it's interesting that neither of the linked papers mentions any that contain finite limits, other than the $\lambda$-small ones. I can't think of any myself--it feels as if basically anything you might add (some discrete infinite categories, some cofiltered ones...) gets you to something that's interchangeable with some $\Phi_\kappa$ for most purposes. $\endgroup$
    – Kevin Carlson
    Commented Feb 22 at 22:30
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    $\begingroup$ Right; as I said, my comment's not an answer to the question in the title. $\endgroup$
    – Tom Leinster
    Commented Feb 22 at 23:33
  • $\begingroup$ My interpretation was that the first bullet point is intended to be read as "other limit-colimit-commutation classes containing the finite limits", and the second bullet point should include "where $\Phi$ contains the finite limit diagrams" (but this should be clarified in the question). $\endgroup$
    – varkor
    Commented Feb 23 at 9:25
  • $\begingroup$ Edited, indeed I was aware of the commutation classes which include filtered colimits, but I'm asking about the ones which are subclasses of filtered colimits (so commute with $\Phi$-diagrams for a limit class $\Phi$ as in the remainder of the question). $\endgroup$
    – Morgan Rogers
    Commented Feb 23 at 9:30

2 Answers 2

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An example of a category with all finite colimits and $\aleph_1$-filetered colimits but not all sequential colimits is the $\aleph_1$-filtered $Ind$-completion $Ind_{\omega_1}(\mathcal C)$ of any category $\mathcal C$ which doesn’t have sequential colimits.

A pretty natural example of the example is the category of Banach spaces and all bounded linear maps (not just the contractive maps — if you take just the contractive maps you get a locally presentable category).

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  • $\begingroup$ I think I can see how one would prove this, at least for an $\omega_1$-indexed diagram: we can construct its colimit in the category of topological vector spaces (I can't immediately see how the result is a Banach space, but continuing), then observe that the maps in the colimit cone must be bounded, since if the norms in the diagram are divergent in $\mathbb{R}$ then there is a countable divergent cofinal subsequence, which is impossible in $\omega_1$? I think I'm missing some details which would make this precise. Can you suggest a reference? $\endgroup$
    – Morgan Rogers
    Commented Mar 28 at 11:05
  • $\begingroup$ math.stackexchange.com/questions/1424777/… $\endgroup$
    – Tim Campion
    Commented Mar 29 at 1:53
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There's one class which I know and which feels natural enough to mention: L-finite diagrams.

Robert Parè proved in his paper Simply connected limits that L-finite limits are precisely the "centraliser" of (finitely) filtered colimits in Set; i. e. they are limits which commute with all filtered colimits.

There's also a direct description of this class of diagrams: they contain initial finite (or, equivalently, initial finitely generated) subcategory.

Overall I feel like this structure of mutual commutation of limits and colimits in some familiar fixed category (maybe a topos, maybe a Grothendieck abelian category) is very much underdeveloped. For example, one can try to find natural closure properties, which are enjoyed by a centraliser of definable (...in which sense? I'd say 'finitely axiomatisable', but I'm not sure) class of diagrams. In which sense L-finite diagrams are the "multiplier algebra" of finite diagrams? Can we generalize this "double commutant theorem for finite limits" replacing Set with arbitrary topos/bicomplete category? For me it's not clear how/whether Parè's arguments can be transferred.

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    $\begingroup$ While this is technically correct, I don't feel it is quite in the spirit of the question, since L-finite diagrams are the saturation of the finite diagrams, and so one doesn't obtain any new limits by considering them over the finite diagrams. $\endgroup$
    – varkor
    Commented Mar 27 at 10:04
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    $\begingroup$ I still don’t know what the spirit of the question is. It seems to me that pointing out that the class of finite categories is not already saturated is extremely relevant in the context of the question. $\endgroup$
    – Tim Campion
    Commented Mar 29 at 1:57
  • $\begingroup$ @varkor I mean, the definition of L-finiteness, i. e. that terminal object in presheaves on C^op is compact (or, to explicate: representable functor which is represented by the terminal object in op-presheaves commutes with filtered colimits) does seem to produce saturated classes of diagrams if you replace "filtered" with other product-closed class of diagrams; but it doesn't seem true that if you begin with a class X, analogous "L-Xness" characterises the "limit centraliser". $\endgroup$
    – Denis T
    Commented Mar 29 at 14:09

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