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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2011

    Is there a term for a category such that for every object xx the poset Sub(x)Sub(x) of subobjects of xx is a complete lattice? Actually, I really only care that it’s a complete inf-lattice; (unless the category is well-powered, sub(x)sub(x) might be large and thus not necessarily cocomplete even if complete).

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 1st 2011
    • (edited Sep 1st 2011)

    Sub(x)Sub(x) is not necessarily equivalent to a set, hence calling it a poset requires some assumptions. Edit: oh I see, you are calling it large. OK.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeSep 1st 2011
    • (edited Sep 1st 2011)

    A related issue is why a geometric category should be well powered, which Mike had asked about in a query box at geometric category. I’ve replied there, suggesting that this (the equivalence between completeness and cocompleteness of Sub(x)Sub(x)) may be the reason.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeSep 2nd 2011

    Re: 1, there might be, but I don’t think I know it. I have heard something like “mono-complete” (and more generally “\mathcal{M}-complete” for a class \mathcal{M} of subobjects) to mean that all (not necessarily small) intersections of subobjects exist — or \mathcal{E}-cocomplete in the dual case.

    Re: 3, yes, that is a nice thing to have — but is it sensible to want that to come along with the adjective “geometric”, since geometric logic includes infinite disjunction but not infinite conjunction?

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeSep 2nd 2011

    Thanks, and upon reflection, mono-completeness is really what I need when well-poweredness fails (so my original question was not quite correct).

    I can’t really justify wanting geometric categories to be well-powered; in fact, I’m inclined to agree with you. I only suggest what might have been somebody’s motivation.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeSep 3rd 2011

    I added some discussion of the advantages and disadvantages of well-poweredness at geometric category, replacing the query box.