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Is there a term for a category such that for every object $x$ the poset $Sub(x)$ of subobjects of $x$ is a complete lattice? Actually, I really only care that it’s a complete inf-lattice; (unless the category is well-powered, $sub(x)$ might be large and thus not necessarily cocomplete even if complete).
$Sub(x)$ 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.
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 $Sub(x)$) may be the reason.
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?
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.
I added some discussion of the advantages and disadvantages of well-poweredness at geometric category, replacing the query box.
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