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I was looking for a place to record a somewhat more global overview of the notion of locally presentable category, its related notions and its generalizations to higher category theory. But somehow all of the existing entries feel too narrow in focus to accomodate this. So I ended up creating now a new entry titled
Think of this as accomodating material such as one might present in a seminar talk that is meant to bring people with some basic background up to speed with the relevant notions, without going into the wealth of technical lemmas.
I only just started. Will continue in a moment after a short break…
Good idea! It might be worth explaining why we use filtered colimits rather than arbitrary ones. It seems to me that the idea is that if one has a compact object, then its image under any morphism should be “small” in the sense that any “cover” of it can be reduced to a finite one. So, if we enlarge the “covering system” so that it is closed under finite “unions”, then we get a filtered system, and the object is compact precisely when every morphism factors through a single object in the “covering system”, and this factorisation is unique modulo a certain equivalence relation which amounts to saying that homs out of a compact object preserve filtered colimits.
Thanks. That’s a good point. Actually, the relation between “compact” as in “compact topological space” as well as “compact topos” on the one hand and in “compact object in a category” on the other is somewhat subtle. The thing is that the former two are really conditions to commute with filtered colimits of truncated objects/truncated morphisms.
I have once made some notes on this in section 3.3.3 here.
I have edited a bit more at locally presentable categories - introduction. Mostly I tried to fill in data in the second and third main sections, but for the moment it remains skeletal.
I have to quit now. Will come back to this in a few hours. Would be happy about whatever comments or even edits you’d have meanwhile.
I have edited a bit more. I think now essentially all the material is there that should go there. But some more glue should still be added.
There’s a table in the ’summary and overview’ section which I don’t know how to fix.
Sorry, could you point directly to what you are looking at? The table which I do see is not broken, but likely I am not looking at the table which you have in mind. Sorry.
Is it just my browser? The table here doesn’t display properly for me.
Fixed. There were some spurious line breaks.
Also, I changed to section link to something more likely to be permanent.
Replaced characters ’ř’ that had turned into ’?’ in links to the pages Jiří Adámek and Jiří Rosický
This sort of ? is a relic of an old mistaken database edit that clobbered some unicode symbols, and should be fixed wherever they appear. Thanks!
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