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I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:
Currently it contains the following table, to be further fine-tuned. Comments are welcome.
| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects | |–|–|–|–|–|—-|–|–| | (0,1)-category theory | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | | | category theory | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories | $\hookrightarrow$ | accessible categories | | model category theory | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | | | (∞,1)-topos theory | (∞,1)-toposes |$\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories | $\hookrightarrow$ |accessible (∞,1)-categories |
You are writing ’topos’ throughout to mean Grothendieck topos.
Yes.
Well, at least the link to ‘topos’ should go to Grothendieck topos, so I have made it do so. And perhaps locale or frame would be a better target for ‘$(0,1)$-topos’?
Sure, I have made “(0,1)-topos” point to locale.
On a more substantial note: has anyone thought about how to bring the part in the table where it says “Porst’s theorem” more in-line with the rest? In the entry to the right of where it says “Porst’s theorem” it it should really read “accessible reflective sublattices of (0,1)-presheaf (0,1)-categories”. Would this be true?
Possibly this is obvious. I haven’t really thought about it yet and can’t do right now. But if somebody has, I’d enjoy hearing it.
Another problem with this table as currently written (which for some reason no longer renders in comment #1, but looks fine at the nLab page locally presentable categories - table) is that algebraic lattices correspond to locally finitely presentable categories, not arbitrary locally presentable categories. I would argue that the posetal analogue of an arbitrary locally presentable category is simply a suplattice. (In particular, not every frame is an algebraic lattice, whereas every topos is a locally presentable category.) If we want to include algebraic lattices in the table, then there should be a column for the locally finitely presentable case of the other rows (but that might make the table too wide).
Thanks for the alert. I would be happy to add a column saying “generated under finite colimits from finite objects” and then to move the entry “algebraic lattices” to that column. If the entry gets too wide, I think the software automatically produces a scrollbar environment.
But how to organize it? If we insert that new column to the left of the existing “generated under colimits” then the compatibility with the “inclusion of left exact localizations breaks.
Any suggestion for how to handle this? Maybe we should make a second copy of the whole table, now for finite generation.
Well, I have a lot of trouble reading the table as-is anyway. I don’t know why, but the top header row doesn’t have column-separator lines, which means that the third and fourth cells run together and for a long time I was reading it as “inclusion of left generated under exact colimits from localizations small objects”, which is just close enough to meaningful to give me a headache trying to figure it out. (-:
Also, I think the headings “generated under colimits from small objects” and “generated under filtered colimits from small objects” are misleading, because the second sounds like a stronger condition than the first (since a filtered colimit is, in particular, a limit) whereas in fact being accessible is weaker than being locally presentable. Moreover, the generation in a locally presentable category is under filtered colimits.
I tried rearranging the table some by moving large blocks of text out of it (which I think don’t look good in tables) into the header above, adding a column for the locally finitely presentable case.
Much better! Thanks.
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