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added to the Properties-section of reflective (infinity,1)-subcategory the statement and detailed proof of the fact that reflective (oo,1)-subcategories are precisely the full subcategories on local objects.
This proof is really not specific to (oo,1)-categories and parallels a corresponding proof for 1-categories essentially verbatim. A similar 1-categorical proof I had once typed into geometric embedding. I should really copy either one of these versions to reflective subcategory.
further detailed proofs of Properties of reflective (infinity,1)-subcategories.
This is actually a nice playground for using some of the machinery of adjoint oo-functors etc.
more Examples
I wanted to type out the details of the proof of the localization lemma which says, effectively, that given a “strongly saturated class” of morphisms $S$, the localization at it is reflective.
I understand this for ordinary categories and $S$ a calculus of fractions. I think I typed that out once at geometric embedding.
But here the HTT-proof (5.5.5.14) with its lemmas is rather more demanding. I thought I could get it under control for the entry, but it has resisted that attempt so far…
I have now finally essentially completed the full discussion of the detailed proof of the localization lemma (HTT 5.5.5.14).
In HTT, towards the end of the proof there are plenty of typos in that the symbols used get mixed up and are renamed without mentioning. Looks like the result of a new version of the writeup overwriting an older version. Now, I have tried to fix these typos. But also, I had the funny idea to invent my own symbols altogether. And with some probability I introduced typos myself, of course…
… and using that, finally completed spelling out the proof of the localization proposition itself.
P.S. there is the cache bug at work. My new material is at reflective sub-(infinity,1)-category.
If instead you go to reflective (infinity,1)-subcategory you see still the old version, unfortunately, instead of being redirected.
a new section on exact reflective localizations
I spelled out more (supposedly: all) details in the proof at Exact localizations of the theorem that says that a reflector is exact precisely if the collection of morphisms that it inverts is stable under pullback.
The proof applies verbatim to the 1-categorical setup, too. Maybe eventually I should copy it over to reflective subcategory or so. But also conversely, there should be more classical theory being relevant here.
Dave Cardechi kindly fixed the sentence (which was wrong and due to me) leading up to the theorem about characterizations of accessible localizations at reflective sub-(infinity,1)-category.
I created at reflective sub-(infinity,1)-category a new subsection Extra conditions to which I moved the previously existing Exact localizations and to which I added a new subsection Accessible reflective subcategories.
added a paragraph Model category presentation (mainly serving as a pointer to the more detailed discussion at Bousfield localization of model categories, but such a pointer was missing).
Proposition 9, which reproduces Prop. 6.2.1.2 from HTT, was removed from the electronic version of the book. I guess there must be an error in the proof.
Right, now that you say this I remember the discussion about this point. Need to do something about the entry…
Speaking of which, I recently tried to read the proof of the new Proposition 6.2.1.2 in HTT, but I fail to understand even the first paragraph. Am I missing something or is the claim “we can assume that $X^\kappa$ is stable under pullbacks” bogus?
Hmm… I also don’t see why it should be true, or how the commutation of pullbacks and filtered colimits helps. And the next claim that we can also assume $X$ is $\kappa$-accessible also seems to me to require justification, since in general, $\kappa\le\lambda$ isn’t sufficient for $\kappa$-accessible to imply $\lambda$-accessible.
I think this first paragraph can simply be replaced by an application of Proposition 5.4.7.4:
Let $\mathcal{C}$ be a $\kappa$-accessible $\infty$-category and let $\tau\gg\kappa$ be an uncountable regular cardinal such that $\mathcal{C}^\kappa$ is essentially $\tau$-small. Then the full subcategory $\mathcal{C}^\tau\subseteq \mathcal{C}$ is stable under all $\kappa$-small limits which exist in $\mathcal{C}$.
(Note that the relation $\tau\gg\kappa$ implies that $\mathcal{C}$ is $\tau$-accessible.) This does not use that the formation of pullbacks commutes with filtered colimits (but this is still used later in the proof of 6.2.1.2).
Marc, thanks for pointing this out. Maybe once you think this has been patched, might you have the time to briefly edit the nLab entry accordingly?
That sounds plausible to me. Maybe you should contact Lurie?
@Urs: Will do asap. The page on (∞,1)-sheaves also needs to be updated: the proof of 6.2.2.7 in HTT was completely changed in the electronic version, since it relied on the flawed Prop. 6.2.1.2. The new proof proceeds by explicitly defining the sheafification functor using transfinite recursion.
@Mike: Done! I took the opportunity to mention that half the proof of Prop 7.2.1.10 is missing.
Thanks, Marc! I am sure glad to know that somebody like you is looking into sorting these things out.
OK, I think I’ve straightened most things out. I’ve written a sketch of proof that sheafification is left exact. I’d appreciate feedback from experts. One part of the proof I’m still not sure how to prove in details: the fact that the functor $Match(U,-)$ is idempotent. I can prove it by hand for presheaves of sets, but I don’t know how to make a categorical enough argument that would work here. Lurie says this is a “simple cofinality argument” in Remark 6.2.2.15.
Other than that I’ve replaced the flawed Proposition 9 by the new Prop. 6.2.1.2 in HTT (which is quite weaker), and I’ve fixed references to that result at topological localization.
I see now that the fact that $Match(U,-)$ is idempotent is really obvious! It’s just that $Match(U,G)(u)=G(u)$ if $u\in U$, because $u^\ast U$ has an initial object…
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