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fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.
added more details on weak equivalences (local epimorphisms, really) in the Cech localization of the projective model structure from Dugger-Hollander-Isaksen
at model structure on simplicial presheaves I have (finally) added a section
Presentation of (oo,1)-toposes
I have also added to the section Homtopy (co)limits the observation that finite homotopy limits in the local model structure may be computed in the global structure.
have added also a section Inclusion of chain complexes of sheaves, so far just observing the obvious Quillen adjunction induced from Dold-Kan
I had had a section on descent for presheaves with coefficients in strict $\infty$-groupoids and how it relates to the descent of these regarded as simplicial presheaves over in the entry on smooth $\infty$-groupoids. Since it did not really belog there specifically at all, I have now moved that over to model structure on simplicial presheaves in a new section:
Descent for values in strict and abelian $\infty$-groupoids.
The main point is there a clear statement of Verity’s result of sufficient conditions under which Street’s definition of descent is actually correct and matches the one of simplicial presheaves.
(I have only copy-and-pasted it for the moment. I should go through this and see if there is need to polish or otherwise improve this.)
I have added publication data to
cross-linked the discussion of local right properness (here) with locally cartesian closed model category and with locally cartesian closed (infinity,1)-category
Proposition 6.3 at model structure on simplicial presheaves claims that the localization of presheaves of spaces at a coverage is a topological localization, but I do not understand the proof. The second commuting square is claimed to be a pushout in which the top morphism is a monomorphism, and I do not see why either claim holds. It further seems to me that the sieve $S(\{U_i\}\cup \{V_{j,k}\})$ is just the same as $S(\{U_i\})$ since each $V_{j,k}$ refines some $U_i$ by definition, so the conclusion of the proof also does not make sense to me.
This is related to a recent answer I wrote on MathOverflow, where I explain the proof of a weaker claim (for pullback-stable coverages).
Thanks for the heads-up. Looks like I wrote this in May 2010 (where I had still a disclaimer which I seem to have later removed), but it is indeed quite mangled. I now went ahead and fixed up that Prop. 6.3, but looking ahead this is not the only edit necessary.
So for the moment I have moved that whole chunk (former Prop. 6.3 up to the Conclusion 6.6) to the Sandbox. Will see what to do with it….
Looking at what’s in the sandbox now, I think that for the square in the proof of Prop 0.1 to be a pushout, one should replace $y(V)$ by the sieve generated by $V$. Otherwise there is no reason for the pushout to be a subobject of $y(U)$.
I was also looking at the proof in Johnstone (C, Lemma 2.1.6), which is basically this: If $R\subset S\subset y(X)$ are sieves such that $R$ is in the (sifted) coverage and if $F$ is a sheaf for the coverage, then $Map(S,F)\to Map(R,F)$ has a section, in particular is an effective epimorphism. On the other hand it is the limit of the maps $F(V) \to Map(R\times_{y(X)}y(V),F)$ for $V$ in the sieve $S$, which by definition of coverage have retractions since each $R\times_{y(X)}y(V)$ contains a covering sieve. For a sheaf of sets this means that these maps are monomorphisms and we can conclude. But already for a sheaf of groupoids I’m not sure what to do…
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