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…

]]>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….

]]>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).

]]>cross-linked the discussion of local right properness (here) with *locally cartesian closed model category* and with *locally cartesian closed (infinity,1)-category*

I have added publication data to

- John F. Jardine,
*Boolean localization, in practice*, Documenta Mathematica, Vol. 1 (1996), 245-275 (documenta:vol-01/13)

Fixed some ?ech which should have been Čech.

]]>added pointer to the MO comment Garner 13 characterizing projectively cofibrant simplicial presheaves.

Somebody should spell out the details here! If only by copy-and-pasting-and-beautifying-a-little the code from the MO answer.

]]>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.)

]]>have added also a section Inclusion of chain complexes of sheaves, so far just observing the obvious Quillen adjunction induced from Dold-Kan

]]>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.

]]>added more details on weak equivalences (local epimorphisms, really) in the Cech localization of the projective model structure from Dugger-Hollander-Isaksen

]]>fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.

]]>