6
$\begingroup$

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.

One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start with the projective model structure on simplicial presheaves and then we take the Left Bousefield localization with respect to the map of the form $j \hookrightarrow x$ where $j$ is a covering sieve of $x$ for the topology $J$, where they are both seen as 'constant' simplicial presheaves.

This is a model for the $(\infty,1)$-topos of (possibly non Hypercomplete) $\infty$-sheaves over $(C,J)$.

It is proved in the appendix of Hypercovers and simplicial presheaves, by Dugger, Hollander and Isaksen that for any simplicial presheaves $S$ the map from $S$ to its levelwise sheafication is a weak equivalence. In particular, any map of simplicial presheaves is a weak equivalence for this model structure if an only if its level sheafication is a weak equivalence.

This mean that one get a nice notion of weak equivalence for maps between simplicial objects of the topos $\mathcal{T}$, that we will call "Cech weak equivalence" (by opposition to the Jardin-Joyal weak equivalence which are easily describe internally as the maps inducing isomorphism on all $\pi_n$).

My question is:

To what extent this notion of "Cech weak equivalence" depends on the choice of a site of definition of the topos $\mathcal{T}$ ?

I'm completely willing to assume that $C$ has all finite limite if it change something (see the next remarks)

The key result which I think is relevant to this question is Lurie's Proposition 6.4.5.7 of Higher topos theory, which (assuming $C$ has finite limits) describe the universal property of the $(\infty,1)$-topos of $\infty$-sheaves over $(C,J)$ purely in terms of $\mathcal{T}$, and hence proves that the model categories of simplicial presheaves over different sites of definitions of $\mathcal{T}$ (admiting finite product) are all canonically Quillen equivalent. This allows to says a few thing, but unfortunately this is not completely enough to conclude that the weak equivalence between simplicial sheaves are the same (only between those which are fibrant/cofibrant) !

My area of expertise being more topos theory than model category I might be missing something, that is why I'm asking this question here.

PS: it also appears that starting with the injective model structure instead of the projective one does not change the weak equivalences at all (even after localization).

$\endgroup$
  • $\begingroup$ Essentially yes. But I don't think an actual "site independante description" is know (a description of these is considered as an open problem) so I'm just looking for a proof that two site yields the same notion of weak equivalence. $\endgroup$ – Simon Henry Nov 27 '14 at 18:10
  • $\begingroup$ Don't you mean finite limits above, not finite products? $\endgroup$ – David Carchedi Nov 27 '14 at 18:47
  • $\begingroup$ Yes, that's corrected. $\endgroup$ – Simon Henry Nov 27 '14 at 18:54
4
$\begingroup$

It is independent, provided the two sites of definition, also yield the same infinity topos (so e.g. when they both have finite limits, or the infinity-topos is hypercomplete). Given two simplicial objects $F$ and $G$ of the topos of $\mathcal{E}$, so $$F,G:\Delta^{op} \to \mathcal{E},$$ consider the embedding $$\theta:\mathcal{E} \hookrightarrow Sh_\infty\left(\mathcal{E}\right),$$ where I am writing $Sh_\infty\left(\mathcal{E}\right)$ for the $1$-localic infinity-topos corresponding to $\mathcal{E}.$ I claim that a map $f:F \to G$ is a weak equivalences if and only if the induced map between $\operatorname{colim} \theta \circ F \to \operatorname{colim} \theta \circ G$ is an equivalence in $Sh_\infty\left(\mathcal{E}\right)$.

Proof: $f$ is a weak equivalence if and only if it is is one when considered as a map of simplicial presheaves, which is if and only if it becomes one in the associated infinity-topos. Choose a site $C$. In simplicial presheaves on $C$ with the global model structure, the homotopy colimit of $\theta \circ F$ is $F$ itself, which can be computed as the diagonal of the bisimplicial presheaf which is constant in one direction. Hence, the colimit of $\theta \circ F$ is the infinity sheaf associated to $F$.

$\endgroup$
  • $\begingroup$ Thanks ! Do you know if there Is a counterexample without additional assumption ? $\endgroup$ – Simon Henry Nov 27 '14 at 18:12
  • 1
    $\begingroup$ There has to be, since assuming there is no counterexample, one could show that both sites yield the same infinity-topos. $\endgroup$ – David Carchedi Nov 27 '14 at 18:16
  • $\begingroup$ More precisely, my question was: "do you know if there is a known example of two sites of definition of T which yields different infini topos ?" $\endgroup$ – Simon Henry Nov 27 '14 at 18:19
  • 4
    $\begingroup$ I learned this from Jacob Lurie: Let $Q=\prod_i I$ be the hilbert cube. Consider all the open subsets which are homeomorphic to $Q \times [0,1)$. These form a basis, but are not closed under finite intersections. Considering them as a subcategory of the poset of all open subsets, one can define an obvious site structure. However, infinity sheaves on this site is different than infinity sheaves on the Hilbert cube. $\endgroup$ – David Carchedi Nov 27 '14 at 18:36

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.