3
$\begingroup$

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose that $M$ is left proper combinatorial, then the Bousfield localisation $M_{\bar{S}}$ is a model topos [Prop.6.2.1.2, HTT, Lurie].

Let $sPre(C)$ be the category of simplicial presheaves of an essentially small category $C$, the fibrant objects of the Bousfield localisation with respect to the set of all hypercovers $S$ are precisely the $S$-local objects, that is, the presheaves of Kan complexes satisfying descent for all hypercovers. Under suitable condition, this localisation is equivalent to the localisation of the set of all bounded hypercovers. And a presheaf satisfies descent for all bounded hypercover is equivalent to that it satisfies Čech descent. Thus the fibrancy condition can be reformulated for Čech descent which is a relatively concrete condition.

In the case of Nisnevich descent, where $C$ is a site whose topology is defined by a complete bounded regular cd structure, the fibrancy condition can be stated for the distinguished square, that is, a simplicial presheaf $F$ is fibrant if and only if $F$ takes values in Kan complexes and it takes every elementary ditinguished square to a homotopy pullback.

Let $\alpha$ be a Nisnevich distinguished square, $\require{AMScd}$ \begin{CD} W @>>> Y \\ @VVV @VpVV\\ U @>i>> X \end{CD} and let $P(\alpha)\to X$ be the morphism from the pushout of the upper part $U\leftarrow W \rightarrow Y$ to $X$.

Question: Consider the set of morphism $Nis=\{P(\alpha)\to X\}_{\alpha}$ and the localisation $sPre(C)_{\overline{Nis}}$ of $\overline{Nis}$, are the fibrant objects of $sPre(C)_{\overline{Nis}}$ the same as $sPre(C)_{{Nis}}$?

Further question after edit: Consider the category $sPre(C)_{\mathbb{A}_1,\overline{Nis}}$ which has a different model category structure than the motivic homotopy category $sPre(C)_{\mathbb{A}_1,{Nis}}$, are the fibrant objects in $sPre(C)_{\mathbb{A}_1,\overline{Nis}}$ the same as that for $sPre(C)_{\mathbb{A}_1,{Nis}}$?

I slightly modified the question as now the question is not generally about hypercover as I thought. What I'm looking for are the fibrant objects in the model topos $sPre(C)_{\mathbb{A}_1,\overline{Nis}}$. In the comments below, Pavlov mentioned the paper by Strunk and Raptis,

Let $\mathcal{C}$ be a small simplicial category. Then there is a bijective correspondence between Grothendieck topology $\bar{\tau}$ on $Ho(\mathcal{C})$ and homotopy left exact left Bousfield localization of $sPSh^{\Delta}(\mathcal{C})$ which are $t$-complete.

where a $\mathcal{U}$-local model topos $sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis}$ is given together with its Grothendieck topology arising from Nisnevich topology on the ordinary category $Sm_S$. $sPSh^{\Delta}(\mathcal{C})$ denotes the functor category of simplicial functors $\mathcal{C}^{op}\to \mathcal{sSet}$ for a small simplicial category $\mathcal{C}$. A Grothendieck topology for a simplicial category $\mathcal{C}$ is by definition a Grothendieck topology on $Ho(\mathcal{C})$.

Given a Grothendieck topology above, the model structure of the localization can be described by hypercovers of this topology. By this theorem, there should be a Grothendieck topology for $Ho(Sm_k)$ that gives the localisation $(sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis})_{\overline{\mathcal{H}(Nis)}}\cong sPsh(Sm_k)_{\mathcal{A_1},\overline{Nis}}$ of $sPsh^{\Delta}(Sm_k)$. The Grothendieck topology of the $\mathcal{U}$-local model topos $sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis}$ arises from the Nisnevich topology of $Sm_k$ as ordinary category. So this topology isn't the one that gives the final localization.

How can one determine this Grothendieck topology hence find the fibrant objects of $sPre(C)_{\mathbb{A}_1,\overline{Nis}}$?

$\endgroup$
  • 1
    $\begingroup$ "This localisation is equivalent to the localisation of the set of all bounded hypercovers.": only for hypercomplete sites. This is false, for example, for the etale site. $\endgroup$ – Dmitri Pavlov Aug 3 at 19:57
  • 1
    $\begingroup$ Concerning the newly added question: what is the meaning of the expression "homotopy theory passes to"? $\endgroup$ – Dmitri Pavlov Aug 4 at 15:51
  • $\begingroup$ @DmitriPavlov I have edited the question to be more specific $\endgroup$ – Nicky Aug 4 at 16:30
  • $\begingroup$ "are the fibrant objects and the A^1-homotopy classes the same": an A^1-homotopy class is by definition a collection of objects. In what sense is a collection of objects is supposed to be "the same" as a fibrant object (i.e., a single object)? $\endgroup$ – Dmitri Pavlov Aug 4 at 17:27
  • $\begingroup$ @DmitriPavlov I just mean if the A^1 homotopy class of the model topos the same as that of the motivic homotopy category. But anyway let me just ask if the fibrant objects are the same. Sorry for the misleading question. I have edited the question $\endgroup$ – Nicky Aug 4 at 18:03
2
$\begingroup$

are the fibrant objects of sPre(C)_Nis the same as sPre(C)_Nis?

Yes. This follows from the characterization of fibrant objects in a left Bousfield localization as fibrant objects in the original model structure that are also local.

The closure of S under homotopy base changes does not change the class of local objects: S is contained in S-local weak equivalences and the class of S-local weak equivalences of simplicial presheaves in any model topos (such as sPre(C)_Nis) is closed under homotopy base changes (computed in simplicial presheaves before localization) because the localization (sheafification) functor sPre(C)→sPre(C)_Nis is right exact, i.e., preserves finite homotopy limits. Since S-local weak equivalences are the same, the S-local objects are also the same.

$\endgroup$
  • $\begingroup$ Do you mean that $hocolim _Imap(X^I,Y)\cong map(holim_I X^I, Y)$ for homotopy function complex $map(-,-)$? $\endgroup$ – Nicky Aug 4 at 12:28
  • $\begingroup$ @Nicky: No. I added details to the answer. $\endgroup$ – Dmitri Pavlov Aug 4 at 14:56
  • $\begingroup$ But $sPre (C)_{Nis}$ is not a model topos. $\endgroup$ – Nicky Aug 4 at 15:17
  • 2
    $\begingroup$ @Nicky: sPre(C)_Nis is very much a model topos, by definition. sPre(C)_{A^1} is not a model topos, but that was not in the original question. A paper by Raptis and Strunk "Model topoi and motivic homotopy theory" is relevant. In particular, they construct some model toposes that are related to the motivic homotopy category and also discuss to what extent the motivic homotopy category satisfies the properties of a model topos. $\endgroup$ – Dmitri Pavlov Aug 4 at 15:36
  • 1
    $\begingroup$ Just to clarify what I mentioned above, sPre(C)_{A^1} (without the Nisnevich localization) is a model topos (since it can be presented as a the category of simplicial presheaves on the simplicial A^1-enrichment of C), whereas sPre(C)_{A^1,Nis) (with the Nisnevich localization) is not a model topos. $\endgroup$ – Dmitri Pavlov Aug 4 at 19:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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