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I'm reading a post about Nisnevich topology and I would like to clarify what the author means in Definition 1.5:

We define $\mathrm{Spc}_S = L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S)$ to be the full subcategory of $\mathcal{P}(\mathrm{Sm}_S)$ consisting of presheaves that satisfy descent with respect to Nisnevich covers. Such presheaves are also said to be Nisnevich local.

I have a general question what does this precisely mean if one says that *something satisfies Nisnevich descent * or satisfy descent with respect to Nisnevich covers.
More generally we can replace Nisnevich by any other Grothendieck site.

The something may be a presheaf. So may I assume that the the meaning of the statement that a presheaf defined over a Grothendieck site satisfies descent means just that that this presheaf satisfies the sheaf axiom for every cover with respect this Grothendieck topology; that is it's just a sheaf with respect this Grothendieck topology?

Is that's what is meant when is said that that a presheaf satisfies descent over a certain cite?

But the something may also be something else, e.g. algebraic K-theory (https://ncatlab.org/nlab/show/Nisnevich+site#idea). What does it mean here that Algebraic K-theory satisfies descent over the Nisnevich site?

user267839
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1 Answers1

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that is it's just a sheaf with respect this Grothendieck topology?

This is exactly what it means, with the proviso that these are simplicial presheaves, and the sheaf condition is formulated in a homotopy coherent way, using homotopy limits over the full Čech nerve. Over the Nisnevich site, this reduces to a homotopy pullback condition for Nisnevich squares.

For a brief overview, see “Hypercovers and simplicial presheaves” by Dugger–Hollander–Isaksen.

For an expository account, see “Sheaves and homotopy theory” by Dugger.

For a book-length treatment, see “Local Homotopy Theory” by Jardine.

Dmitri P.
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  • And what means for a presheaf which is not simplicial the condition "to satisfy descent for certain Grothendieck site"? Clearly we can always associate the nerve to it, but I found several times this formulation in sources which not introduce the terminology of simplicial (pre)sheaves. – user267839 Sep 08 '21 at 19:43
  • @user7391733: In the context of the link you provided (https://dec41.user.srcf.net/exp/motivic/sect0001.html), all presheaves are simplicial, by assumption. A presheaf of sets, however, can be converted into a simplicial presheaf by turning each set into a discrete simplicial set. – Dmitri P. Sep 08 '21 at 19:54
  • Yes you are right, in my context these were indeed simplicial. But in general if one deals with presheaves valued in sets, abelian groups, etc, one always must turn them implicitely into simplicial by in the way you explain it in order to be able to deside if they "satisfy descent or not", right? Otherwise the phrase on descent make no sense? – user267839 Sep 08 '21 at 20:03
  • @user7391733: You don't actually have to convert it, the homotopy descent condition simply reduces to the ordinary descent condition for presheaves of sets, abelian groups, etc. So you can simply use the usual definition of a sheaf in this case. – Dmitri P. Sep 08 '21 at 20:07
  • I'm confused a bit. If we deal with presheaves valued in simplicial ste (or other "homotopical" objects like spectra etc.), then we work with homotopy descent condition described in “Hypercovers and simplicial presheaves” paper. Now assume we work with a non simplicial presheaf $F$ over some site $C$. As you said that in that case it is not neccessary to convert it to simplcial one. – user267839 Sep 08 '21 at 20:27
  • And then for every site $C$ then $F$ satisfies descent with respect to $C$ iff for every $C$-cover $U_i \to U$ the sequence $F(U) \to \prod_i F(U_i) \to \prod_{i,j}F(U_i \times U_j)$ is exact? So in this non simplicial case we don't need to care about higher multiple "intersection" $F(U_{i_1} \cap U_{i_2} \cap ... )$? – user267839 Sep 08 '21 at 20:27
  • @user7391733: That's right, for any ordinary 1-category V, the homotopy descent condition for V reduces to the ordinary descent condition. But you do need homotopy descent for simplicial sets or spectra. – Dmitri P. Sep 08 '21 at 20:57
  • alright, that is the homotopy descent becomes neccessary to define descent property only when the presheaves have values in $(\infty,1)$-categories. For values in ordinary category the three term sequence caries all relevant information one needs to check if $F$ satisfies descent, right? – user267839 Sep 08 '21 at 22:01
  • In some kind checking that $(\infty,1)$-presheaves are sheaves allows homotopy methods (all conditions become weaker) but on the other hand requires finer resolution, that's the "deal"? – user267839 Sep 08 '21 at 22:06
  • @user7391733: Yes. A general statement: for presheaves valued in an n-category, we need to consider the diagram with (n+1)-fold intersections. Thus, for ordinary presheaves of sets or abelian groups we need twofold intersections, for presheaves of groupoids we need 3-fold intersections, and for presheaves of (∞,1)-categories or (∞,n)-categories we need arbitrary finite intersections. – Dmitri P. Sep 08 '21 at 23:42