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?
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.