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I'm trying to understand the notion of a Čech Nerve in simplicial presheaves. Suppose we have a site $C$, and we consider its category of simplicial presheaves $\mathsf{sPshv}(C)$. This is a simplicially enriched category, tensored and cotensored over simplicial sets. Let $r: C \to \mathsf{Pshv}(C)$ denote the Yoneda embedding (this is Dugger's notation). Given a covering sieve $\mathcal{U} = \{ U_i \to U \}$, we can consider the simplicial presheaf $r(\mathcal{U})$ which is defined in degree $k$ as $\coprod_{i_0, \dots, i_k} r(U_{i_0}) \times_{r(U)} \dots \times_{r(U)} r(U_{i_k})$. In DCCT, Definition 3.1.10, Schreiber writes that the Čech nerve of the cover $\mathcal{U}$ is then the geometric realization: $$ \check{C}(\mathcal{U}) = |r(\mathcal{U})| = \int^{[k] \in \mathsf{\Delta}} \Delta^k \otimes \coprod_{i_0, \dots, i_k} r(U_{i_0}) \times_{r(U)} \dots \times_{r(U)} r(U_{i_k}) $$ Now I'm confused by this, because doesn't geometric realization of a simplicial presheaf return a presheaf? (Unless maybe I am misinterpreting and this is not geometric realization?) Schreiber then writes that this is a simplicial presheaf, does this mean we take the resulting constant simplicial presheaf? Meanwhile, older sources seem to just assume that the site $C$ has pullbacks and coproducts and call the Čech Nerve the simplicial $C$ object given in degree $k$ by $\coprod_{i_0, \dots, i_k} U_{i_0} \times_{U} \dots \times_{U} U_{i_k}$. In DCCT Definition 3.1.26, we define a covering sieve $\mathcal{U} = \{U_i \to U\}$ to be a good cover if $\check{C}(\mathcal{U})$ is degreewise a coproduct of representables. Schrieber writes this is equivalent (I think?) to $r(U_{i_0}) \times_{r(U)} \dots \times_{r(U)} r(U_{i_k}) = r(U_{i_0} \times_U \dots \times_U U_{i_k})$, but isn't this always true, since the Yoneda embedding preserves limits?

Descent for $\infty$-stacks is defined using the Čech nerve, so I would appreciate any clarification.

Emilio Minichiello
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  • I think "geometric realisation" here refers to an operation that turns bisimplicial things into simplicial things. (In the case of bisimplicial sets, it simply extracts the diagonal.) – Zhen Lin Mar 16 '21 at 22:35
  • I see, so we think of each $r(U_i)$ as a constant simplicial presheaf. I'm just not quite sure what this gains us in this case, since this would then be the diagonal, and that means we just have $r(\mathcal{U})$, right? – Emilio Minichiello Mar 17 '21 at 12:36
  • Yes. So the two definitions are the same, as you would hope. – Zhen Lin Mar 17 '21 at 12:38
  • Okay, I guess this is just a roundabout definition then. What about the good cover part of the question? I assume I must be wrong about something and that not every cover is good? – Emilio Minichiello Mar 17 '21 at 12:39
  • I don't think you should refer (solely) to that definition of "good cover". "Goodness" depends on context. For algebraic topology of manifolds, I think one usually defines "good cover" to be a cover where every element of the cover and every intersection of finitely many elements of the cover is a disjoint union of contractible subspaces. This is consistent with Urs's definition if you use the site consisting of cartesian spaces only. – Zhen Lin Mar 17 '21 at 14:24

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