Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\bullet$ with $C_n = \bigsqcup Y_{i_1} \times_X \cdots \times_X Y_{i_n}$. One could instead refine this by choosing any cover of the intersection $D_1 \to \bigsqcup Y_i \times_X Y_j$ and then iteratively picking a cover of the coskeleton of the piece of $D_\bullet$ constructed so far to get a hypercover.
I want to do the same thing with sieves. I have an actual application in mind -- a functor that is compatible with injections but not surjections. The naive guess doesn't seem to work: $R \times_X R = R$, so a cover is just another sieve $D_1 \subseteq R$ that also covers $X$. The goal is to compute cohomology by taking the limit of cohomology of the hypercovers as in Verdier's hypercovering theorem.
Since hypercovers modify the fiber products $Y_i \times_X Y_j$, maybe the right thing to do is to let $R$ no longer be a presheaf. Rather than demand $Y_i \times_X Y_j$ is in $R$, the cover $D_1$ would be in $R$. The condition ``if $S \to T$ and $T \in R$, then $S$ is covered by objects of $R$'' would be a replacement in degree 1. I'm not sure what the higher degree conditions should be, but maybe a compatibility of the choices of covering objects from $R$ for finite diagrams.
Q: Can one formulate the notion of hypercovers and descent with respect to them using sieves instead of covering families? Is there a version of the same hypercovering theorem?
I apologize if this is well known. I'm a little lost in the abstract nonsense and really just trying to get a workable version of Verdier's hypercovering theorem.
