Descent theory for higher sheaves

Question

I am trying to understand the descent condition for sheaves from presheaves. Let Presh(S) be the $(\infty,1)$ category of presheaves on an $(\infty,1)$ site S, and Sh(S) be the corresponding category of sheaves. In PreSh(S), we have morphisms W: Y $\rightarrow$ X, the $(\infty,1)$ sheaves A are those $(\infty,1)$ presheaves in PreSh(S) which are local objects with respect to this morphism in this category; these local objects induce bijection between PreSh(S)(X, A) and PreSh(S)(Y, A) in the homotopy category of $\infty$-Top. Y can be regarded as cover or hypercover with respect to the $(\infty,1)$ presheaf X. Thus, the $(\infty,1)$ sheaf condition descends from Y to X. But, I have a doubt here. X and Y are objects in PreSh(S); I do not get how Y can be a hypercover which is a morphism in the category Simplicial PreSh(S), not PreSh(S). Can someone kindly clarify this? I have just completed my undergrad in physics; apologies if this doubt sounds trivial.

Answer 1

If you start with a hypercover represented as a morphism $Z_\bullet \to c_\bullet X$ in simplicial presheaves, where $c_\bullet X$ denotes the constant simplicial presheaf, then your $Y$ is the colimit (= geometric realization) of $Z_\bullet$. Since the colimit of $c_\bullet X$ is just $X$, the colimit of $Z_\bullet \to c_\bullet X$ gives you your morphism $Y\to X$ of presheaves.

The point of taking the colimit is that mapping out of colimits takes them to limits, so you have $\mathrm{PreSh}(S)(Y,A) \simeq \lim \mathrm{PreSh}(S)(Z_\bullet, A)$, where the latter is the more straightforwardly defined space of "descent data" in $A$ over the hypercover $Z$.