The Verdier hypercovering theorem (Q2888785)

The author studies some generalisations of the classical Verdier hypercovering theorem. This theorem approximates the morphisms \([X,Y]\) in the category of simplicial sheaves and presheaves by simplicial homotopy classes of maps. In its standard form this approximation is given by the comparison function \(\lim_{[p] : Z \rightarrow X} {\pi}(Z,Y) \rightarrow [X,Y]\) defined by mapping the element NEWLINE\[CARRIAGE_RETURNNEWLINE\begin{tikzcd} X & Z \lar["{[p]}"]\rar["f"] & Y \end{tikzcd}CARRIAGE_RETURNNEWLINE\]NEWLINE to the morphism \(f\cdot p^{-1}\) . Here, \({\pi}(Z,Y )\) denotes simplicial homotopy classes of maps corresponding to a hypercover \(p : Z\rightarrow X\). The theorem asserts that the comparison function is an isomorphism provided \(X\) an \(Y\) are locally fibrant. Based on results developed by him in [Algebraic topology. The Abel symposium 2007. Proceedings of the fourth Abel symposium, Oslo, Norway, August 5--10, 2007. Berlin: Springer. Abel Symposia 4, 185--218 (2009; Zbl 1182.55006)] the author gives a new easier proof of the theorem. This new proof yields a pointed generalisation which does not require the assumption that \(X\) is fibrant.