The Stacks project

Remark 20.23.5. This means that if we have two total orderings $<_1$ and $<_2$ on the index set $I$, then we get an isomorphism of complexes $\tau = \pi _2 \circ c_1 : \check{\mathcal{C}}_{ord\text{-}1}(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{ord\text{-}2}(\mathcal{U}, \mathcal{F})$. It is clear that

where $i_0 <_1 i_1 <_1 \ldots <_1 i_ p$ and $i_{\sigma (0)} <_2 i_{\sigma (1)} <_2 \ldots <_2 i_{\sigma (p)}$. This is the sense in which the ordered Čech complex is independent of the chosen total ordering.