The Čech nerve | Higher Topos Theory notes

Construction 2.24. (Čech nerve)

Let $$C$$ be a category with pullbacks. For a morphism $f\colon U \to X$ in $$C$$ , we define its Čech nerve $\check{C}_{\bullet}(f) \in \Fun(\simp\catop,C)$ as the image of $f \in \Ar(C)$ under the following functor:

\Fun([1],C) \simeq \Fun((\simp^{\leq 0}_+)\catop,C) \xrightarrow{j_*} \Fun(\simp\catop_+,C) \xrightarrow{i^*} \Fun(\simp\catop,C).

Here $\simp_+$ is the augmented simplex category, $i\colon \simp \hookrightarrow \simp_+$ is the canonical inclusion, $j\colon \simp_+^{\leq 0} \hookrightarrow \simp_+$ is the inclusion of the objects $$[-1]$$ and $$[0]$$ , and $[1] \simeq (\simp^{\leq 0}_+)\catop$ is the canonical equivalence sending $$0$$ to $$[0]$$ and $$1$$ to $$[-1]$$ . The pointwise formula for the right Kan extension $$j_*$$ shows that the Čech nerve is given by

\check{C}_n(f) \, \iso \, U^{\times^{n+1}_X} \, = \, \underbrace{U \times_{X} U \times_X \dots \times_X U}_{n + 1 \text{ times}}.

This computation also shows that the right Kan extension functor $$j_*$$ exists.