Lemma 2.20. ([Lurie 2009, Proposition 6.1.2.11])
Let \(C\) be a category and consider an augmented simplicial object \(U_{\bullet}^+\colon\simp\catop_+ \to C\). The following conditions are equivalent:
The diagram \(U_{\bullet}^+\) is right Kan extended from \((\simp_+^{\leq 0})\catop \subseteq \simp\catop_+\);
The simplicial object \(U_{\bullet} := U_{\bullet}^+\vert_{\simp}\) is a groupoid object and the square
is a pullback square.
Proof
Assume first that \(U_{\bullet}^+\) is right Kan extended from \((\simp_+^{\leq 0})\catop\). The pointwise formula for right Kan extension gives canonical isomorphisms
\[U_n \iso \underbrace{U_0 \times_{U_{-1}} \dots \times_{U_{-1}} U_0}_{n+1\text{ factors}}\]
for every \(n \geq 0\). Thus \(U_{\bullet}\) is the Čech nerve of \(U_0 \to U_{-1}\). The displayed square in the statement is the case \(n=1\), and the remaining pullback identities show that \(U_{\bullet}\) is a groupoid object.Conversely, assume that \(U_{\bullet}\) is a groupoid object and that the displayed square is a pullback square. The latter identifies \[U_1 \iso U_0 \times_{U_{-1}} U_0.\]
Applying the groupoid condition successively to the ordered decompositions of \([n]\) into adjacent intervals gives \[U_n \iso \underbrace{U_1 \times_{U_0} \dots \times_{U_0} U_1}_{n\text{ factors}}
\iso \underbrace{U_0 \times_{U_{-1}} \dots \times_{U_{-1}} U_0}_{n+1\text{ factors}}.\]
These isomorphisms are induced by the vertex maps \([0] \to [n]\) and are therefore compatible with the simplicial structure. Hence \(U_{\bullet}^+\) agrees with the Čech nerve of \(U_0 \to U_{-1}\), which is precisely the right Kan extension of its restriction to \((\simp_+^{\leq 0})\catop\).References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.