Proposition 5.1.15. Assume that \(C\) admits both finite limits and geometric realizations. Then the functors \(\bB \) and \(\bOmega \) define adjunctions \[ \bB \colon \Mon (C) \rightleftarrows C_* \noloc \bOmega \qquadtext { and } \bB \colon \Grp (C) \rightleftarrows C_* \noloc \bOmega . \]
Proof. Consider the following two adjunctions:
The bottom composite \(i^*j_*\) is precisely the Δech nerve functor \(\check {C}_{\bullet }\). The functor \(i_!\) is given by left Kan extension along \(i\). From the pointwise formula for left Kan extensions, we see that the top composite \(j^*i_!\) sends a simplicial object \(X\) in \(C\) to the map \(X_0 \to \abs {X} = \colim _n X_n\).
Now, note that the top composite \(j^*i_!\) sends the full subcategory \(\Mon (C) \subseteq \Fun (\simp \catop ,C)\) into the full subcategory \(C_* \subseteq \Ar (C)\), and that the resulting functor \(\Mon (C) \to C_*\) is precisely \(\bB \). Similarly, the bottom composite \(i^*j_*\) restricts to \(\bOmega \colon C_* \to \Mon (C)\). It follows that the adjunction \(j^*i_! \dashv i^*j_*\) restricts to an adjunction \(\bB \dashv \bOmega \) between \(\Mon (C)\) and \(C_*\). Since \(\bOmega \) takes values in \(\Grp (C) \subseteq \Mon (C)\), this adjunction further restricts to an adjunction between \(\Grp (C)\) and \(C_*\). β‘
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