A comparison of categories of structured objects can often be reduced to a comparison of free objects. We record Lurie’s criterion for doing so, which will be used in Section 19.6 to recognize monadic functors.
We first recall the class of simplicial objects for which the relevant geometric realizations exist formally. Let \(\simp _{-\infty }\) be the category whose objects are the symbols \([n]\) for \(n \geq -1\) and whose morphisms \([m] \to [n]\) are the order-preserving maps \[ \{-\infty \} \sqcup [m] \longrightarrow \{-\infty \} \sqcup [n] \] which preserve \(-\infty \). There are inclusions \(\simp \subseteq \simp _+ \subseteq \simp _{-\infty }\), where a morphism belongs to \(\simp _+\) precisely when the inverse image of \(-\infty \) is \(\{-\infty \}\).
Definition 21.7.1 ([Lurie (2017), Definition 4.7.2.2]). An augmented simplicial object \(X\colon \simp _+\catop \to C\) is split if it extends to a functor \(\simp _{-\infty }\catop \to C\). A simplicial object is split if it extends to a split augmented simplicial object.
Let \(G\colon D \to C\) be a functor. A simplicial object \(X_\bullet \colon \simp \catop \to D\) is \(G\)-split if \(GX_\bullet \) is split in \(C\).
Lemma 21.7.2 (Absolute realization of split simplicial objects). Every split augmented simplicial object is a colimit diagram. In particular, every split simplicial object admits a geometric realization, and every functor preserves this realization.
Proof. See Reference ? of [Cisinski et al. (2026)]. □
The following form of Lurie’s comparison theorem does not require the language of monads.
Theorem 21.7.3 (Comparison over a base). Let
be a triangle equipped with a natural isomorphism \[ \theta \colon G'K \iso G \] Assume that \(G\) and \(G'\) admit left adjoints \(F\) and \(F'\), and that the mate \(\theta ^\flat \colon F' \to KF\) is an isomorphism. Assume furthermore that:
- (1)
-
The functors \(G\) and \(G'\) are conservative;
- (2)
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The \(\infty \)-category \(D\) admits geometric realizations of \(G\)-split simplicial objects, and \(G\) preserves them;
- (3)
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The \(\infty \)-category \(D'\) admits geometric realizations of \(G'\)-split simplicial objects, and \(G'\) preserves them.
Then \(K\) is an equivalence of \(\infty \)-categories.
Proof. Apply [Lurie (2017), Corollary 4.7.3.16] to the displayed triangle. Its condition on free objects is precisely the invertibility of \(\theta ^\flat \), and conservativity of \(G\) gives the equivalence conclusion. □
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