Lemma 13.1.16. Let \(C_L,C_R \subseteq C\) be wide subcategories. Then the span categories of the adequate triples \((C,C^{\simeq },C_R)\) and \((C,C_L,C^{\simeq })\) from Example 13.1.3 are given by: \[ \Span _{\simeq ,R}(C) \simeq C_R \qquadtext {and}\Span _{L,\simeq }(C) \simeq C_L\catop . \] In particular, there are canonical inclusions \[ C_R \simeq \Span _{\simeq ,R}(C) \hookrightarrow \Span _{L,R}(C) \qquadtext {and} C_L\catop \simeq \Span _{L,\simeq }(C) \hookrightarrow \Span _{L,R}(C). \]
Proof. It suffices to prove the first equivalence; the second one follows by combining the first one with Lemma 13.1.15. For \([n] \in \simp \catop \), consider the source functor \(s\colon \Tw ^r([n]) \to [n], \, (i \leq j) \mapsto i\). Note that this functor admits a fully faithful left adjoint \([n] \to \Tw ^r([n])\) sending \(i \in [n]\) to \((i \leq n) \in \Tw ^r([n])\). Indeed, there is a morphism \((i \leq n) \to (k \leq l)\) in \(\Tw ^r([n])\) if and only if \(i \leq k\). It thus follows from Lemma 21.8.8 that restriction along \(s\) induces an inclusion of animae \[ \Hom _{\Cat _{\infty }}([n],C_R) \subseteq \Hom _{\Cat _{\infty }}([n],C) \hookrightarrow \Hom _{\Cat _{\infty }}(\Tw ^r([n]),C) \] whose essential image consists of those functors \(\Tw ^r[n] \to C\) that send backwards morphisms to isomorphisms and forward morphisms into \(C_R\). Since these are precisely the morphisms of adequate triples into \((C,C^{\simeq },C_R)\), we obtain a natural equivalence \[ N(C_R)_n = \Hom _{\Cat _{\infty }}([n],C_R) \iso \Hom _{\AdTrip }(\Tw ^r([n]),(C,C^{\simeq },C_R)) = \NSpan _{\simeq ,R}(C)_n. \] This shows that \(N(C_R) \simeq \NSpan _{\simeq ,R}(C)\), and hence that \(C_R \simeq \Span _{\simeq ,R}(C)\) by full faithfulness of the nerve functor. □
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