Proposition 22.2.7 (Presentability of structured objects). Let \(C\) be a presentable \(\infty \)-category.
- (1)
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The \(\infty \)-categories \(\CMon (C)\) and \(\CGrp (C)\) are presentable, and the inclusions \[ \CGrp (C)\hookrightarrow \CMon (C)\hookrightarrow \Fun (\Span (\Fin ),C) \] admit left adjoints.
- (2)
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If \(C\) is stable and \(\Gg \) is a small collection of objects of \(C\), then the smallest full stable subcategory \(\Loc (\Gg )\subseteq C\) that contains \(\Gg \) and is closed under small colimits is presentable. Its inclusion into \(C\) preserves small colimits and consequently admits a right adjoint.
Proof. For (1), choose a small collection \(\Gg \) of objects of \(C\) whose corepresentable functors jointly detect isomorphisms. Evaluation at a finite set \(S\) has a left adjoint \[ \gamma _S\colon C\longrightarrow \Fun (\Span (\Fin ),C). \] Localizing the presentable functor category \(\Fun (\Span (\Fin ),C)\) at the maps \[ \emptyset \longrightarrow \gamma _{\emptyset }(G) \qquad \text {and}\qquad \gamma _S(G)\sqcup \gamma _T(G)\longrightarrow \gamma _{S\sqcup T}(G) \] for \(G\in \Gg \) cuts out precisely the product-preserving functors, hence \(\CMon (C)\). This is an instance of the accessible-localization theorem [Lurie (2009), Proposition 5.5.4.15]. Localizing in addition at the endomorphisms induced by the shear \(\left (\begin {smallmatrix}1&0\\1&1\end {smallmatrix}\right )\) cuts out precisely the grouplike objects. This proves the assertion for \(\CGrp (C)\) and gives both reflectors.
For (2), apply [Lurie (2017), Proposition 1.4.4.11(2)] to the small collection of all shifts \(G[n]\) with \(G\in \Gg \) and \(n\in \Z \). The resulting presentable subcategory is \(\Loc (\Gg )\). Its inclusion preserves small colimits by definition, so it admits a right adjoint by Theorem 22.2.5. □
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