Corollary 19.1.17 ([Lurie (2017), Corollaries 4.2.3.3 and 4.2.3.5]). Let \(C\) be a monoidal \(\infty \)-category, let \(D\) be left-tensored over \(C\), and let \(A\in \Alg (C)\). Let \(\fm ^*\colon \LMod _A(D)\to D\) be the forgetful functor, and let \(I\) be a small \(\infty \)-category.
- (1)
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If \(D\) admits \(I\)-indexed limits, then \(\fm ^*\) creates and preserves \(I\)-indexed limits.
- (2)
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If \(D\) admits \(I\)-indexed colimits and the functor \(X\otimes -\colon D\to D\) preserves \(I\)-indexed colimits for every \(X\in C\), then \(\fm ^*\) creates and preserves \(I\)-indexed colimits.
Proof. We prove (1). Equip \(\Fun (I,D)\) with the pointwise left tensoring over \(C\). The identity functor of \(C\) and the constant-diagram functor \(\const \colon D\to \Fun (I,D)\) assemble to an \(\oLMod \)-monoidal functor between the corresponding left tensorings. Since \(\const \) has right adjoint \(\lim _I\), Proposition 14.4.7 equips the pair \[ (\id _C,\lim _I) \] with the right-adjoint lax \(\oLMod \)-monoidal structure. Passing to \(\oLMod \)-algebras and taking the fiber over \(A\) therefore gives an adjunction \[ \const \colon \LMod _A(D) \rightleftarrows \LMod _A(\Fun (I,D)) \simeq \Fun (I,\LMod _A(D)) \noloc \lim _I. \] The underlying object of the right adjoint is the limit in \(D\). Thus \(\fm ^*\) preserves these limits. Since it detects isomorphisms, comparison with the constructed limit also shows that it creates them.
Part (2) is [Lurie (2017), Corollary 4.2.3.5]. Its proof uses the simplicial model for modules to lift an operadic colimit of the underlying diagram; we briefly recall this model in Section 19.3. □
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