Proposition 14.3.10 (Limits of \(\Oo \)-algebras). Let \(C\) be a symmetric monoidal \(\infty \)-category, let \(\Oo \) be an \(\infty \)-operad, and let \(I\) be a small \(\infty \)-category such that \(C\) admits \(I\)-indexed limits. Then \(\Alg _{\Oo }(C)\) admits \(I\)-indexed limits, and the evaluation functors \[ \ev _x\colon \Alg _{\Oo }(C)\to C, \qquad A\mapsto A_x \qquad (x\in \Oo ^{\simeq }), \] jointly create them.
Proof. By Corollary 21.2.2, the constant-diagram functor fits into an adjunction \[ \const \colon C\rightleftarrows \Fun (I,C)\noloc \lim _I. \] Being given by restriction along \(I \to *\), the functor \(\const \) is symmetric monoidal for the pointwise symmetric monoidal structure, so Proposition 14.3.6 makes this a symmetric monoidal adjunction. By Lemma 14.3.9, it induces an adjunction \[ \Alg _{\Oo }(C) \rightleftarrows \Alg _{\Oo }(\Fun (I,C)) \simeq \Fun (I,\Alg _{\Oo }(C)). \] Under the equivalence of Lemma 14.2.12, the left adjoint is the constant-diagram functor. The right adjoint consequently computes limits in \(\Alg _{\Oo }(C)\), and its value at every color \(x\) is \(\lim _I A_x\). Thus all the evaluation functors preserve these limits. They jointly detect isomorphisms: every object of \(\Oo ^{\otimes }\) is a finite product of colors, and algebra maps preserve these products, so a natural transformation that is invertible at every color is invertible at every object. Consequently the evaluation functors jointly reflect limit cones and hence jointly create the limits. □
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