Remark 19.1.6. All the definitions we just made also make sense for modules over commutative algebras, where we would replace the \(\infty \)-operad \(\oLMod \) by its commutative version \(\oMod \) from Example 12.2.8. It sits in a commutative diagram of \(\infty \)-operads as follows:
In particular, restriction along the inclusion \(\Comm \hookrightarrow \oMod \) defines a forgetful functor \(\Mod (C) := \Alg _{\oMod }(C) \to \CAlg (C)\), allowing us to define an \(\infty \)-category \(\Mod _A(C)\) for every commutative algebra \(A\) in \(C\). It turns out this \(\infty \)-category only depends on the underlying associative algebra of \(A\):
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