Free module functor

Proposition 19.1.13 ([Lurie (2017), Proposition 4.2.4.2, Corollary 4.2.4.4]). Let \(C\) be a monoidal \(\infty \)-category and let \(D\) be an \(\infty \)-category left tensored over \(C\).

(1)

The forgetful functor \(\LMod (D) \xrightarrow {(\fa ^*,\fm ^*)} \Alg (C) \times D\) admits a left adjoint \[ F\colon \Alg (C) \times D \to \LMod (D), \qquad (A,M_0) \mapsto F_A(M_0) \] satisfying \(\fa ^*F_A(M_0) \simeq A\) and \(\fm ^*F_A(M_0) \simeq A \otimes M_0\).

(2)

For an associative algebra \(A \in \Alg (C)\), the forgetful functor \(\fm ^*\colon \LMod _A(D) \to D\) admits a left adjoint \[ F_A\colon D \to \LMod _A(D), \] where the underlying object of \(F_A(M_0)\) is \(A \otimes M_0\).