Modes as properties

Proposition 18.5.2 (Modes as properties, [Lurie (2017), Propositions 4.8.2.4, 4.8.2.9 and 4.8.2.10]). Let \(M\) be a mode.

(1)

A presentable \(\infty \)-category admits an \(M\)-module structure if and only if it is \(M\)-local, and such a structure is unique.

(2)

Tensoring with \(M\) defines a Bousfield localization \[ M\otimes -\colon \PrL \longrightarrow (\PrL )^{M\text {-}\mathrm {loc}}, \] where \((\PrL )^{M\text {-}\mathrm {loc}}\) is the full subcategory of \(M\)-local presentable \(\infty \)-categories.

(3)

If \(D\) is \(M\)-local, then precomposition with the unit \(C\to M\otimes C\) induces an equivalence \[ \FunL (M\otimes C,D)\iso \FunL (C,D). \]