Definition 7.37.
An analytic ring is a pair \(A = (A^{\triangleright}, \Mod_A)\) consisting of a light condensed animated commutative ring \(A^{\triangleright}\) together with a reflective subcategory \(\iota\colon \Mod_A \hookrightarrow \Mod_{A^{\triangleright}}\), with left adjoint \(L\colon \Mod_{A^{\triangleright}} \to \Mod_A\), satisfying the following conditions:
\(\Mod_A\) is closed under colimits (also limits, but this is automatic).
\(\Mod_A\) is closed under the internal hom \(\iHom(M,-)\) for every \(M \in \Mod_{A^{\triangleright}}\).
The functor \(\iota L\colon \Mod_{A^{\triangleright}} \to \Mod_{A^{\triangleright}}\) preserves connective objects with respect to the canonical t-structure.
We have \(A^{\triangleright} \in \Mod_A\).