Monogenic Morita theorem

Proof. By Corollary 18.5.6, the unit functor \(-\otimes \unit \colon \Sp \to C\) is the essentially unique colimit-preserving symmetric monoidal functor out of \(\Sp \), and its right adjoint is the mapping-spectrum functor \(\hom _C(\unit ,-)\) (Corollary 4.4.10). By Proposition 14.3.6 this right adjoint is canonically lax symmetric monoidal. In particular, it carries the commutative algebra \(\unit \in \CAlg (C)\) to the commutative algebra \[ A\coloneqq \hom _C(\unit ,\unit )\in \CAlg (\Sp ). \] Applying the functoriality of commutative module categories under lax symmetric monoidal functors gives a lax symmetric monoidal functor \[ \Phi \colon C\simeq \Mod _{\unit }(C)\longrightarrow \Mod _A(C), \qquad X\longmapsto \hom _C(\unit ,X), \] whose underlying \(A\)-module-valued functor is that of Proposition 19.5.5. Since \(\unit \) is compact, \(\Phi \) preserves filtered colimits; being exact, it preserves finite colimits, and hence all colimits.

Using the right-module equivalence of Observation 19.2.4, we may identify \(\Mod _A(C)\) with \(\RMod _A(C)\). Under this identification, the two-sided bar construction used in the proof of Proposition 19.5.5 defines a colimit-preserving functor \[ \Psi \colon \Mod _A(C)\longrightarrow C, \qquad M\longmapsto M\otimes _A\unit , \] where \(\unit \) carries the left \(A\)-module structure of Proposition 19.5.5. The same construction exhibits \(\Psi \) as a left adjoint to \(\Phi \). On the free module we have \(\Psi (A)\simeq \unit \), so the counit and unit of \(\Psi \dashv \Phi \) restrict to isomorphisms on the generators: the counit at \(\unit \) is the canonical isomorphism \(\Psi \Phi (\unit )=A\otimes _A\unit \simeq \unit \), and the unit at \(A\) is the isomorphism \(A\to \Phi \Psi (A)=\hom _C(\unit ,\unit )=A\). Both \(\Phi \) and \(\Psi \) are exact and preserve colimits, so the full subcategories of \(C\) and of \(\Mod _A(C)\) on which the counit, respectively the unit, is an isomorphism are stable and closed under colimits. Since \(\unit \) is a compact generator of \(C\) and \(A\) is a compact generator of \(\Mod _A(C)\) (Example 22.3.10), these subcategories are everything. Hence \(\Phi \) is an equivalence.

It remains to promote the lax symmetric monoidal structure on \(\Phi \) to a strong one. Its unit comparison \(A\to \Phi (\unit )\) is an isomorphism, and its multiplication comparison \[ \Phi (X)\otimes _A\Phi (Y)\longrightarrow \Phi (X\otimes Y) \] is an isomorphism for \(X=Y=\unit \), where both sides are \(A\). Fixing \(X=\unit \), the full subcategory of those \(Y\) for which the comparison is an isomorphism is stable, closed under colimits, and contains \(\unit \), hence is all of \(C\). Running the same argument in the second variable, now for each fixed \(Y\in C\), shows that the comparison is an isomorphism for all \(X\) and \(Y\). Thus \(\Phi \) is a symmetric monoidal equivalence. □