Barr Beck monogenic Morita

Remark 19.6.4. Barr–Beck gives an alternative proof of the underlying equivalence in the monogenic Morita theorem, Theorem 19.5.6. In its notation, consider the adjunction \[ -\otimes \unit \colon \Sp \rightleftarrows C\noloc \hom _C(\unit ,-). \] The right adjoint is conservative because \(\unit \) is a generator. It preserves all colimits: compactness of \(\unit \) gives preservation of filtered colimits, exactness gives preservation of finite colimits, and arbitrary coproducts are filtered colimits of finite coproducts. Barr–Beck therefore identifies \(C\) with the Eilenberg–Moore category of the associated monad on \(\Sp \).

If \(A=\hom _C(\unit ,\unit )\), the canonical natural transformation \[ -\otimes A\longrightarrow \hom _C(\unit ,-\otimes \unit ) \] is an isomorphism: both sides preserve colimits and it is an isomorphism on the sphere spectrum. This identification respects the monad structures by the construction of multiplication on the endomorphism spectrum \(A\). Hence the Eilenberg–Moore category is \(\Mod _A(C)\), and its comparison functor is the functor \(\Phi \) of Theorem 19.5.6. The proof given there additionally shows directly that this equivalence is symmetric monoidal.