A note on triangulated monads and categories of module spectra

TL;DR

This paper investigates conditions under which categories of module spectra over ring spectra are triangulated, showing that such structures are essentially monadic and equivalent for naive and strict modules.

Contribution

It establishes that a monad on a triangulated category with a triangulated module category is 'intrinsically monadic' after certain completions, linking naive and strict module categories.

Findings

Categories of naive modules are triangulated iff equivalent to strict modules.

Any triangulated monad's Eilenberg-Moore category inherits the triangulation.

Monadicity becomes 'essential' after Verdier quotient and idempotent completion.

Abstract

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is 'essentially monadic', i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is 'intrinsically monadic'. It follows that for any highly structured ring spectrum, its category of homotopy (a.k.a. naive) modules is triangulated if and only if it is equivalent to its category of highly structured (a.k.a. strict) modules.