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Modules over a fixed unital ring are strictly monadic over sets
Statement
For every fixed unital ring , the underlying-set functor is strictly monadic, and hence monadic.
Facts & Assumptions
Given: A fixed unital ring and the free-left--module adjunction.
The Eilenberg–Moore category of the free-left--module monad is isomorphic over to the category of left -modules (For a unital ring R, the free-R-module monad on sets has left R-modules as its Eilenberg–Moore algebras).
A functor is strictly monadic when its comparison functor is an isomorphism of categories (Monadic and strictly monadic functors).
The comparison functor is and acts on morphisms by (The comparison functor to the Eilenberg–Moore category exists and is unique).
Proof
The isomorphism in [L1] sends a module to its underlying set with its finite-linear-combination algebra structure and acts identically on underlying functions. By the comparison formula in [L3], it is the comparison for the free-module adjunction, including for the zero ring and zero module.
The comparison is therefore an isomorphism over , so the underlying-set functor is strictly monadic by [L2].
Depends on
Used by
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Sources
- E. Riehl, Category Theory in Context, 2nd ed., Example 5.1.4(iii) and Corollary 5.5.3(ii) (standard reference, not scraped)