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Monadic and strictly monadic functors
Definition
Let have a left adjoint , let be the induced monad, and let be the comparison functor of The comparison functor to the Eilenberg–Moore category exists and is unique. The functor is monadic when is an equivalence of categories (Equivalence, quasi-inverse, and adjoint equivalence of categories).
It is strictly monadic when is an isomorphism of categories, so its object and morphism correspondences, inverse, and equations hold on the nose. Strict monadicity implies monadicity; the converse is not part of the definition.
Depends on
Used by
- Every category monadic over Set is complete Corollary
- G-sets are strictly monadic over sets Corollary
- Torsion-free abelian groups give a conservative right adjoint that is not monadic Counterexample
- Categories of models for algebraic theories Definition
- The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
- FALSE: every conservative right adjoint is monadic False statement
- FALSE: ordinary Beck creation characterizes strict monadicity False statement
- Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers Proposition
- Beck's monadicity theorem in data-supplied form Theorem
- Every monadic functor is conservative Theorem
- Groups are strictly monadic over sets Theorem
- Modules over a fixed unital ring are strictly monadic over sets Theorem
- Monoids and unital rings are strictly monadic over sets Theorem
- Strict Beck monadicity theorem Theorem
- The inclusion of a reflective full subcategory is monadic Theorem
- Under the ultrafilter lemma, compact Hausdorff spaces are monadic over sets Theorem
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, 2nd ed., Definition 5.3.1 (standard reference, not scraped)
- S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter VI (standard reference, not scraped)