Theorems · Definition · category theory

CategoryTheory.Monad.monadicOfHasPreservesReflexiveCoequalizersOfReflectsIsomorphisms

{C : Type u₁} →
  {D : Type u₂} →
    [inst : CategoryTheory.Category.{v₁, u₁} C] →
      [inst_1 : CategoryTheory.Category.{v₁, u₂} D] →
        {G : CategoryTheory.Functor D C} →
          {F : CategoryTheory.Functor C D} →
            (F ⊣ G) →
              [CategoryTheory.Limits.HasReflexiveCoequalizers D] →
                [G.ReflectsIsomorphisms] →
                  [CategoryTheory.Monad.PreservesColimitOfIsReflexivePair G] → CategoryTheory.MonadicRightAdjoint G

Reflexive (crude) monadicity theorem. If G has a right adjoint, D has and G preserves reflexive coequalizers and G reflects isomorphisms, then G is monadic.

Defined in

Mathlib.CategoryTheory.Monad.Monadicity

Cited by

0 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasReflexiveCoequalizersCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Monad.PreservesColimitOfIsReflexivePair