Theorems · Definition · category theory

CategoryTheory.Monad.comparison

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {L : CategoryTheory.Functor C D} →
          {R : CategoryTheory.Functor D C} → (h : L ⊣ R) → CategoryTheory.Functor D h.toMonad.Algebra

Given any adjunction L ⊣ R, there is a comparison functor CategoryTheory.Monad.comparison R sending objects Y : D to Eilenberg-Moore algebras for L ⋙ R with underlying object R.obj X. We later show that this is full when R is full, faithful when R is faithful, and essentially surjective when R is reflective.

Defined in

Mathlib.CategoryTheory.Monad.Adjunction

Cited by

14 results in Mathlib

Foundations

Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category