Theorems · Theorem · category theory

CategoryTheory.isRightAdjoint_square_lift_monadic

∀ {A : Type u₁} {B : Type u₂} {C : Type u₃} [inst : CategoryTheory.Category.{v₁, u₁} A]
  [inst_1 : CategoryTheory.Category.{v₂, u₂} B] [inst_2 : CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄}
  [inst_3 : CategoryTheory.Category.{v₄, u₄} D] (Q : CategoryTheory.Functor A B) (V : CategoryTheory.Functor B D)
  (U : CategoryTheory.Functor A C) (R : CategoryTheory.Functor C D) (comm : U.comp R ≅ Q.comp V) [U.IsRightAdjoint]
  [CategoryTheory.MonadicRightAdjoint V] [R.IsRightAdjoint] [CategoryTheory.Limits.HasReflexiveCoequalizers A],
  Q.IsRightAdjoint

Suppose we have a commutative square of functors `` Q A → B U ↓ ↓ V C → D R ` where U has a left adjoint, A has reflexive coequalizers and V is monadic. Then Q has a left adjoint if R` has a left adjoint. See https://ncatlab.org/nlab/show/adjoint+lifting+theorem

Defined in

Mathlib.CategoryTheory.Adjunction.Lifting.Left

Cited by

0 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsRightAdjointCategoryTheory.MonadicRightAdjointCategoryTheory.Functor.IsRightAdjointCategoryTheory.Limits.HasReflexiveCoequalizers