Theorems · Theorem · category theory

CategoryTheory.isLeftAdjoint_square_lift

∀ {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) (L : CategoryTheory.Functor C D) (comm : U.comp L ≅ Q.comp V) [U.IsLeftAdjoint]
  [inst_5 : V.IsLeftAdjoint] [L.IsLeftAdjoint]
  (h : (X : B) → CategoryTheory.RegularMono ((CategoryTheory.Adjunction.ofIsLeftAdjoint V).unit.app X))
  [CategoryTheory.Limits.HasCoreflexiveEqualizers A], Q.IsLeftAdjoint

Suppose we have a commutative square of functors `` Q A → B U ↓ ↓ V C → D L ` where U has a right adjoint, A has coreflexive equalizers and V has a right adjoint such that each component of the counit is a regular mono. Then Q has a right adjoint if L` has a right adjoint. See https://ncatlab.org/nlab/show/adjoint+lifting+theorem

Defined in

Mathlib.CategoryTheory.Adjunction.Lifting.Right

Cited by

0 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLeftAdjointCategoryTheory.Functor.IsLeftAdjointCategoryTheory.Functor.IsLeftAdjointCategoryTheory.Limits.HasCoreflexiveEqualizers