Theorems · Theorem · category theory

CategoryTheory.Functor.regularEpiOfPreserves_isColimit

∀ {C : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_3, u_3} C]
  [inst_1 : CategoryTheory.Category.{v_4, u_4} D] {X Y : C} (f : X ⟶ Y) [inst_2 : CategoryTheory.EffectiveEpi f]
  (F : CategoryTheory.Functor C D) [inst_3 : F.PreservesEffectiveEpis]
  [inst_4 : CategoryTheory.Limits.PreservesLimitsOfShape CategoryTheory.Limits.WalkingCospan F]
  (c : CategoryTheory.Limits.PullbackCone f f) (hc : CategoryTheory.Limits.IsLimit c),
  (CategoryTheory.Functor.regularEpiOfPreserves f F c hc).isColimit =
    CategoryTheory.isColimitCoforkOfEffectiveEpi (F.map f)
      (CategoryTheory.Limits.PullbackCone.mk (F.map c.fst) (F.map c.snd) ⋯)
      ((CategoryTheory.Limits.IsLimit.equivOfNatIsoOfIso
          (CategoryTheory.Limits.cospanIsoMk
            (CategoryTheory.Iso.refl
              (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.one))
            (CategoryTheory.Iso.refl
              (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.left))
            (CategoryTheory.Iso.refl
              (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.right))
            ⋯ ⋯)
          (F.mapCone c) (CategoryTheory.Limits.PullbackCone.mk (F.map c.fst) (F.map c.snd) ⋯)
          (CategoryTheory.Limits.Cone.ext
            (CategoryTheory.Iso.refl
              ((CategoryTheory.Limits.Cone.postcompose
                      (CategoryTheory.Limits.cospanIsoMk
                          (CategoryTheory.Iso.refl
                            (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.one))
                          (CategoryTheory.Iso.refl
                            (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.left))
                          (CategoryTheory.Iso.refl
                            (((CategoryTheory.Limits.cospan f f).comp F).obj CategoryTheory.Limits.WalkingCospan.right))
                          ⋯ ⋯).hom).obj
                  (F.mapCone c)).pt)
            ⋯))
        (CategoryTheory.Limits.isLimitOfPreserves F hc))

Defined in

Mathlib.CategoryTheory.EffectiveEpi.Preserves

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.EffectiveEpiCategoryTheory.Functor.PreservesEffectiveEpisCategoryTheory.Limits.PreservesLimitsOfShape