Theorems · Definition · category theory

CategoryTheory.Limits.colimitOpIsoOpLimit

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {J : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} J] →
        (F : CategoryTheory.Functor J C) →
          [inst_2 : CategoryTheory.Limits.HasLimit F] →
            CategoryTheory.Limits.colimit F.op ≅ Opposite.op (CategoryTheory.Limits.limit F)

The colimit of F.op is the opposite of limit F.

Defined in: Mathlib.CategoryTheory.Limits.Opposites
Cited by: 4 results in Mathlib
Foundations: Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Category CategoryTheory.Limits.HasLimit

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites13

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Functorstatement and proof · cited by 16,252
Oppositestatement · cited by 8,081
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.Functor.opstatement · cited by 997
CategoryTheory.Limits.colimitstatement · cited by 453
CategoryTheory.Limits.limitstatement · cited by 346
CategoryTheory.Limits.HasLimitstatement and proof · cited by 226
CategoryTheory.Limits.limit.isLimitproof · cited by 146
CategoryTheory.Limits.limit.coneproof · cited by 97
CategoryTheory.Limits.Cone.opproof · cited by 22
CategoryTheory.Limits.colimit.isoColimitCoconeproof · cited by 18

Cited by4

Results whose statement or proof uses this declaration.

CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_homstatement · cited by 2
CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_invstatement and proof · cited by 1
CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv_assocstatement and proof · cited by 0
CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom_assocstatement and proof · cited by 0