Theorems · Theorem · category theory

CategoryTheory.Functor.initial_equivalence_comp

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D)
  (G : CategoryTheory.Functor D E) [F.IsEquivalence] [G.Initial], (F.comp G).Initial

See also the strictly more general initial_comp below.

Defined in: Mathlib.CategoryTheory.Limits.Final
Cited by: 3 results in Mathlib
Foundations: Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Category CategoryTheory.Category CategoryTheory.Functor.IsEquivalence CategoryTheory.Functor.Initial

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites9

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
CategoryTheory.Functor.compstatement · cited by 6,529
CategoryTheory.Equivalence.symmproof · cited by 195
CategoryTheory.Functor.IsEquivalencestatement and proof · cited by 111
CategoryTheory.Functor.Initialstatement and proof · cited by 84
CategoryTheory.Functor.asEquivalenceproof · cited by 58
CategoryTheory.CostructuredArrow.preproof · cited by 36
CategoryTheory.isConnected_of_equivalentproof · cited by 11

Cited by3

Results whose statement or proof uses this declaration.

CategoryTheory.Functor.initial_iff_equivalence_compproof · cited by 1
CategoryTheory.Comma.final_fst_of_isConnected_structuredArrowproof · cited by 0
CategoryTheory.Comma.final_snd_of_isFiltered_structuredArrowproof · cited by 0