Theorems · Definition · category theory

AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    AugmentedSimplexCategory.equivAugmentedSimplicialObject.functor.comp
        CategoryTheory.SimplicialObject.Augmented.toArrow ≅
      (CategoryTheory.Functor.mapArrowFunctor AugmentedSimplexCategoryᵒᵖ C).comp
        ((CategoryTheory.evaluation (CategoryTheory.Arrow AugmentedSimplexCategoryᵒᵖ) (CategoryTheory.Arrow C)).obj
          (CategoryTheory.Arrow.mk (CategoryTheory.WithInitial.homTo { len := 0 }).op))

Through the equivalence (AugmentedSimplexCategory ⥤ C) ≌ CosimplicialObject.Augmented C, the arrow attached to the cosimplicial object is the one obtained by evaluation at the unique arrow star ⟶ of [0].

Defined in

Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic

Cited by

4 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category