Theorems · Theorem · category theory

SimplexCategoryGenRel.hom_induction

∀ (P : CategoryTheory.MorphismProperty SimplexCategoryGenRel),
  (∀ {n : ℕ}, P (CategoryTheory.CategoryStruct.id (SimplexCategoryGenRel.mk n))) →
    (∀ {n m : ℕ} (u : SimplexCategoryGenRel.mk n ⟶ SimplexCategoryGenRel.mk m) (i : Fin (m + 2)),
        P u → P (CategoryTheory.CategoryStruct.comp u (SimplexCategoryGenRel.δ i))) →
      (∀ {n m : ℕ} (u : SimplexCategoryGenRel.mk n ⟶ SimplexCategoryGenRel.mk (m + 1)) (i : Fin (m + 1)),
          P u → P (CategoryTheory.CategoryStruct.comp u (SimplexCategoryGenRel.σ i))) →
        ∀ {a b : SimplexCategoryGenRel} (f : a ⟶ b), P f

An unrolled version of the induction principle obtained in the previous lemma.

Defined in: Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites18

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

Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.CategoryStruct.idstatement and proof · cited by 6,235
CategoryTheory.MorphismPropertystatement and proof · cited by 2,179
CategoryTheory.Category.id_compproof · cited by 1,998
top_le_iffproof · cited by 175
SimplexCategoryGenRelstatement and proof · cited by 39
SimplexCategoryGenRel.mkstatement and proof · cited by 39
SimplexCategoryGenRel.σstatement and proof · cited by 26
SimplexCategoryGenRel.δstatement and proof · cited by 20
CategoryTheory.MorphismProperty.multiplicativeClosureproof · cited by 18
SimplexCategoryGenRel.facesproof · cited by 5

Cited by1

Results whose statement or proof uses this declaration.

SimplexCategoryGenRel.exists_P_σ_P_δ_factorizationproof · cited by 0