Theorems · Definition · algebraic topology

SSet.horn.IsCompatible.lift

{X : SSet} →
  {n : ℕ} →
    {i : Fin (n + 2)} →
      {f : (j : Fin (n + 2)) → j ≠ i → (SSet.stdSimplex.obj { len := n } ⟶ X)} →
        SSet.horn.IsCompatible f →
          {Y : SSet} →
            (p : X ⟶ Y) →
              [HomotopicalAlgebra.Fibration p] →
                (b : SSet.stdSimplex.obj { len := n + 1 } ⟶ Y) →
                  (∀ (j : Fin (n + 2)) (hj : j ≠ i),
                      CategoryTheory.CategoryStruct.comp (f j hj) p =
                        CategoryTheory.CategoryStruct.comp (SSet.stdSimplex.δ j) b) →
                    (SSet.stdSimplex.obj { len := n + 1 } ⟶ X)

If f : ∀ (j : Fin (n + 2)) (_ : j ≠ i), Δ[n] ⟶ X is a compatible family of morphisms (which defines a morphism Λ[n + 1, i] ⟶ X), p : X ⟶ Y a Kan fibration and b : Δ[n + 1] ⟶ Y such that for all j ≠ i, f j _ ≫ p = stdSimplex.δ j ≫ b, then this is a lifting Δ[n + 1] ⟶ X.

Defined in

Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations

Cited by

5 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

HomotopicalAlgebra.Fibration