Theorems · Definition · category theory

Homotopy.ofExtend

{ι : Type u_1} →
  {ι' : Type u_2} →
    {c : ComplexShape ι} →
      {c' : ComplexShape ι'} →
        {C : Type u_3} →
          [inst : CategoryTheory.Category.{v_1, u_3} C] →
            [inst_1 : CategoryTheory.Limits.HasZeroObject C] →
              [inst_2 : CategoryTheory.Preadditive C] →
                {K L : HomologicalComplex C c} →
                  {f g : K ⟶ L} →
                    {e : c.Embedding c'} →
                      [e.IsRelIff] →
                        Homotopy (HomologicalComplex.extendMap f e) (HomologicalComplex.extendMap g e) → Homotopy f g

If e : c.Embedding c' is an embedding of complex shapes, f and g are morphism between cochain complexes of shape c, and h is an homotopy between the extensions extendMap f e and extendMap g e, then this is the corresponding homotopy between f and g.

Defined in

Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Cited by

4 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveComplexShape.Embedding.IsRelIff