Theorems · Definition · category theory

CategoryTheory.Abelian.Ext.hom

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      [inst_2 : CategoryTheory.HasExt C] →
        {X Y : C} →
          [inst_3 : HasDerivedCategory C] →
            {a : ℕ} →
              CategoryTheory.Abelian.Ext X Y a →
                CategoryTheory.ShiftedHom ((DerivedCategory.singleFunctor C 0).obj X)
                  ((DerivedCategory.singleFunctor C 0).obj Y) ↑a

The morphism in the derived category which corresponds to an element in Ext X Y a.

Defined in: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
Cited by: 42 results in Mathlib
Foundations: Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Abelian CategoryTheory.HasExt HasDerivedCategory

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Around this declaration

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

Cites11

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

DFunLike.coeproof · cited by 62,936
CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.Functor.objstatement · cited by 19,642
CategoryTheory.Abelianstatement and proof · cited by 1,753
CategoryTheory.HasExtstatement and proof · cited by 218
CategoryTheory.Abelian.Extstatement and proof · cited by 191
HasDerivedCategorystatement and proof · cited by 190
DerivedCategorystatement · cited by 165
CategoryTheory.ShiftedHomstatement · cited by 88
DerivedCategory.singleFunctorstatement · cited by 78
CategoryTheory.Abelian.Ext.homEquivproof · cited by 8

Cited by43

Results whose statement or proof uses this declaration.

CategoryTheory.Abelian.Ext.extstatement and proof · cited by 31
CategoryTheory.Abelian.Ext.comp_homstatement · cited by 27
CategoryTheory.Abelian.Ext.mk₀_homstatement · cited by 20
CategoryTheory.Abelian.Ext.mk₀_comp_mk₀proof · cited by 10
CategoryTheory.Abelian.Ext.zero_compproof · cited by 10
CategoryTheory.Abelian.Ext.comp_zeroproof · cited by 7
CategoryTheory.ShortComplex.ShortExact.extClass_homstatement · cited by 6
CategoryTheory.Abelian.Ext.zero_homstatement and proof · cited by 6
CategoryTheory.Abelian.Ext.mk₀_id_compproof · cited by 5
CategoryTheory.Abelian.Ext.comp_mk₀_idproof · cited by 5
CategoryTheory.Abelian.Ext.hom'proof · cited by 4
CategoryTheory.Abelian.Ext.mapExactFunctor_homstatement and proof · cited by 4