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FC (γ) Σ(X ) ⊆ X and if XC −−→ C → − B→ − Σ(XC ) is a triangle in ∇, then C(XC , Ω(B)) = 0. (iii) The pair (X , Y) is a torsion pair in C. Proof. We prove that (i) ⇔ (iii). The proof of (ii) ⇔ (iii) is similar. 3, it remains to show that (i) ⇒ (iii). For any C ∈ C, let fC : R(C) → − C be the counit of the adjoint pair (i, R), and fC gC hC β α let (∗): R(C) −−→ C −−→ Y C −−→ ΣR(C) be a triangle in ∇. If Ω(Y C ) − →A− → g C C −−→ Y C is a triangle in ∆, then since fC ◦ g C = 0, there exists a morphism κ : R(C) → − A such that: κ ◦ α = fC .

Define a functor F ∗ : X → − D by F ∗ = F i. Then F ∗ is obviously left exact and the above isomorphism shows that ∼ = → F . If G : X → − D is another exact functor endowed with a natural F (f ) : F ∗ R − ∼ ∼ = = → F i = F ∗ . Hence F ∗ is the isomorphism ξ : GR − → F , then ξi : G ∼ = GRi − 5. LIFTING TORSION PAIRS 42 unique up to isomorphism left exact functor which extends F . Then R : C → − X represents X as the stabilization of C. Conversely assume that R represents X as the stabilization of C with respect to its left triangulation.

Hence f is a weak kernel of g. The parenthetical case is dual. 3. TORSION PAIRS 37 The following characterizes when a pair of subcategories forms a torsion pair. 7. Let (X , Y) be a pair of subcategories of a pretriangulated category C. If C(X , Y) = 0, then the following are equivalent: (i) (α) The inclusion i : X → C has a right adjoint R and the counit fC : R(C) → − C is a weak kernel, ∀C ∈ C. gC fC hC (β) If R(C) −−→ C −−→ Y C −−→ ΣR(C) is a triangle in ∇, then Y C ∈ Y. gC (γ) Ω(Y) ⊆ Y and if Ω(Y C ) → − A→ − C −−→ Y C is a triangle in ∆, then C C(Σ(A), Y ) = 0.

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Comprehensive Introduction To Differential Geometry, 2nd Edition, Volume 4 by Michael Spivak

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