13.4. StabilityCondition: Defs🔗
13.4.1. PreStabilityCondition.WithClassMap🔗
[Definition 1.1] restated as Def 5.1
[Definition 5.1]
A Bridgeland prestability condition on a triangulated category \mathcal{C} with respect to a class map v \colon K_0(\mathcal{C}) \to \Lambda consists of a slicing \mathcal{P} together with a group homomorphism Z \colon \Lambda \to \mathbb{C} (the central charge) such that for every nonzero \mathcal{P}(\phi)-semistable object E, one has Z(v([E])) = m \cdot e^{i\pi\phi} for some m > 0. This is Definition 5.1 of Bridgeland (2007), lifted to an arbitrary class lattice.
Construction: The structure packages the slicing, the central charge on \Lambda, and the compatibility proof that the image of each semistable class lies on the correct ray in \mathbb{C}.
🔗structure
Fields
Z : Λ →+ ℂ
The central charge on the class lattice Λ.
compat' : ∀ (φ : ℝ) (E : C),
self.slicing.P φ E →
¬CategoryTheory.Limits.IsZero E →
∃ m, 0 < m ∧ self.Z (v (CategoryTheory.Triangulated.K₀.of C E)) = ↑m * Complex.exp (↑(Real.pi * φ) * Complex.I)
Compatibility (raw). Use σ.compat instead.
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13.4.8. PreStabilityCondition🔗
A Bridgeland prestability condition on \mathcal{C}, defined as the specialization of PreStabilityCondition.WithClassMap to the identity class map \operatorname{id} \colon K_0(\mathcal{C}) \to K_0(\mathcal{C}).
Construction: An abbreviation for PreStabilityCondition.WithClassMap C (AddMonoidHom.id (K₀ C)).
🔗defCategoryTheory.Triangulated.PreStabilityCondition.{v, u} (C : Type u)
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ]
[CategoryTheory.Preadditive C]
[∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
[CategoryTheory.Pretriangulated C] : Type u CategoryTheory.Triangulated.PreStabilityCondition.{v,
u}
(C : Type u)
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Limits.HasZeroObject C]
[CategoryTheory.HasShift C ℤ]
[CategoryTheory.Preadditive C]
[∀ (n : ℤ),
(CategoryTheory.shiftFunctor C
n).Additive]
[CategoryTheory.Pretriangulated C] :
Type u
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13.4.9. StabilityCondition.WithClassMap🔗
[Definition 5.7]
A Bridgeland stability condition on a triangulated category \mathcal{C} with respect to a class map v \colon K_0(\mathcal{C}) \to \Lambda is a prestability condition whose underlying slicing is locally finite: for every \phi \in \mathbb{R} and every bounded interval of phases, every object in the corresponding quasi-abelian category has finite length. This is Definition 5.3 of Bridgeland (2007).
Construction: Extends PreStabilityCondition.WithClassMap with a proof that the slicing satisfies the IsLocallyFinite condition.
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13.4.10. StabilityCondition🔗
A Bridgeland stability condition on \mathcal{C}, the specialization of StabilityCondition.WithClassMap to the identity class map. This is the classical object \operatorname{Stab}(\mathcal{C}) from Bridgeland (2007).
Construction: An abbreviation for StabilityCondition.WithClassMap C (AddMonoidHom.id (K₀ C)).
🔗defCategoryTheory.Triangulated.StabilityCondition.{v, u} (C : Type u)
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.HasShift C ℤ]
[CategoryTheory.Preadditive C]
[∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
[CategoryTheory.Pretriangulated C] [CategoryTheory.IsTriangulated C] :
Type u CategoryTheory.Triangulated.StabilityCondition.{v,
u}
(C : Type u)
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.Limits.HasZeroObject C]
[CategoryTheory.HasShift C ℤ]
[CategoryTheory.Preadditive C]
[∀ (n : ℤ),
(CategoryTheory.shiftFunctor C
n).Additive]
[CategoryTheory.Pretriangulated C]
[CategoryTheory.IsTriangulated C] :
Type u
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13.4.11. stabilityCondition_compat_apply🔗
The compatibility condition for a stability condition with identity class map: for every nonzero \mathcal{P}(\phi)-semistable object E, the central charge satisfies Z([E]) = m \cdot e^{i\pi\phi} for some m > 0.
Proof: Immediate by simplifying the general compatibility statement when v = \operatorname{id}.
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13.4.12. im_ofReal_mul_exp_mul_exp_neg🔗
The phase rotation identity: \operatorname{Im}(m \cdot e^{i\pi\psi} \cdot e^{-i\pi\phi}) = m \cdot \sin(\pi(\psi - \phi)). This factorization underlies all sign arguments in Bridgeland's deformation theory (Lemmas 6.1--6.4).
Proof: Combines the exponentials via e^{i\pi\psi} \cdot e^{-i\pi\phi} = e^{i\pi(\psi - \phi)}, then extracts the imaginary part using Euler's formula.
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13.4.13. slicingDist🔗
The Bridgeland generalized metric on slicings. For slicings \mathcal{P}_1 and \mathcal{P}_2, this is d(\mathcal{P}_1, \mathcal{P}_2) = \sup_{E \neq 0} \max(|\phi_1^+(E) - \phi_2^+(E)|,\, |\phi_1^-(E) - \phi_2^-(E)|) where \phi^{\pm} denote the maximum/minimum HN phases.
Construction: Defined as an \mathbb{R}_{\geq 0}^{\infty}-valued supremum over all nonzero objects of the maximum of the absolute phase differences.
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13.4.14. stabSeminorm🔗
The Bridgeland seminorm \|U\|_\sigma on \operatorname{Hom}(\Lambda, \mathbb{C}). For a stability condition \sigma = (Z, \mathcal{P}) with class map v and a group homomorphism U \colon \Lambda \to \mathbb{C}, this is \|U\|_\sigma = \sup \{\, |U(v[E])| / |Z(v[E])| : E \text{ is } \sigma\text{-semistable and nonzero}\,\}.
Construction: Defined as an \mathbb{R}_{\geq 0}^{\infty}-valued supremum indexed over objects, phases, semistability witnesses, and nonzero witnesses.
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13.4.15. basisNhd🔗
The basis neighborhood B_\varepsilon(\sigma) for the Bridgeland topology on \operatorname{Stab}_v(\mathcal{D}): B_\varepsilon(\sigma) = \{\, \tau : \|\tau.Z - \sigma.Z\|_\sigma < \sin(\pi\varepsilon) \text{ and } d(\mathcal{P}_\sigma, \mathcal{P}_\tau) < \varepsilon \,\}.
Construction: Defined as the set of stability conditions satisfying both the seminorm bound on the central charge difference and the slicing distance bound.
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13.4.16. topologicalSpace🔗
The Bridgeland topology on \operatorname{Stab}_v(\mathcal{D}), generated by \{\, B_\varepsilon(\sigma) : \sigma \in \operatorname{Stab}_v(\mathcal{D}),\, 0 < \varepsilon < 1/8 \,\}. This is the BLMNPS topology: the coarsest making both the charge map \sigma \mapsto \sigma.Z and the slicing map continuous.
Proof: Constructed via TopologicalSpace.generateFrom applied to the family of basis neighborhoods.
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13.4.17. ComponentIndex🔗
The set of connected components of \operatorname{Stab}_v(\mathcal{D}), i.e., the quotient \pi_0(\operatorname{Stab}_v(\mathcal{D})).
Construction: An abbreviation for ConnectedComponents (StabilityCondition.WithClassMap C v).
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13.4.18. Component🔗
The type of stability conditions in a fixed connected component \Sigma of \operatorname{Stab}_v(\mathcal{D}).
Construction: Defined as the subtype of stability conditions whose ConnectedComponents.mk equals the given component index.
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13.4.19. CentralChargeIsLocalHomeomorphOnConnectedComponents🔗
[Theorem 1.2] statement only; proof is componentTopologicalLinearLocalModel
The Bridgeland Theorem 1.2 statement in existential form: for every connected component \Sigma of \operatorname{Stab}_v(\mathcal{D}), there exists a complex normed subspace V(\Sigma) \subseteq \operatorname{Hom}(\Lambda, \mathbb{C}) such that every central charge in \Sigma lies in V(\Sigma), and the restricted central charge map \sigma \mapsto Z(\sigma) is a local homeomorphism from \Sigma onto V(\Sigma).
Construction: A proposition quantifying over component indices, asserting existence of a \mathbb{C}-normed submodule and an IsLocalHomeomorph instance.
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