GA Derived Category Skill

Derived categories and derived functors for graded Clifford modules.

Trit: +1 (PLUS) — Extension/generation of derived structures

Mathematical Foundation

Derived Category of Clifford Modules

D(Cl-Mod) = localization of Ch(Cl-Mod) at quasi-isomorphisms

Objects: Chain complexes of Cl(V,Q)-modules
Morphisms: Roof diagrams (spans via quasi-iso)

Graded Structure Preserved

D(Cl-Mod) inherits grading from Cl:

D(Cl⁰-Mod) ──→ D(Cl¹-Mod) ──→ D(Cl²-Mod) ──→ ...
     ↓              ↓              ↓
   D^{gr}(Cl-Mod) = ⊕ₖ D(Clᵏ-Mod)

Derived Functors

RHom and ⊗^L

RHom_Cl(M, N) = Hom in D(Cl-Mod)
             = total right derived of Hom

M ⊗^L_Cl N = derived tensor product
           = total left derived of ⊗

Key: H^i(RHom(M,N)) = Ext^i(M,N)
     H_i(M ⊗^L N) = Tor_i(M,N)

Derived Wedge and Contraction

# Derived wedge product
L∧ : D(Cl-Mod) × D(Cl-Mod) → D(Cl-Mod)
(M•, N•) ↦ Tot(M• ⊗ N•) with wedge differential

# Derived contraction  
R⌋ : D(Cl-Mod) × D(Cl-Mod) → D(Cl-Mod)
(M•, N•) ↦ RHom(M•, N•) with contraction structure

# GF(3) trit assignment:
trit(L∧) = +1  (extension)
trit(R⌋) = -1  (contraction)
trit(id) = 0   (ergodic)

ACSet Schema for Derived Structures

@present SchDerivedClifford(FreeSchema) begin
  # Chain complex objects
  (ChainObj, Morphism, Differential)::Ob
  
  # Complex structure
  source::Hom(Morphism, ChainObj)
  target::Hom(Morphism, ChainObj)
  diff::Hom(Differential, Morphism)  # d: Cₙ → Cₙ₋₁
  
  # Grading
  degree::Attr(ChainObj, Int)        # homological degree
  cl_grade::Attr(ChainObj, Int)      # Clifford grade
  
  # Quasi-isomorphism marking
  is_quasi_iso::Attr(Morphism, Bool)
  
  # Derived functor tracking
  derived_from::Attr(Morphism, Symbol)  # :wedge, :contract, :hom
  
  # GF(3)
  trit::Attr(Morphism, GF3Trit)
  
  # Constraints
  compose(diff, diff) == zero  # d² = 0
end

Triangulated Structure

D(Cl-Mod) is triangulated:

Distinguished triangles: A → B → C → A[1]
Shift functor: [1] shifts complex by 1

Clifford-specific: grade shift interacts with homological shift
  Clᵏ-Mod[n] involves both k (Clifford) and n (homological)

Octahedral Axiom for GA

Given morphisms in D(Cl-Mod):
  f: A → B (wedge)
  g: B → C (wedge)
  
Octahedron relates cones:
  Cone(f), Cone(g), Cone(g∘f), Cone(f')[1]
  
GF(3): Each face contributes trit, total = 0

t-Structures and Hearts

Standard t-structure on D(Cl-Mod):
  D≤0 = {M• : Hⁱ(M•) = 0 for i > 0}
  D≥0 = {M• : Hⁱ(M•) = 0 for i < 0}

Heart = D≤0 ∩ D≥0 ≅ Cl-Mod (abelian!)

Clifford t-structure (grade-aware):
  D≤k = {M• : Hⁱ(M•) is Cl≤k-module}

Integration with GA Skills

Skill	Derived Role	Trit
ga-abelian-extensions	Ext = H*(RHom)	-1
ga-central-extensions	Lie algebra cohomology	0
ga-derived-category	D(Cl-Mod) framework	+1

Triad: (-1) + 0 + (+1) = 0 ✓

Spectral Sequences

Grade spectral sequence:
E₁^{p,q} = H^q(Clᵖ-Mod) ⟹ H^{p+q}(Cl-Mod)

Converges: d_r : E_r^{p,q} → E_r^{p+r, q-r+1}

GF(3) tracking: trit(d_r) computed from page structure

Open Games: Derived as Strategy Space

Play:    Cl-Mod → D(Cl-Mod)     -- "derive module"
Coplay:  D(Cl-Mod) → H*(−)      -- "take cohomology"

Equilibrium: Quasi-isomorphic resolutions
             (multiple strategies, same outcome)

Specter Navigation

;; Navigate derived category
(def DERIVED-PATH
  (sp/path [:complex ALL :differential]))

;; Compute RHom via resolution
(defn rhom-navigate [M N]
  (sp/transform [PROJECTIVE-RESOLUTION]
    #(hom-complex % N)
    M))

;; Extract Ext groups
(sp/select [DERIVED-PATH :cohomology] rhom-complex)

Koszul Duality

Cl(V) is Koszul algebra (when V finite-dim)
Koszul dual: Cl(V)! ≅ ∧*(V*) (exterior algebra)

Duality functor: D(Cl-Mod) ≃ D(∧*-Mod)
Preserves GF(3) trit structure

Commands

# Compute derived wedge
julia -e 'L_wedge(projective_res(M), N)'

# RHom computation
julia -e 'RHom_Cl(M, N) |> cohomology'

# Spectral sequence page
bb -e '(spectral-seq-page clifford-complex 2)'

# Triangulated structure check
julia -e 'is_distinguished_triangle(A, B, C)'

References

• Weibel: Intro to Homological Algebra (Ch. 10: Derived Categories)
• Gelfand & Manin: Methods of Homological Algebra
• ga-abelian-extensions skill (Ext computation)
• sheaf-cohomology skill (derived sheaf functors)

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