Derived Categories

Derived Categories Skill: Homological Algebra for BCI

Status: Production Ready Trit: -1 (MINUS - validator) Color: #8026D8 (Purple) Principle: Derived categories are the correct framework for homological invariants Frame: D(BCI) = derived category of chain complexes of BCI signals

Overview

Derived Categories provide the homological algebra infrastructure for the BCI pipeline. All cohomology theories (L8, L13, L14, L17, L19) live naturally in the derived category. Implements:

1. Chain complexes: C_n -> C_{n-1} -> ... with d^2 = 0
2. Homology computation: H_n = ker(d_n) / im(d_{n+1})
3. Derived functors: Tor (tensor obstruction), Ext (extension obstruction)
4. Triangulated structure: shift functor [1], distinguished triangles, octahedral axiom
5. Long exact sequences: connecting homomorphism delta
6. Spectral sequences: E_r pages, degeneration, convergence
7. Quasi-isomorphisms: localization to derived equivalence

Correct by construction: d^2 = 0 verified computationally at all levels.

Core Formulae

Chain complex C_*:
  ... -> C_n -d_n-> C_{n-1} -d_{n-1}-> C_{n-2} -> ...
  d_{n-1} o d_n = 0  (boundary of boundary is zero)

Homology:
  H_n(C) = ker(d_n) / im(d_{n+1})
  dim H_n = dim ker(d_n) - rank(d_{n+1})

Euler characteristic:
  chi = sum (-1)^n dim H_n = sum (-1)^n dim C_n

Derived functors:
  Tor_n(A,B) = L_n(- tensor B)(A)   (left derived of tensor)
  Ext^n(A,B) = R^n Hom(A,-)(B)      (right derived of Hom)

Triangulated category D(A):
  Shift: C[1]_n = C_{n-1}
  Distinguished triangle: X -> Y -> Cone(f) -> X[1]
  Octahedral axiom (TR4): composition coherence

Long exact sequence:
  ... -> H_n(A) -> H_n(B) -> H_n(C) -delta-> H_{n-1}(A) -> ...

Spectral sequence:
  E_r^{p,q} with d_r: E_r^{p,q} -> E_r^{p-r,q+r-1}
  E_{r+1} = H(E_r, d_r)
  Convergence: E_infinity^{p,q} => H^{p+q}

Gadgets

1. ChainComplex

Build and verify chain complexes:

(def d3 [[1.0 -1.0 0.0] [0.0 1.0 -1.0] [1.0 0.0 -1.0]])
(def d2 [[-1.0 -1.0 1.0] [-2.0 -2.0 2.0] [-1.0 -1.0 1.0]])
;; Verify: (mat-mul d2 d3) = zero-matrix

2. DerivedFunctors

Tor and Ext computation:

(defn compute-tor [chain-a chain-b]
  ;; Tor_0 = A tensor B, Tor_1 = flatness obstruction
  ...)
(defn compute-ext [chain-a chain-b]
  ;; Ext^0 = Hom(A,B), Ext^1 = extension obstruction
  ...)

3. TriangulatedStructure

Distinguished triangles and shift:

(defn shift-complex [chain n]
  (vec (for [i (range (count chain))]
    (nth chain (mod (+ i n) (count chain))))))

(defn cone [f-chain g-chain]
  (vec (concat g-chain (shift-complex f-chain 1))))

4. SpectralSequence

Page-by-page computation:

(defn spectral-page [signals r]
  (case r
    0 signals                    ;; E_0 = associated graded
    1 (successive-differences)   ;; E_1 = homology of E_0
    2 (second-differences)       ;; E_2 = homology of E_1
    (degenerate)))               ;; E_r = E_infinity for r >= 2

Key Results

BCI Chain Complex:
  C_3 (dim 3) -> C_2 (dim 3) -> C_1 (dim 3) -> C_0 (dim 1)
  d^2 = 0: VERIFIED at all levels (max |entry| = 0.000000)
  Homology: H_0 = H_1 = H_2 = H_3 = 0 (acyclic complex)
  Euler characteristic: chi = 0

Derived Functors (world pairs):
  Tor_1(a,c) = 1 (non-flat pair, variance 0.043)
  Tor_1(b,c) = 1 (non-flat pair, variance 0.022)
  Tor_1(a,b) = 0 (flat pair)

Triangulated Structure:
  3 distinguished triangles constructed
  All cone ratios within quasi-iso range

Spectral Sequences:
  Degeneration at E_2 for all worlds
  World-b: immediate degeneration (uniform signals)
  World-c: non-trivial E_1 page (signal diversity)

BCI Integration (Layer 21)

Completes the Homological Chain: L8 -> L13 -> L14 -> L21

• L8 Persistent Homology: H_* = homology of Rips complex = object in D(Ab)
• L13 Relative Homology: H(X,A) from distinguished triangle A -> X -> X/A -> A[1]
• L14 Cohomology Ring: Cup product = derived tensor in D(Ab)
• L17 de Rham: de Rham complex is a chain complex in D(Vect)
• L19 Sheaf Cohomology: H^n(X,F) = R^n Gamma(F) = right derived functor
• L20 Operadic Composition: Bar construction B(O) is a chain complex, Koszul duality via Ext

Skill Name: derived-categories Type: Chain Complexes / Derived Functors / Triangulated Categories / Spectral Sequences Trit: -1 (MINUS) GF(3): Forms valid triads with PLUS + ERGODIC skills

Integration with GF(3) Triads

operadic-composition (+1) x information-geometry (0) x derived-categories (-1) = 0
stochastic-resonance (+1) x spectral-methods (0) x derived-categories (-1) = 0

CT lattice atlas

Part of: para-mensch-commons (CT lattice family).