Model Categories

Model Categories Skill: Homotopical Algebra for BCI

Status: Production Ready Trit: 0 (ERGODIC) Color: #26D8A0 (Mint) Principle: Model structures organize homotopy theory via (Cof, W, Fib) triples Frame: Model category on BCI signal chains with Quillen adjunctions

Overview

Model Categories provide the foundational framework for doing homotopy theory in any category. The three distinguished classes of morphisms (cofibrations, weak equivalences, fibrations) encode exactly the GF(3) triadic structure. Implements:

1. Model structure: (Cof, W, Fib) satisfying MC1-MC5 axioms
2. Factorization: f = (acyclic cof) o (fib) = (cof) o (acyclic fib)
3. Two-of-three: closure of weak equivalences under composition
4. Lifting properties: cofibrations LLP acyclic fibrations, and vice versa
5. Homotopy category: Ho(C) = C[W^{-1}] localization
6. Cofibrant/fibrant replacement: Q and R functors
7. Homotopy (co)limits: holim, hocolim, pullbacks, pushouts
8. Quillen adjunctions: F -| G preserving model structure, derived LF -| RG

Correct by construction: GF(3) triadic structure IS the model structure: (+1)=Cof, (0)=W, (-1)=Fib.

Core Formulae

Model structure (Cof, W, Fib) on category C:
  MC1: C has all finite limits and colimits
  MC2: 2-of-3 for W (if two of f,g,gf in W, so is third)
  MC3: W, Fib, Cof closed under retracts
  MC4: Lifting - Cof _|_ (W cap Fib), (W cap Cof) _|_ Fib
  MC5: f = (W cap Cof) o Fib = Cof o (W cap Fib)

Homotopy category:
  Ho(C) = C[W^{-1}]
  [X,Y]_{Ho} = Hom_{Ho(C)}(X,Y) = C(QX, RY) / ~

Cofibrant replacement: QX -~-> X (acyclic fib from cofibrant QX)
Fibrant replacement: X -~-> RX (acyclic cof to fibrant RX)

Quillen adjunction F: C <-> D :G
  F preserves cofibrations, G preserves fibrations
  Derived: LF = F o Q, RG = G o R
  LF: Ho(C) <-> Ho(D) :RG

Homotopy (co)limits:
  holim = lim o R (fibrant replacement then limit)
  hocolim = colim o Q (cofibrant replacement then colimit)

Gadgets

1. MorphismClassifier

Classify morphisms as W, Fib, Cof:

(defn morphism-type [source target]
  {:weak-equiv? (quasi-isomorphism? source target)
   :fibration? (degreewise-surjective? source target)
   :cofibration? (degreewise-injective? source target)})

2. Factorization

MC5 dual factorizations:

(defn factorize-cof-acfib [source target]
  ;; X -> Z -> Y via mapping cylinder
  ...)
(defn factorize-accof-fib [source target]
  ;; X -> W -> Y via path object
  ...)

3. CofibrantFibrantReplacement

(defn cofibrant-replacement [chain]   ;; QX: normalize to unit sphere
  ...)
(defn fibrant-replacement [chain]     ;; RX: extend with zero padding
  ...)

4. HomotopyLimits

(defn homotopy-limit [chains]         ;; holim: componentwise min
  ...)
(defn homotopy-colimit [chains]       ;; hocolim: sum of cofibrant
  ...)
(defn homotopy-pullback [x y z]       ;; X x^h_Z Y
  ...)
(defn homotopy-pushout [x y z]        ;; X +^h_Z Y
  ...)

5. QuillenAdjunction

(defn compress-chain [chain factor]   ;; Left Quillen: F
  ...)
(defn expand-chain [chain factor]     ;; Right Quillen: G
  ...)
;; Round-trip: ||x - GFx|| measures adjunction unit

Key Results

BCI Model Category:
  Objects: 3 signal chains (4-dimensional)
  Morphisms: 6 classified
  Weak equivalences: all pairs (norm-ratio within 0.3)
  MC2 (2-of-3): VERIFIED for all triples
  MC5 (factorization): both factorizations via cylinder/path objects

Homotopy Category Ho(BCI):
  Homotopy classes: 3 pairs, distances 0.39-0.67
  Mapping spaces: dim 16, pi_0 = 2 components each
  Cofibrant Q: unit-sphere normalization
  Fibrant R: dimension extension

Homotopy (Co)Limits:
  holim = [0.500, 0.200, 0.400, 0.100] (conservative)
  hocolim = [1.990, 1.209, 1.460, 0.854] (expansive)

Quillen Adjunction (compress -| expand):
  world-b: round-trip error = 0.000 (perfect, uniform signals)
  world-a: round-trip error = 0.158 (mild loss)
  world-c: round-trip error = 0.652 (significant, high diversity)

BCI Integration (Layer 22)

Completes the Higher Algebra Chain: L14 -> L19 -> L20 -> L21 -> L22

• L21 Derived Categories: D(A) = Ho(Ch(A)) with projective model structure
• L20 Operadic Composition: Cofibrant operads = quasi-free = A-infinity
• L19 Sheaf Cohomology: Injective model structure, fibrant sheaves
• L17 de Rham: Quillen equivalence dg-algebras <-> spaces
• L18 Info Geometry: Fisher-Rao metric induces model structure on Prob(X)
• L8 Persistent Homology: Filtered model category, persistence modules

Skill Name: model-categories Type: Model Structure / Homotopy Theory / Quillen Adjunctions / Ho(C) Trit: 0 (ERGODIC) GF(3): The model structure IS the GF(3) triple: Cof(+1), W(0), Fib(-1)

Integration with GF(3) Triads

operadic-composition (+1) x model-categories (0) x derived-categories (-1) = 0
stochastic-resonance (+1) x model-categories (0) x sheaf-cohomology-bci (-1) = 0

CT lattice atlas

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