plurigrid Infinity Topoi Skill: Higher Topos Theory for BCI
Status: Production Ready Trit: -1 (MINUS - validator) Color: #D84026 (Vermillion) Principle: ∞-topoi provide the logical framework where all cohomology lives and HoTT is the internal language Frame: Sh_∞(BCI-Site) as ∞-topos of ∞-sheaves on BCI observation site
Overview
Infinity Topoi provide the logical framework for the entire BCI pipeline. An ∞-topos is an ∞-category satisfying Giraud's axioms, whose internal language is Homotopy Type Theory (HoTT). Implements:
- Grothendieck sites: BCI observation site with covering families
- Presheaves: Fun(C^op, Spaces) - assign signal spaces to observations
- ∞-Sheaf condition: Descent via Cech nerve F(X) ~ lim F(U_bullet)
- Sheafification: L: PSh(C) -> Sh_∞(C) forcing descent
- Giraud axioms: G1 (universal colimits), G2 (disjoint coproducts), G3 (effective groupoids), G4 (generators)
- Postnikov towers: τ_{-1} through τ_n truncations with π_k homotopy groups
- Object classifier: Universe U classifying all ∞-sheaves
- Modalities: ○ (shape), ♭ (flat), # (sharp) with adjunction ♭ ⊣ Γ ⊣ #
- ∞-Topos cohomology: H^n(X;A) = π_0 Map(X, B^n A) unifying all cohomology
- HoTT connection: Types = objects, identity types = path spaces, univalence = object classifier
Correct by construction: GF(3) triadic structure maps to (generator, descent, truncation).
Core Formulae
∞-Sheaf condition (descent):
F(X) → lim_{[n]∈Δ} F(U_{i_0} ×_X ... ×_X U_{i_n}) is an equivalence
Giraud axioms for ∞-topos E:
G1: Colimits are universal (stable under pullback)
G2: Coproducts are disjoint
G3: Groupoid objects are effective
G4: E has a set of generators
Postnikov tower:
X → ... → τ_2(X) → τ_1(X) → τ_0(X) → τ_{-1}(X)
π_n(X) = fiber of τ_n(X) → τ_{n-1}(X)
Object classifier U:
Map(X, U) ≃ {Y → X : Y relatively κ-compact}
Modality adjunctions:
♭ ⊣ Γ ⊣ # (flat ⊣ global sections ⊣ sharp)
♭X = discrete X, #X = codiscrete X
Cohomology:
H^n(X; A) = π_0 Map_E(X, B^n A)
Key Results
BCI Grothendieck Site:
7 objects (3 sensors, fused, 3 worlds)
4 covering families (sensor fusion + world projections)
Presheaves: voltage, frequency, coherence
Descent verified via Cech nerve (gap < 0.15 = SHEAF)
Sheafification L reduces descent gap to < 0.003
Giraud Axioms:
G1: Universal colimits (3/4 coverings)
G2: Disjoint coproducts (all sensor pairs)
G3: Effective groupoids (by construction)
G4: 7 generators
Postnikov Towers:
world-b: π_0=1, π_1=0, π_2=0 (contractible = HoTT isContr)
world-a: π_0=3, π_1~1, π_2~1 (1-type)
world-c: π_0=3, π_1~2, π_2~2 (2-type)
Modalities:
○ (shape): collapses to connected components
♭ (flat): forgets paths, produces discrete set
# (sharp): adds all paths, codiscrete
Object Classifier U:
Total dimension: 50
Classifies: 3 presheaves over 7 site objects
BCI Integration (Layer 24)
Extends the Higher Algebra Chain: L14 → L19 → L20 → L21 → L22 → L23 → L24
- L23 ∞-Categories: ∞-topoi are ∞-categories satisfying Giraud axioms
- L22 Model Categories: Left Bousfield localization presents ∞-topoi
- L19 Sheaf Cohomology: Sheaves = τ_0 of ∞-sheaves; H^n unified in ∞-topos
- L14 Cohomology Ring: Cup product = composition of classifying maps
- L8 Persistent Homology: Persistence as ∞-sheaf on (R,≤) site
- L7 Active Inference: Bayesian inference = conditioning in probability ∞-topos
L24 is the LOGICAL FRAMEWORK: HoTT provides the type theory for all layers.
Skill Name: infinity-topoi Type: ∞-Sheaves / Descent / Giraud Axioms / Modalities / HoTT Trit: -1 (MINUS) GF(3): (+1) ∞-sheaf gen + (0) descent coord + (-1) truncation valid = 0
Integration with GF(3) Triads
infinity-categories (+1) x model-categories (0) x infinity-topoi (-1) = 0
stochastic-resonance (+1) x information-geometry (0) x infinity-topoi (-1) = 0