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infinity-topoi skill

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This skill helps you apply higher topos theory concepts to BCIs, enabling ∞-topos, HoTT, and descent reasoning for robust data interpretation.

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---
name: infinity-topoi
description: Higher topos theory via ∞-sheaves, descent, Giraud axioms, modalities, Postnikov towers, object classifiers, and HoTT connection
version: 1.0.0
---

# 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:

1. **Grothendieck sites**: BCI observation site with covering families
2. **Presheaves**: Fun(C^op, Spaces) - assign signal spaces to observations
3. **∞-Sheaf condition**: Descent via Cech nerve F(X) ~ lim F(U_bullet)
4. **Sheafification**: L: PSh(C) -> Sh_∞(C) forcing descent
5. **Giraud axioms**: G1 (universal colimits), G2 (disjoint coproducts), G3 (effective groupoids), G4 (generators)
6. **Postnikov towers**: τ_{-1} through τ_n truncations with π_k homotopy groups
7. **Object classifier**: Universe U classifying all ∞-sheaves
8. **Modalities**: ○ (shape), ♭ (flat), # (sharp) with adjunction ♭ ⊣ Γ ⊣ #
9. **∞-Topos cohomology**: H^n(X;A) = π_0 Map(X, B^n A) unifying all cohomology
10. **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
```

Overview

This skill implements higher topos theory for the BCI pipeline using ∞-sheaves, descent, and the internal language of Homotopy Type Theory (HoTT). It provides machinery for sheafification, Postnikov truncations, object classification, modalities, and a unified notion of cohomology inside an ∞-topos. The goal is a correct-by-construction logical framework that organizes observations, sensor fusion, and multi-layer inference.

How this skill works

The skill builds an ∞-topos from a Grothendieck observation site, forming presheaves valued in spaces and imposing the ∞-sheaf (descent) condition via Cech nerves. Sheafification enforces descent, while Giraud axioms guarantee the topos structure. It exposes Postnikov towers, object classifier universes, and three modalities (shape, flat, sharp) and interprets cohomology as mapping into iterated classifying objects B^n A.

When to use it

Model multi-sensor observation systems where locality and glueing matter
Formalize inference and learning layers with HoTT as the internal language
Compute sheaf cohomology and classify parameterized families of observations
Apply truncation or Postnikov approximations to manage higher homotopy information
Define and use modalities (shape/flat/sharp) to control path information and discretization

Best practices

Define a clear Grothendieck site of observations with explicit covering families before building presheaves
Verify descent numerically or symbolically via Cech nerves and apply sheafification to fix gaps
Use Postnikov truncations to simplify objects when only low-degree homotopy matters
Leverage the object classifier to parametrize and manage families of relative objects uniformly
Prefer modality adjunctions to transition between discrete, homotopical, and codiscrete viewpoints

Example use cases

BCI sensor fusion: model sensor observations as presheaves and enforce fusion via covering families and descent
Cohomology-based diagnostics: compute H^n(X;A) as π_0 Map(X,B^n A) to detect global obstructions
Type-theoretic specifications: express system invariants in HoTT and use univalence via the object classifier
Layered inference: map logical layers (L14..L24) into the ∞-topos to unify persistent homology, cohomology rings, and Bayesian conditioning
Dimensionality control: apply Postnikov towers to reduce a complex world-object to a 1- or 0-type for downstream algorithms

FAQ

What does sheafification fix in practice?

Sheafification forces the descent condition so presheaves that fail glueing become genuine ∞-sheaves; in practice it closes small descent gaps and restores consistency across covers.

How do modalities help in applications?

Modalities let you move between discrete data (♭), full homotopical information (#), and shape/collapse (○), enabling controlled abstraction, discretization, or path completion depending on the task.