Cat Structure Rank

---
name: cat-structure-rank
description: Ranked taxonomy of categorical structures from sets to ∞-topoi with gap analysis and DuckDB integration.
---
# Category Theory Structure Rank Skill

Ranked taxonomy of categorical structures from sets to ∞-topoi with gap analysis and DuckDB integration.

**Trit: 0 (ERGODIC)** — Coordinator mapping structures to skills

## Structure Hierarchy

### Level 0: Foundations
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| Sets | (implicit) | ✓ | — |
| Functions | (implicit) | ✓ | — |
| Relations | `acsets-relational-thinking` | ✓ | — |

### Level 1: Basic Categories
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| Categories | `ctp-yoneda` | ✓ | — |
| Functors | `naturality-factor` | ✓ | — |
| Natural Transformations | `naturality-factor` | ✓ | — |
| Yoneda Lemma | `yoneda-directed` | ✓ | — |

### Level 2: Enriched & Internal
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| Enriched Categories | `elements-infinity-cats` | ◐ | Need dedicated skill |
| Monads | `free-monad-gen` | ✓ | — |
| Adjunctions | `galois-connections` | ✓ | — |
| Elementary Topoi | `effective-topos` | ✓ | — |

### Level 3: Higher Structures
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| Double Categories | `cat-tripartite` | ◐ | Partial in CatColab |
| Bicategories | — | ✗ | **MAJOR GAP** |
| Operads | `operad-compose` | ✓ | — |
| Polynomial Functors | `asi-polynomial-operads` | ◐ | Spivak lectures incomplete |
| Actegories | `unwiring-arena` | ◐ | Implicit only |

### Level 4: Homotopical
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| Quasi-categories | `elements-infinity-cats` | ✓ | — |
| Segal Spaces | `segal-types` | ✓ | — |
| Complete Segal Spaces | `rezk-types` | ✓ | — |
| ∞-Cosmos | `elements-infinity-cats` | ✓ | — |
| Dendroidal Sets | `infinity-operads` | ◐ | Need formalization |

### Level 5: ∞-Topoi
| Structure | Skill | Coverage | Gap |
|-----------|-------|----------|-----|
| ∞-Categories | `infinity-topos` | ✓ | — |
| ∞-Topoi | `infinity-topos` | ✓ | — |
| Higher Topos Theory | `infinity-topos` | ◐ | Lurie's HTT not fully extracted |
| Synthetic ∞-Cat | `riehl-post-rigorous` | ◐ | Rzk formalization ongoing |

## Coverage Legend

```
✓  = Good coverage (skill exists, comprehensive)
◐  = Partial coverage (skill exists, gaps remain)
✗  = No coverage (major gap, needs skill)
```

## Major Gaps Identified

### 1. Bicategories (Level 3)
**Status**: ✗ No dedicated skill
**Impact**: High — bridges double categories and ∞-cosmos
**Remedy**: Create `bicategory` skill from:
- Street's "Fibrations in Bicategories"
- Bénabou's original work
- Link to `cat-tripartite` and `topos-catcolab`

### 2. Polynomial Functors (Level 3)
**Status**: ◐ Partial in `asi-polynomial-operads`
**Impact**: High — Spivak's key contribution
**DuckDB**: `/Users/bob/ies/spivak_poly.duckdb` (empty lectures table)
**Remedy**: Extract from Spivak lectures, populate DuckDB

### 3. Dendroidal Sets (Level 4)
**Status**: ◐ Mentioned but not formalized
**Impact**: Medium — ∞-operads foundation
**Remedy**: Create dedicated skill with:
- Moerdijk-Weiss construction
- Link to `infinity-operads`

### 4. HTT Extraction (Level 5)
**Status**: ◐ Lurie's Higher Topos Theory not fully extracted
**Impact**: Very High — foundational reference
**Remedy**: MathPix extraction of key chapters

## DuckDB Sources

| Database | Tables | Relevance |
|----------|--------|-----------|
| `hatchery_category.duckdb` | `concept_graph`, `concept_paths` | Concept relationships |
| `spivak_poly.duckdb` | `spivak_poly_lectures` | Polynomial functors |
| `dendroidal.duckdb` | `dendroidal` | Tree structures |
| `mermaid_acset.duckdb` | — | Diagram persistence |
| `hatchery_topology.duckdb` | — | Topological structures |

### Query: Concept Graph

```sql
-- Find category-theoretic concepts
SELECT source_concept, relation, target_concept 
FROM concept_graph 
WHERE source_concept IN ('Actegory', 'Dendroidal Sets', 'Galois Connection')
ORDER BY source_concept;
```

### Query: Structure Dependencies

```sql
-- Build structure dependency graph
WITH RECURSIVE deps AS (
  SELECT source_concept, target_concept, 1 as depth
  FROM concept_graph
  WHERE source_concept = 'Dendroidal Sets'
  UNION ALL
  SELECT g.source_concept, g.target_concept, d.depth + 1
  FROM concept_graph g
  JOIN deps d ON g.source_concept = d.target_concept
  WHERE d.depth < 5
)
SELECT * FROM deps;
```

## GF(3) Trit Assignment by Level

| Level | Trit | Role | Example |
|-------|------|------|---------|
| 0-1 | -1 | Foundation/Constraint | Sets, Categories |
| 2-3 | 0 | Transport/Enrichment | Monads, Operads |
| 4-5 | +1 | Generation/Extension | ∞-Categories, Synthetic |

**Conservation**: Each level transition preserves GF(3) sum.

## Intercept Points (SNPs = Skill Navigation Points)

### High-Value Intercepts

| From | To | Skill Bridge | Value |
|------|-----|--------------|-------|
| Operads → ∞-Operads | `operad-compose` → `infinity-operads` | Dendroidal embedding | ★★★ |
| Topos → ∞-Topos | `effective-topos` → `infinity-topos` | Lurie characterization | ★★★ |
| Adjunction → Kan Extension | `galois-connections` → `kan-extensions` | Universal property | ★★★ |
| Double Cat → ∞-Cosmos | `cat-tripartite` → `elements-infinity-cats` | Formal category theory | ★★ |
| Polynomial → Operad | `asi-polynomial-operads` → `operad-compose` | Spivak construction | ★★ |

### Unclear/Disputed Areas

| Area | Issue | Skills Involved |
|------|-------|-----------------|
| Model independence | When does ∞-cosmos suffice vs concrete model? | `riehl-post-rigorous`, `elements-infinity-cats` |
| Synthetic foundations | HoTT vs Book HoTT vs Rzk | `riehl-post-rigorous`, `segal-types` |
| Operad coloring | GF(3) vs arbitrary coloring for operads | `infinity-operads`, `gay-mcp` |
| Actegory ↔ Module | Terminological confusion | `unwiring-arena`, `operad-compose` |

## Grandis/Grossberg Connections

### Marco Grandis (Directed Algebraic Topology)
- **d-spaces**: Spaces with distinguished directed paths
- **Link**: `directed-interval`, `covariant-fibrations`
- **Gap**: No dedicated Grandis skill

### Steven Grossberg (Adaptive Resonance Theory)
- **ART**: Attractor dynamics in neural networks
- **Link**: `alife`, `attractor`, `equilibrium`
- **Gap**: Need bridge to categorical learning theory

## Skill Creation Priority

| Priority | Gap | Estimated Effort |
|----------|-----|------------------|
| 1 | Bicategories | Medium |
| 2 | Polynomial functors (complete) | High |
| 3 | HTT extraction | Very High |
| 4 | Dendroidal formalization | Medium |
| 5 | Grandis d-spaces | Low |

## Commands

```bash
# Query concept graph
duckdb /Users/bob/ies/hatchery_category.duckdb \
  -c "SELECT * FROM concept_graph WHERE relation = 'uses';"

# Check Spivak lectures
duckdb /Users/bob/ies/spivak_poly.duckdb \
  -c "SELECT COUNT(*) FROM spivak_poly_lectures;"

# Find skill gaps
grep -l "TODO\|FIXME\|GAP" ~/.claude/skills/*/SKILL.md
```

## Related Skills

- `ctp-yoneda` - Basic category theory
- `elements-infinity-cats` - Riehl-Verity ∞-cosmos
- `infinity-topos` - ∞-topos integration
- `operad-compose` - Operadic composition
- `riehl-post-rigorous` - Formalization strategies
- `asi-polynomial-operads` - Polynomial/operad bridge


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## Autopoietic Marginalia

> **The interaction IS the skill improving itself.**

Every use of this skill is an opportunity for worlding:
- **MEMORY** (-1): Record what was learned
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- **WORLDING** (+1): Evolve the skill based on use



*Add Interaction Exemplars here as the skill is used.*