Category Theory Structure Rank Skill

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

Level 1: Basic Categories

Level 2: Enriched & Internal

Level 3: Higher Structures

Level 4: Homotopical

Level 5: ∞-Topoi

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

Query: Concept Graph

-- 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

-- 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

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

Intercept Points (SNPs = Skill Navigation Points)

High-Value Intercepts

Unclear/​Disputed Areas

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

Commands

# 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
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.