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

-- 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

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

# 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.