Category Theory Structure Rank Skill 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

| 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

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.

Agent Skills: Category Theory Structure Rank Skill

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