Asi cat-structure-rank

Category Theory Structure Rank Skill

install

source · Clone the upstream repo

git clone https://github.com/plurigrid/asi

Claude Code · Install into ~/.claude/skills/

T=$(mktemp -d) && git clone --depth=1 https://github.com/plurigrid/asi "$T" && mkdir -p ~/.claude/skills && cp -r "$T/skills/cat-structure-rank" ~/.claude/skills/plurigrid-asi-cat-structure-rank && rm -rf "$T"

manifest: skills/cat-structure-rank/SKILL.md

source content

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.