Cat# Three Homes Skill
"All Concepts are Cat#" — Spivak (ACT 2023)
Trit: 0 (ERGODIC)
Color: #49EE54
Source: Spivak, Lynch, Shapiro - "All concepts are Cat#" ACT 2023
Core Definition
Cat# = Comod(Poly, y, ◁)
The double category of polynomial comonads where:
- Poly = free completely distributive category on one object
- y = identity polynomial
- ◁ = composition (substitution) of polynomials
The Three Homes
Home 1: Polynomial Comonads (Objects of Cat#)
Categories ARE the objects of Cat#.
A category C becomes polynomial: Σ_{A:Ob(C)} y^{C[A]}
where C[A] = Σ_{B:Ob(C)} C(A,B) = "maps out of A"
• Counit ε: c → y supplies identities
• Comult δ: c → c◁c supplies codomains and composition
Skill mapping: gay-mcp (+1) — inject deterministic state
Home 2: Monads in Span (Linear restriction)
Comod(Set, 1, ×) ≅ Span
Mod(Span) ≅ Prof(Cat)
Linear polynomials only: c = Cy (just a set of objects)
Bicomodules: Cy ◁──Py──▷ Dy (spans of sets)
Categories = monads in Span
Skill mapping: acsets (0) — schema as span
Home 3: Path Algebras (Most familiar)
Graph category G = (• ⇉ •) with polynomial g = y³ + y
g-Set ≅ Grph (category of graphs)
path: g◁ ──→ ◁g is a monad (prafunctor Grph → Grph)
Categories = path-algebras = path-complete graphs
Skill mapping: bisimulation-game (-1) — validate path equivalence
GF(3) Triad
bisimulation-game (-1) ⊗ cat-three-homes (0) ⊗ gay-mcp (+1) = 0 ✓
| Trit | Home | Skill | Role | |------|------|-------|------| | -1 | Path Algebras | bisimulation-game | Validate equivalences | | 0 | Span/Prof | acsets, cat-three-homes | Schema bridge | | +1 | Poly Comonads | gay-mcp | State injection |
Key Structures
Bicomodules (Horizontal morphisms in Cat#)
c ◁ p ◁ d with maps satisfying laws w.r.t. ε, δ
These are precisely prafunctors d-Set → c-Set (data migrations).
The Mod Construction
If D has nice local coequalizers → Mod(D)
If P has nice local equalizers → Comod(P)
Poly has ◁-preserved local equalizers: e → p ⟹ q
So we can form Comod(Poly) = Cat#
Org (Dynamic Arrangements)
Org ↪ Cat# (fully faithful)
Objects: p : Poly → cofree comonoid 𝔠_p
Horizontals: [p,q]-coalgebras (dynamic arrangements)
Models neural networks, prediction markets, rewiring diagrams.
Multivariate Extension
For any category E with pullbacks:
Poly_E embeds into Cat#
by sending I : E to slice category A/I
Discrete categories in Cat# ≅ multivariate polynomials in Set.
Commands
# Query homes
bb cat-three-homes.bb --home 1 # Polynomial comonads
bb cat-three-homes.bb --home 2 # Span/Prof
bb cat-three-homes.bb --home 3 # Path algebras
# Dispatch concept to home
bb cat-three-homes.bb --dispatch "functor"
# Show all triads
bb cat-three-homes.bb --triads
DuckDB Schema
CREATE TABLE cat_homes (
home_id INT PRIMARY KEY,
name VARCHAR,
structure VARCHAR,
skill VARCHAR,
trit TINYINT
);
INSERT INTO cat_homes VALUES
(1, 'Polynomial Comonads', 'Comod(Poly,y,◁)', 'gay-mcp', 1),
(2, 'Monads in Span', 'Mod(Span)≅Prof', 'acsets', 0),
(3, 'Path Algebras', 'path-complete graphs', 'bisimulation-game', -1);
Related Skills
| Skill | Trit | Relation |
|-------|------|----------|
| catsharp | 0 | Core Cat# skill |
| kan-extensions | 0 | Universal property source |
| infinity-operads | +1 | Higher categorical extension |
| topos-catcolab | 0 | CatColab double theories |
| polynomial-functors | 0 | Poly foundation |
References
- Spivak, Lynch, Shapiro - "All concepts are Cat#" (ACT 2023)
- Niu & Spivak - "Polynomial Functors" (Cambridge 2024)
- Shulman - "Framed Bicategories and Monoidal Fibrations"
- Ahman-Uustalu - Polynomial comonads are categories
- Garner - Prafunctors (HoTTEST video)
2-Torial Integration
Concepts from Topos 2-torials map to homes:
| 2-Torial | Concept | Home | |----------|---------|------| | Doctrinal Adjunctions | doctrine, lax | Home 2 (Span) | | Instances of Models | double category, model | Home 1 (Poly) | | Coalgebraic-Modal | coalgebra, monad | Home 3 (Path) |
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.