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

1. Spivak, Lynch, Shapiro - "All concepts are Cat#" (ACT 2023)
2. Niu & Spivak - "Polynomial Functors" (Cambridge 2024)
3. Shulman - "Framed Bicategories and Monoidal Fibrations"
4. Ahman-Uustalu - Polynomial comonads are categories
5. 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)

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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.