cats.for" (Categories for AI)

"Category theory is compositionality made formal" — Bruno Gavranović, cats.for.ai

URL: https://cats.for.ai Trit: 0 (ERGODIC) Color: #26D826 (Green) Status: ✅ Production Ready

Overview

cats.for.ai is the definitive lecture series connecting category theory to machine learning, organized by DeepMind, Qualcomm AI, and academic researchers.

┌─────────────────────────────────────────────────────────────────────────────┐
│                         cats.for.ai INTEGRATION                             │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                             │
│   POLY INACCESSIBLE WORLDS AS COLORS MAXIMAL                               │
│                                                                             │
│   ┌───────────────┐     ┌───────────────┐     ┌───────────────┐            │
│   │   Poly(y)     │     │  Inaccessible │     │    Colors     │            │
│   │   Functors    │────▶│    Worlds     │────▶│    Maximal    │            │
│   │    (-1)       │     │     (0)       │     │     (+1)      │            │
│   └───────────────┘     └───────────────┘     └───────────────┘            │
│         │                      │                      │                     │
│         │    Optics/Lenses     │   Modal Semantics    │   GF(3) Coloring   │
│         │                      │                      │                     │
│         └──────────────────────┴──────────────────────┘                     │
│                                 │                                           │
│                    GF(3): (-1) + 0 + (+1) ≡ 0                              │
│                                                                             │
└─────────────────────────────────────────────────────────────────────────────┘

Organizing Committee

Name	Affiliation	Role
Bruno Gavranović	Strathclyde / CyberCat	Optics, Lenses, Backprop
Petar Veličković	DeepMind / Cambridge	GNNs, Categories
Pim de Haan	Amsterdam / Qualcomm	Geometric DL, Equivariance
Andrew Dudzik	DeepMind	Monads, RNNs
João G. Araújo	Cohere / USP	LLMs, Semantics

Guest Speakers

Speaker	Topic	Trit
David Spivak	Poly, Dynamic Systems	0 (ERGODIC)
Jules Hedges	Categorical Cybernetics	0 (ERGODIC)
Tai-Danae Bradley	CT for LLMs	+1 (PLUS)
Taco Cohen	Causal Abstraction	-1 (MINUS)
Pietro Vertechi	Parametric Spans	0 (ERGODIC)
Thomas Gebhart	Sheaves for AI	-1 (MINUS)

Lecture Program

Part 1: Introductory Lectures (Completed)

Week	Topic	Speaker	Key Concepts
1	Why Category Theory?	Gavranović	Compositionality, Modularity
2	Categories & Functors	Veličković	Objects, Morphisms, Functors
3	Optics & Lenses	Gavranović	Bidirectional flow, Backprop
4	Geometric DL & Naturality	de Haan	Equivariance, Natural transformations
5	Monoids, Monads, LSTMs	Dudzik	Recurrence, State

Part 2: Research Seminars (Ongoing)

Date	Topic	Speaker
Nov 14	Neural network layers as parametric spans	Vertechi
Nov 21	Causal Model Abstraction	Cohen
Dec 12	Category Theory Inspired by LLMs	Bradley
Mar 20	Categorical Cybernetics	Hedges
Mar 27	Dynamic organizational systems	Spivak
May 29	Sheaves for AI	Gebhart

Core Concepts

1. Polynomial Functors (Py)

-- Poly = polynomial functors on Set
-- y = representable functor Hom(1, -)
-- ◁ = composition of polynomials

data Poly where
  Poly :: { positions :: Type
          , directions :: positions -> Type
          } -> Poly

-- Key operations
(⊗) :: Poly -> Poly -> Poly  -- parallel (tensor)
(◁) :: Poly -> Poly -> Poly  -- sequential (composition)
(+) :: Poly -> Poly -> Poly  -- coproduct

2. Inaccessible Worlds (Modal Semantics)

-- Kripke frame with inaccessibility
record KripkeFrame : Set₁ where
  field
    World : Set
    _≺_ : World → World → Set  -- accessibility
    inaccessible : World → Set  -- worlds with no predecessors

-- Modal operators
□ : (World → Prop) → (World → Prop)  -- necessity
□ φ w = ∀ v → w ≺ v → φ v

◇ : (World → Prop) → (World → Prop)  -- possibility
◇ φ w = ∃ v → w ≺ v × φ v

-- Inaccessible world: ¬∃v. v ≺ w
isInaccessible : World → Prop
isInaccessible w = ∀ v → ¬(v ≺ w)

3. Colors Maximal (GF(3) Saturation)

(ns cats4ai.colors
  (:require [gay.core :as gay]))

(defn color-world [world-seed trit]
  "Assign maximal color to world based on trit"
  (let [hue (case trit
              :MINUS   (+ 180 (mod (* world-seed 37) 120))  ; cold
              :ERGODIC (+ 60 (mod (* world-seed 41) 120))   ; neutral
              :PLUS    (mod (* world-seed 43) 60))          ; warm
        saturation 1.0  ; MAXIMAL
        lightness 0.5]
    {:hue hue :sat saturation :lit lightness
     :hex (gay/hsl->hex hue saturation lightness)}))

(defn worlds-palette [n]
  "Generate n inaccessible worlds with maximal colors"
  (for [i (range n)
        :let [trit (case (mod i 3) 0 :ERGODIC 1 :PLUS 2 :MINUS)]]
    {:world-id i
     :accessible? false  ; inaccessible
     :color (color-world i trit)
     :trit trit}))

Optics for Machine Learning

Lens = Backpropagation

-- A lens captures forward/backward pass
data Lens s t a b = Lens
  { view :: s -> a        -- forward pass
  , update :: s -> b -> t  -- backward pass (gradient)
  }

-- Backprop is lens composition
backprop :: Lens s t a b -> Lens a b c d -> Lens s t c d
backprop l1 l2 = Lens
  { view = view l2 . view l1
  , update = \s d ->
      let a = view l1 s
          b = update l2 a d
      in update l1 s b
  }

-- Chain rule emerges from lens composition!
-- d(g∘f)/dx = dg/df · df/dx

Para = Parametric Morphisms

-- Parametric morphism (neural network layer)
data Para p a b = Para
  { params :: p           -- learnable parameters
  , forward :: p -> a -> b  -- parameterized forward
  }

-- Para composition
(>>>) :: Para p a b -> Para q b c -> Para (p, q) a c
(Para p f) >>> (Para q g) = Para (p, q) (\(p,q) a -> g q (f p a))

Optic = Generalized Bidirectional

-- Optic generalizes lens, prism, traversal
type Optic p s t a b = p a b -> p s t

-- Specific optics for ML:
type Lens s t a b = Optic (Star ((->) s)) s t a b      -- deterministic
type Prism s t a b = Optic (Costar (Either a)) s t a b -- optional
type Affine s t a b = Optic (Star Maybe) s t a b       -- at most one

Integration Map

With Existing Skills

┌─────────────────────────────────────────────────────────────────────────────┐
│                      cats.for.ai SKILL INTEGRATION                          │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                             │
│  cats.for.ai                                                                │
│       │                                                                     │
│       ├──▶ open-games (Hedges)                                             │
│       │         └── Play/Coplay bidirectional                               │
│       │                                                                     │
│       ├──▶ catsharp (Spivak)                                               │
│       │         └── Cat# = Comod(P) equipment                              │
│       │                                                                     │
│       ├──▶ topos-catcolab (Patterson)                                      │
│       │         └── Double theories, ACSets                                 │
│       │                                                                     │
│       ├──▶ cybernetic-open-game                                            │
│       │         └── Agent-O-Rama ↔ Worldnet ↔ STC                          │
│       │                                                                     │
│       ├──▶ elements-infinity-cats (Riehl-Verity)                           │
│       │         └── ∞-cosmos model independence                             │
│       │                                                                     │
│       ├──▶ sheaf-laplacian-coordination (Gebhart)                          │
│       │         └── Sheaf neural networks                                   │
│       │                                                                     │
│       ├──▶ crdt (Automerge)                                                │
│       │         └── Join-semilattice merge                                  │
│       │                                                                     │
│       └──▶ derangement-crdt                                                │
│                 └── Colorable permutations                                  │
│                                                                             │
└─────────────────────────────────────────────────────────────────────────────┘

GF(3) Triads

# Core cats.for.ai triads
optics-lenses (-1) ⊗ cats-for-ai (0) ⊗ backprop (+1) = 0 ✓
sheaves (-1) ⊗ cats-for-ai (0) ⊗ gnns (+1) = 0 ✓
cybernetics (-1) ⊗ cats-for-ai (0) ⊗ open-games (+1) = 0 ✓
poly-functors (-1) ⊗ cats-for-ai (0) ⊗ para-morphisms (+1) = 0 ✓
causal-abstraction (-1) ⊗ cats-for-ai (0) ⊗ llm-semantics (+1) = 0 ✓

Inaccessible Worlds Protocol

For each inaccessible world (no predecessor in accessibility relation):

class InaccessibleWorld:
    """A world with no accessible predecessors - epistemic ground truth"""

    def __init__(self, seed: int, trit: int):
        self.seed = seed
        self.trit = trit  # -1, 0, +1
        self.color = self._maximal_color()
        self.accessible_from = set()  # empty = inaccessible

    def _maximal_color(self) -> dict:
        """Assign maximally saturated color based on trit"""
        hue_base = {-1: 240, 0: 120, 1: 0}[self.trit]  # blue/green/red
        hue = (hue_base + (self.seed * 37) % 60) % 360
        return {"h": hue, "s": 1.0, "l": 0.5}  # MAXIMAL saturation

    def poly_action(self, p: "Poly") -> "InaccessibleWorld":
        """Apply polynomial functor action"""
        new_seed = (self.seed * p.positions + sum(p.directions)) % (2**64)
        new_trit = (self.trit + p.trit) % 3
        return InaccessibleWorld(new_seed, new_trit)

YouTube Resources

All lectures available: YouTube Playlist

Lecture	Views	Key Insight
Why Category Theory?	24K+	Compositionality = modularity
Optics and Lenses	15K+	Chain rule = lens composition
Categorical Cybernetics	10K+	Agency via parametrised optics
Sheaves for AI	8K+	Local-to-global inference

Commands

# Watch lecture
just cats4ai-watch LECTURE_NUM

# Run optics demo
just cats4ai-optics-demo

# Generate inaccessible worlds
just cats4ai-worlds N

# Color palette
just cats4ai-colors --maximal

# Integration check
just cats4ai-integrate SKILL_NAME

Babashka Integration

#!/usr/bin/env bb
;; cats4ai.bb - Categories for AI utilities

(ns cats4ai
  (:require [babashka.http-client :as http]
            [cheshire.core :as json]))

(def LECTURES
  [{:week 1 :title "Why Category Theory?" :speaker "Gavranović" :trit 0}
   {:week 2 :title "Categories & Functors" :speaker "Veličković" :trit 0}
   {:week 3 :title "Optics & Lenses" :speaker "Gavranović" :trit -1}
   {:week 4 :title "Geometric DL" :speaker "de Haan" :trit +1}
   {:week 5 :title "Monads & LSTMs" :speaker "Dudzik" :trit 0}])

(defn gf3-conservation? [lectures]
  (zero? (mod (reduce + (map :trit lectures)) 3)))

(defn inaccessible-world [seed]
  {:seed seed
   :trit (case (mod seed 3) 0 :ERGODIC 1 :PLUS 2 :MINUS)
   :accessible-from #{}
   :color {:h (mod (* seed 37) 360) :s 1.0 :l 0.5}})

(defn -main [& args]
  (println "cats.for.ai - Categories for AI")
  (println "GF(3) conserved?" (gf3-conservation? LECTURES))
  (println "Inaccessible worlds:"
           (map inaccessible-world (range 3))))

(when (= *file* (System/getProperty "babashka.file"))
  (apply -main *command-line-args*))

References

Primary

• cats.for.ai - Official site
• YouTube Playlist
• Zulip Chat

Papers

• Gavranović et al. "Categorical Foundations of Gradient-Based Learning" (2024)
• Hedges "Compositional Game Theory" (2016)
• Spivak "Poly: An abundant categorical setting" (2020)
• Bradley "Language Models as Semantic Functors" (2023)
• Gebhart "Sheaf Neural Networks" (2022)

Related Skills

• open-games - Hedges' compositional games
• catsharp - Spivak's Cat# = Comod(P)
• topos-catcolab - Topos collaborative CT
• cybernetic-open-game - Cybernetic feedback
• sheaf-laplacian-coordination - Gebhart sheaves
• crdt - Join-semilattice CRDTs
• derangement-crdt - Colorable derangements

---

Skill Name: cats-for-ai Type: Categorical Machine Learning Trit: 0 (ERGODIC) Poly: Inaccessible worlds as colors maximal GF(3): Conserved across all triads