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 gamescatsharp- Spivak's Cat# = Comod(P)topos-catcolab- Topos collaborative CTcybernetic-open-game- Cybernetic feedbacksheaf-laplacian-coordination- Gebhart sheavescrdt- Join-semilattice CRDTsderangement-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