Course Materials

Welcome to Categorical Foundations of Machine Learning. This graduate course builds the mathematical language — category theory, algebraic topology, and statistical mechanics — that transforms machine learning from empirical fitting into structural reasoning. Each module includes lecture notes, a glossary of key terms, self-check questions, working code examples, and primary literature assignments.

Modules

Module 1

Why Structure Beats Statistics

Extrapolation failure, two paradigms, manifold hypothesis, morphisms as inductive bias

Module 2

Categories: The Language of Structure

Objects, morphisms, composition, identity laws, Set/Graph/Poset/Vect, databases as categories

Module 3

Functors and Natural Transformations

Structure-preserving maps, functor laws, natural transformations, naturality square, functor categories

Module 4

The Yoneda Lemma

Representable functors, hom-functor, Yoneda lemma and proof, Yoneda embedding, specification by interface

Module 5

Universal Constructions I: Limits

Diagrams, cones, limits, products, pullbacks, equalizers, SQL joins as pullbacks, type unification

Module 6

Universal Constructions II: Colimits and Adjunctions

Cocones, colimits, coproducts, pushouts, adjunctions, unit and counit, free/forgetful pairs

Module 7

The Adjoint Triple and Data Migration

Schema morphisms, base change Δ, left pushforward Σ, right pushforward Π, delta lenses, Catlab.jl lab

Module 8

Kan Extensions

Universal extension of functors, pointwise colimit formula, left and right Kan, EBMs as approximate Kan extensions

Module 9

Coends and the Integration Calculus

Ends and coends, wedge condition, Fubini theorem, tensor contraction, marginalization, belief propagation, partition functions

Module 10

Algebraic Topology: Invariants as Hard Constraints

Homotopy, fundamental group π₁, singular homology, persistent homology, barcodes, protein knots, Burgers vectors

Module 11

Sheaves and Local-to-Global Reasoning

Presheaves, sheaf condition, Čech cohomology, H¹ as obstruction, spin glass connection, symmetric lenses

Module 12

Statistical Mechanics and the Renormalization Group

Ising model, partition function, phase transitions, renormalization group as functor, fixed points, training as phase transition

Module 13

Categorical Machine Learning: Synthesis

Geometric deep learning, equivariance by construction, JEPA as approximate RG, sheaf resolution of Z, research presentations

Resources

Guide

Categorical Foundations of Machine Learning

Course Portal