ideasthesia

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Ryan Collison

I write about cool science and share it here.

Sheaf over topological space—local data assembling into global meaning

Sheaves and Contextuality: How Category Theory Models Context-Dependent Meaning

Series: Applied Category Theory | Part: 7 of 10 The word “bank” has no fixed meaning. Whether it refers to a financial institution or a riverbank depends entirely on context. Your brain handles this effortlessly—seamlessly integrating surrounding words, conversational history, and situational cues to disambiguate meaning. But how does meaning

String diagram with flowing wires and boxes—visual categorical calculus

String Diagrams: Drawing Your Way to Mathematical Insight

Series: Applied Category Theory | Part: 6 of 10 The mathematician draws a picture. Not a decoration, not a visual aid—a proof. Lines flow, boxes connect, wires cross and merge. What looks like a circuit diagram or a subway map is actually doing mathematics. Real mathematics. The kind that lands

Neural network architecture as functor diagram—categorical deep learning

Neural Networks as Functors: Category Theory Meets Deep Learning

Series: Applied Category Theory | Part: 5 of 10 In 2014, a paper by Brendan Fong, David Spivak, and Rémy Tuyéras quietly appeared on arXiv with an audacious claim: neural networks aren’t just computational graphs. They’re functors—structure-preserving maps between categories. Most of the deep learning community ignored

Natural transformation connecting two functors—morphisms between translations

Natural Transformations: When Translations Talk to Each Other

Series: Applied Category Theory | Part: 4 of 10 You have two translation apps on your phone. Both translate English to French. Both work. But here’s the question that breaks most people’s brains: what does it mean for one translation to be “better” than another? Not more accurate—that’

Functor mapping between two category diagrams—structure-preserving translation

Functors: The Maps Between Mathematical Worlds

Series: Applied Category Theory | Part: 3 of 10 You’ve learned that a category is a collection of objects and arrows between them. You’ve seen how morphisms preserve structure, how composition works, and why identity matters. But here’s where category theory gets interesting: what happens when you want

Abstract objects connected by morphisms—structure over substance

Objects, Morphisms, and the Art of Not Caring What Things Are

Series: Applied Category Theory | Part: 2 of 10 In 1945, two mathematicians independently realized they’d been working on the same problem in completely different mathematical worlds. Samuel Eilenberg was studying algebraic topology. Saunders Mac Lane was working in algebra. They discovered something unsettling: the structure of their problems was

Category theory diagrams spreading across disciplines—mathematics of composition

The New Linear Algebra: Why Applied Category Theory Is Eating Everything

Series: Applied Category Theory | Part: 1 of 10 In 1949, Claude Shannon published a paper that changed everything. “A Mathematical Theory of Communication” didn’t just solve the problem of reliable signal transmission—it revealed that information was a thing you could measure, a substance with its own geometry. Within