An illustrated journeyfrom first principles
How mathematics learned to count holes — and to measure what cannot be done. A complete course for the curious, with every prerequisite taught along the way.
Mathematics is the art of giving the same name to different things.
A loop drawn on a shape either encloses a region of that shape — it is a boundary — or it goes around something missing. Homology turns this into algebra: it collects all the loops (and their higher-dimensional cousins), declares the boundaries to be zero, and counts what is left. What is left are the holes.
Now attach a measurement to every step — a height difference, a voltage, an exchange rate. Each step can be perfectly sensible on its own, yet going all the way around can leave you higher than you started, like Escher’s impossible staircase. Cohomology measures exactly these obstructions: local data that cannot be made globally consistent.
Rotate tori, glue squares into Klein bottles, click edges to build cycles, scrub numbers in the text, and watch homology groups computed live as you edit a shape.
Sets, groups, linear algebra, topology, manifolds, differential forms and categories are all taught here, slowly and from the beginning, exactly as far as the destination needs.
Intuition first, but never at the price of truth: precise definitions, real proofs, worked examples, exercises with full solutions, and a glossary that is one hover away.
The road ahead
Each part builds on the last. Read straight through, or follow the map to the ideas you need.
Why holes? And how to read the language they are written in.
The language of structure: sets, functions, groups, and linear algebra.
Topology: what survives when you stretch, bend and squeeze.
Measuring obstructions: from local to global.
Categories, homological algebra, and where the road leads.
Start with the first questionThe measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.