Basic Category Theory
by Tom Leinster
Cambridge textbook released free by the author, organised around universal properties and the three ways of expressing them: adjoint functors, representable functors, and limits. Assumes only linear algebra and group theory, and explains the Yoneda lemma at unusual length.
More resources on Category Theory
Category Theory in Context
Modern introduction to category theory covering functors, natural transformations, limits, adjunctions, and categorical constructions with examples.
Function (nLab)
Wiki entry defining a function as a rule sending each domain element to a unique codomain element, then comparing set-theoretic, type-theoretic and category-theoretic formulations, total versus partial functions, and function sets. Readers see how the elementary notion generalises to morphisms in arbitrary categories.
Category Theory for Scientists (MIT 18.S996)
An open textbook presenting category theory as a language for scientific modeling: sets, monoids, graphs, orders, databases and ologs, categories, functors, natural transformations, limits, colimits and adjunctions. Includes instructor insights and example student projects; suits readers comfortable with proofs.
Category Theory
Recorded lecture series by Bartosz Milewski introducing category theory for programmers: categories, morphisms, initial and terminal objects, products and coproducts, functors, exponentials, natural transformations and monads, illustrated with Haskell. Viewers learn to connect categorical constructions to type systems and functional programming.
Applied Category Theory (MIT 18.S097)
Seven sketches pair applications such as databases, collaborative design, signal flow graphs, electrical circuits and resource theories with categorical ideas including Galois connections, monoidal and enriched categories, operads and toposes. Provides the complete open textbook and problem sets, assuming no prior category theory.
Category Theory for Programmers — Video Lecture Series
Milewski teaching his own text to a live Seattle audience, whiteboard-first. Covers composition, types as sets, universal constructions, functors, natural transformations and the Yoneda lemma, with the questions and false starts a written book removes.