--- title: Categories description: Categories are mathematical structures consisting of objects and arrows that preserve structure. You will understand how to analyze abstract mathematical systems, identify universal properties, and use functors and natural transformations to map relationships between different branches of mathematics. category: mathematics subcategory: category-theory difficulty: beginner, intermediate, advanced url: /subject/categories --- # Categories Categories are mathematical structures consisting of objects and arrows that preserve structure. You will understand how to analyze abstract mathematical systems, identify universal properties, and use functors and natural transformations to map relationships between different branches of mathematics. ## Available Resources 3 Books • 1 Courses • 5 Websites ## Websites ### 1. Kerodon An open, continuously updated online reference for homotopy-coherent mathematics, covering simplicial sets, Kan complexes, and infinity-categories chapter by chapter with tagged, cross-linked proofs readers can cite and follow from foundations through higher category theory. **Difficulty:** Beginner | **Price:** Free **Link:** https://kerodon.net/ **Tags:** higher-category-theory, infinity-categories, simplicial-sets, homotopy-theory, reference ### 2. Category Theory for Programmers Bartosz Milewski's blog series translating categorical ideas into Haskell and C++ terms, with a free companion book. Readers finish able to reason about functors, natural transformations, and monads as everyday programming abstractions rather than abstract algebra. **Difficulty:** Beginner | **Price:** Free **Link:** https://bartoszmilewski.com/2014/10/28/category-theory/ **Tags:** functional-programming, haskell, functors, monads, category-theory ### 3. nLab Collaboratively edited wiki written by working mathematicians, covering category theory, higher categories, and mathematical physics from an nPOV. Entries are dense and heavily cross-referenced, useful for chasing definitions and connections once basic vocabulary is already in place. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/category **Tags:** wiki, reference, higher-category-theory, mathematical-physics, category-theory ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 5. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki ## Courses ### 1. Category Theory for Programmers Learn category theory fundamentals with Bartosz Milewski's engaging course for programmers. Master categories and their applications! **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ **Tags:** category-theory, functional-programming, haskell, lecture-series, programming ## Books ### 1. Category Theory **Author:** Steve Awodey Oxford Logic Guides text building category theory from scratch for readers with some logic or algebra background. Working through the exercises gives command of limits, adjoints, monads, and the categorical treatment of propositional and higher-order logic. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0199237182?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, category-theory ### 2. Categories for the Working Mathematician **Author:** Saunders Mac Lane The standard graduate reference by one of the field's founders, assuming comfort with algebra and topology. Covers limits, adjoints, monads, Kan extensions, and monoidal categories, and remains the source most later texts and papers cite. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387900365?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, category-theory ### 3. Basic Category Theory **Author:** Tom Leinster A short Cambridge textbook covering only the core: categories, functors, natural transformations, adjoints, limits, and the Yoneda lemma. Assumes little beyond undergraduate algebra, and is freely available from the author as an arXiv preprint. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1107044243?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, category-theory --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/categories