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arXiv:1612.09375math.CT

Basic Category Theory

Tom Leinster

Abstract

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

arXiv comments
Textbook, vii+183 pages, 107 exercises. Version 2: small number of minor corrections
Journal reference
Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, 2014
DOI
Subjects
math.CTCategory Theory math.ATAlgebraic Topology math.LOLogic

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Linked by 4 identifiable commenters in 4 comments across 4 threads, from Aug 2019 to Apr 2026. Found in 4 HN submissions from Jan 2017 to Jun 2024. Open submission discussions.

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Discussed in comments4 threads

  1. Category Theory Illustrated – Orders 1 comment

    • If you want to learn category theory in a way that is more orthodox, a lot of people recommend Tom Leinster’s Basic Category Theory, which is free[1]. I’m going to be working through it soon, but the bit I’ve skimmed through looks really good if more “mathsy” than things like…

      seanhunterView on HNParent context

  2. Algebra, Topology, Differential Calculus, and Optimization Theory [pdf] 1 comment

    • > And actually.. since a vector space has an underlying set, it seems to me that to do a direct sum on a vector space, you must first do a direct sum on the set (that is, do a disjoint union), and then do other things to map the rest of the structure of a vector space into the…

      bmachoView on HNParent context

  3. Graphical Linear Algebra 1 comment

    • feanaro: fine, if you think so. I find it interesting that actual mathematicians working in Category Theory, such as Tom Leinster (who wrote a lovely little introductory text on Category Theory [not covering monads though] and made it freely available on arXiv…

      stablenodeView on HNParent context

  4. Linguistics Using Category Theory (2018) 1 comment

    • It sounds like you would be interested in the book / course '7 Sketches in Compositionality' by David Spivak and Brendan Fong, which studies precisely those ideas categorically, with a particular focus on systems that you might broadly call 'computational'…

      autopoiesisView on HNParent context

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