nLab category theory -- references

References on category theory:

History

The concepts of category, functor and natural transformation were introduced in

• Samuel Eilenberg, Saunders MacLane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 2 (1945) 231-294 [doi:10.1090/S0002-9947-1945-0013131-6, jstor:1990284]

apparently (see there) taking inspiration from:

• Hans Freudenthal, Entwicklungen von Räumen und ihren Gruppen, Comp. Math. 4 (1937) 145-234 [numdam:CM_1937__4__145_0]

The rational for introducing the concept of categories was to introduce the concept of functors, and the reason for introducing functors was to introduce the concept of natural transformations (more specifically natural equivalences) in order to make precise the meaning of “natural” in mathematics and specifically in algebraic topology, concretely for the example case of dual objects:

$[$Freyd 64, page 1:] If topology were publicly defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology. The mathematical correctness of such a definition reveals nothing about topology except that its basic axioms can be made quite simple. And with category theory we are confronted with the same pedagogical problem. The basic axioms, which we will shortly be forced to give, are much too simple.

A better (albeit not perfect) description of topology is that it is the study of continuous maps; and category theory is likewise better described as the theory of functors. Both descriptions are logically inadmissible as initial definitions, but they more accurately reflect both the present and the historical motivations of the subjects.

It is not too misleading, at least historically, to say that categories are what one must define in order to define natural transformations.

and

$[$Freyd 65, beginning of Part Two]: Category theory is an embodiment of Klein’s dictum that it is the maps that count in mathematics. If the dictum is true, then it is the functors between categories that are important, not the categories. And such is the case. Indeed, the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations. And it was in order to define these that Eilenberg and MacLane first defined functors.

The article by Eilenberg & Maclane 1945 was a clash of ideas from abstract algebra (Mac Lane) and topology/homotopy theory (Eilenberg). It was first rejected on the ground that it had no content. Since then category theory has flourished into almost all areas of mathematics, has found many applications outside mathematics and even attempts to build a foundations of mathematics.

This and much more history is recalled in:

• Saunders MacLane: Concepts and Categories in Perspective, in: A Century of Mathematics in America, Part I, Am. Math. Soc. (1988) 323–366. [ISBN 0-8218-0124-4, pdf]

• Ralf Krömer: Tool and object: A history and philosophy of category theory, Science Networks. Historical Studies 32 (2007) [doi:10.1007/978-3-7643-7524-9]

see specifically

• Ralf Krömer: Category theory in Algebraic Topology, chapter 3 of: Krömer 2007 [doi:10.1007/978-3-7643-7524-9_2]

See also:

• Jean-Pierre Marquis: What is Category Theory?, in G. Sica (ed.) What is Category Theory?, Polimetrica (2006) 221-256 $[$pdf, SemanticScholar]

• Jean-Pierre Marquis: From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer (2009) [doi:10.1007/978-1-4020-9384-5]

The definition of categories as internal categories in Set is due to:

• Alexander Grothendieck, Section 4 of: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212 [numdam:SB_1960-1961__6__99_0 pdf]

For the definition of categories in the foundations of homotopy type theory see at internal category in homotopy type theory.

Textbooks

Basic category theory

Monographs on category theory:

• Peter Freyd: Abelian Categories – An Introduction to the theory of functors, Harper and Row (1964), Reprints in Theory and Applications of Categories, 3 (2003) [tac:tr2, pdf]

• Peter Freyd: The theories of functors and models, in: Proceedings of Symposium on the Theory of Models, North Holland (1965) [doi:10.1016/C2013-0-11897-1]

• Barry Mitchell: Theory of categories, Pure and Applied Mathematics 17, Academic Press (1965) [ISBN:978-0-12-499250-4]

• Saunders MacLane (notes by Ellis Cooper): Lectures on category theory, Bowdoin Summer School (1969) $[$pdf]

• Peter Hilton (ed.): Category Theory, Homology Theory and Their Applications,

  vol 1: Lecture Notes in Mathematics 86, Springer (1969) [doi:10.1007/BFb0079380]

  vol 2: Lecture Notes in Mathematics 92, Springer (1969) [doi:10.1007/BFb0080761]

  vol 3: Lecture Notes in Mathematics 99, Springer (1969) [doi:10.1007/BFb0081959]

• Bodo Pareigis: Categories and Functors, Pure and Applied Mathematics 39, Academic Press (1970) [doi:10.5282/ubm/epub.7244, pdf]

• Saunders MacLane: Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer (1971, second ed. 1998) [doi:10.1007/978-1-4757-4721-8]

• John Gray: Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, 391, Springer (1974) [doi:10.1007/BFb0061280]

  (formal category theory in the 2-category Cat)

• Horst Schubert: Categories, Springer (1972) [doi:10.1007/978-3-642-65364-3]

• Nicolae Popescu, Liliana Popescu: Theory of categories, Sijthoff & Noordhoff International Publishers (1979) [doi:10.1007/978-94-009-9550-5]

• Ioan Mackenzie James; §1 in: General Topology and Homotopy Theory, Springer (1984) [doi:10.1007/978-1-4613-8283-6]

• Robert Geroch: Mathematical Physics, University of Chicago Press (1985) [ISBN:9780226223063, ark:/13960/t10p8v264]

  (introduces categories by examples arising in mathematical physics)

• Jiří Adámek, Horst Herrlich, George Strecker: Abstract and Concrete Categories – The Joy of Cats, Wiley (1990), reprinted as: Reprints in Theory and Applications of Categories 17 (2006) 1-507 [tac:tr17, book webpage, pdf]

• Peter Freyd, Andre Scedrov: Categories, Allegories, Mathematical Library 39, North-Holland (1990) [ISBN 978-0-444-70368-2]

• Benjamin Pierce: Basic category theory for computer scientists (1991) [ISBN: 9780262660716]

• Francis Borceux: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

  Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

  Vol. 2: Categories and Structures [doi:10.1017/CBO9780511525865]

  Vol. 3: Categories of Sheaves [doi:10.1017/CBO9780511525872]

• Roy L. Crole: Categories for types, Cambridge University Press (1994) [doi:10.1017/CBO9781139172707]

  (categorical semantics of type theory)

• Michael Barr, Charles Wells: Category theory for computing science, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories 22 (2012) 1-538 [tac:tr22, pdf]

  (aimed at computer science, see computational trilogy)

• Steve Awodey: Category theory, Oxford University Press (2006, 2010) [doi:10.1093/acprof:oso/9780198568612.001.0001, ISBN:9780199237180, pdf]

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• F. William Lawvere, Stephen Schanuel: Conceptual Mathematics: A first introduction to categories, Cambridge University Press (2009P) [pdf, pdf]

• Harold Simmons: An Introduction to Category Theory, Cambridge (2011) [ISBN:9781107010871]

• David Spivak: Category theory for scientists, MIT Press (2014) [ISBN:9780262028134, arXiv:1302.6946]

• Tom Leinster: Basic Category Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107360068, ]

• Emily RiehlL Category Theory in Context, Dover Publications (2017) [ISBN:https://store.doverpublications.com/products/9780486809038, pdf, webpage]

• Martin Brandenburg, Einführung in die Kategorientheorie, Springer (2017) [doi:10.1007/978-3-662-53521-9]

• Brendan Fong, David Spivak: An invitation to applied category theory (2018) [doi:10.1017/9781108668804, arXiv:1803.05316, web, pdf]

• Chris Heunen, Jamie Vicary: Categories for Quantum Theory, Oxford University Press (2019) [ISBN:9780198739616]

  (emphasis on monoidal category-theory with an eye towards quantum information via dagger-compact categories)

• Marco Grandis: Category Theory and Applications: A Textbook for Beginners, World Scientific (2021) [doi:10.1142/12253]

• Paolo Perrone: Starting Category Theory, World Scientific (2024) [doi:10.1142/13670, arXiv:1912.10642]

On category theory in computer science/programming languages (such as for monads in computer science):

• Bartosz Milewski (compiled by Igal Tabachnik), Category Theory for Programmers, Blurb (2019) [ISBN:9780464243878, pdf, github, webpage]

Topos theory

Monographs with focus on topos theory:

• Peter Johnstone, Topos theory (1977)

• Michael Barr, Charles Wells: Toposes, Triples, and Theories, Grundlehren der math. Wissenschaften 278, Springer (1983) [tac:tr12]

  (Here “triple” means“ monad”).

• Robert Goldblatt: Topoi, the categorial analysis of logic (1984) [GBooks]

• Colin McLarty: Elementary Categories, Elementary Toposes, Oxford University Press (1992) [ISBN:9780198514732]

• Ieke Moerdijk, Saunders Mac Lane, Sheaves in Geometry and Logic Springer (1992) [doi:10.1007/978-1-4612-0927-0]

• Peter Johnstone, Sketches of an Elephant, vol 1-2, Oxford University Press (2002)

  Vol 1: [ISBN:9780198534259]

  Vol 2: [ISBN:9780198515982]

Higher category theory

• Tom Leinster, Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series 298, Cambridge University Press (2004) [ISBN:0 521 53215 9, doi:10.1017/cbo9780511525896, arXiv:math/0305049]

• Eugenia Cheng, Aaron Lauda: Higher-dimensional categories: an illustrated guide book (2004) [pdf]

• Jacob Lurie, Higher Topos Theory, Princeton University Press (2009) [pup:8957, pdf]

• Carlos Simpson, Homotopy Theory of Higher Categories, Cambridge University Press (2011) [ISBN:9780521516952, hal:00449826, arXiv:1001.4071]

Towards homotopy theory:

• Birgit Richter, From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press (2020) [doi:10.1017/9781108855891, book webpage, pdf]

Course notes

• Daniele Turi: Category Theory Lecture Notes (2001) [pdf]

• Jaap van Oosten: Basic category theory (2002) [pdf]

• Thomas Streicher: Introduction to Category Theory and Categorical Logic (2003) [pdf, pdf]

• Bodo Pareigis: Category theory (2004) [dvi, ps, pdf]

• Tom Leinster: Notes on basic category theory (2007) [web]

• Peter Johnstone: Category Theory , Lecture notes taken by David Mehrle, University of Cambridge (2015) [pdf]

• André Joyal, Categories

• Benedikt Ahrens, Kobe Wullaert: Category Theory for Programming (2022) [arXiv:2209.01259]

Videos

• The Catsters: Videos on various topics in category theory [YouTube]

• Tom LaGatta: Category theory [YouTube]

Relation to philosophy

Discussion of the relation to and motivation from the philosophy of mathematics:

• Joachim Lambek: The Influence of Heraclitus on Modern Mathematics, in: Joseph Agassi, Robert S. Cohen (eds.) Scientific Philosophy Today: Essays in Honor of Mario Bunge, , 111-21. Boston: D. Reidel Publishing Co. (1981) 111-121 [doi:10.1007/978-94-009-8462-2_6, pdf]

• Colin McLarty: The Last Mathematician from Hilbert’s Göttingen: Saunders Mac Lane as Philosopher of Mathematics,Brit. J. Phil. Sci. (2007) [pdf]

Further resources and links

• networks of category theorists.