Editing Category theory (section)

== Historical notes ==
{{more citations needed section|date=November 2015}}
{{main|Timeline of category theory and related mathematics}}
{{rquote|right|It should be observed first that the whole concept of a  category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation [...]|[[Samuel Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]] (1945) <ref name="Eilenberg-1945">{{cite journal |last1=Eilenberg |first1=Samuel |last2=Mac Lane |first2=Saunders |date=1945 |title=General theory of natural equivalences |url=https://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf |archive-url=https://ghostarchive.org/archive/20221010/https://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf |archive-date=2022-10-10 |url-status=live |journal=Transactions of the American Mathematical Society |volume=58 |pages=247 |doi=10.1090/S0002-9947-1945-0013131-6 |issn=0002-9947}}</ref>}}
Whilst specific examples of functors and natural transformations had been given by [[Samuel Eilenberg]] and [[Saunders Mac Lane]] in a 1942 paper on [[group theory]],<ref>{{cite journal |last1=Eilenberg |first1=S. |last2=Mac Lane |first2=S. |date=1942 |title=Group Extensions and Homology |url=https://www.jstor.org/stable/1968966 |journal=Annals of Mathematics |volume=43 |issue=4 |pages=757–831 |url-access=registration |doi=10.2307/1968966 |jstor=1968966 |issn=0003-486X }}</ref> these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors<ref name="Eilenberg-1945" /> (who  discussed applications of category theory to the field of [[algebraic topology]]).<ref name="Marquis-2019">{{cite web |last=Marquis |first=Jean-Pierre |date=2019  |title=Category Theory |url=https://plato.stanford.edu/entries/category-theory/ |website=[[Stanford Encyclopedia of Philosophy]] |publisher=Department of Philosophy, [[Stanford University]] |access-date=26 September 2022}}</ref> Their work was an important part of the transition from intuitive and geometric [[Homology (mathematics)|homology]] to [[homological algebra]], Eilenberg and Mac Lane later writing that their goal was to understand natural transformations, which first required the definition of functors, then categories.

[[Stanislaw Ulam]], and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.{{citation needed|reason=Who claimed this and where? Please see discussion on talk page. This claim of a claim needs sources.|date=June 2024}} Eilenberg was Polish, and studied mathematics in Poland in the 1930s.<ref>{{cite web | url=https://mathshistory.st-andrews.ac.uk/Biographies/Eilenberg/ | title=Samuel Eilenberg – Biography }}</ref> Category theory is also, in some sense, a continuation of the work of [[Emmy Noether]] (one of Mac Lane's teachers) in formalizing abstract processes;<ref>{{Cite book |last=Reck |first=Erich |title=The Prehistory of Mathematical Structuralism |publisher=Oxford University Press |year=2020 |isbn=9780190641221 |edition=1st |pages=215–219 |language=en}}</ref> Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure ([[homomorphism]]s).{{citation needed|date=February 2020}} Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes ([[functor]]s) that relate [[topology|topological structures]] to  algebraic structures ([[topological invariant]]s) that characterize them.

Category theory was originally introduced for the need of [[homological algebra]], and widely extended for the need of modern [[algebraic geometry]] ([[scheme theory]]). Category theory may be viewed as an extension of [[universal algebra]], as the latter studies [[algebraic structure]]s, and the former applies to any kind of [[mathematical structure]] and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to [[mathematical logic]] and [[semantics (computer science)|semantics]] ([[categorical abstract machine]]) came later.

Certain categories called [[topos|topoi]] (singular ''topos'') can even serve as an alternative to [[axiomatic set theory]] as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, [[Constructivism (mathematics)|constructive mathematics]]. [[Topos|Topos theory]] is a form of abstract [[Sheaf (mathematics)|sheaf theory]], with geometric origins, and leads to ideas such as [[pointless topology]].

[[Categorical logic]] is now a well-defined field based on [[type theory]] for [[intuitionistic logic]]s, with applications in [[functional programming]] and [[domain theory]], where a [[cartesian closed category]] is taken as a non-syntactic description of a [[lambda calculus]]. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some [[wikt:abstract|abstract]] sense).

Category theory has been applied in other fields as well, see [[applied category theory]]. For example, [[John Baez]] has shown a link between [[Feynman diagrams]] in [[physics]] and monoidal categories.<ref name="Baez09">{{cite book |first1=J.C. |last1=Baez |first2=M. |last2=Stay |chapter=Physics, Topology, Logic and Computation: A Rosetta Stone |title=New Structures for Physics |series=Lecture Notes in Physics |date=2010 |volume=813 |pages=95–172 |doi=10.1007/978-3-642-12821-9_2 |arxiv=0903.0340 |isbn=978-3-642-12820-2 |s2cid=115169297 }}</ref> Another application of category theory, more specifically topos theory, has been made in mathematical music theory, see for example the book ''The Topos of Music, Geometric Logic of Concepts, Theory, and Performance'' by [[Guerino Mazzola]].

More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of [[William Lawvere]] and Rosebrugh (2003) and Lawvere and [[Stephen Schanuel]] (1997) and Mirroslav Yotov (2012).