View source for Category theory

{{Short description|General theory of mathematical structures}}
{{More sources needed|date=November 2024}}

[[File:Commutative diagram for morphism.svg|right|thumb|200px|Schematic representation of a category with objects ''X'', ''Y'', ''Z'' and morphisms ''f'', ''g'', {{nowrap|1=''g'' ∘ ''f''}}. (The category's three identity morphisms 1<sub>''X''</sub>, 1<sub>''Y''</sub> and 1<sub>''Z''</sub>, if explicitly represented, would appear as three arrows, from the letters ''X'', ''Y'', and ''Z'' to themselves, respectively.)]]

'''Category theory''' is a general theory of [[mathematical structure]]s and their relations. It was introduced by [[Samuel Eilenberg]] and [[Saunders Mac Lane]] in the middle of the 20th century in their foundational work on [[algebraic topology]].<ref>{{Citation |last=Marquis |first=Jean-Pierre |title=Category Theory |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/fall2023/entries/category-theory/ |access-date=2024-04-23 |edition=Fall 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref> Category theory is used in almost all areas of mathematics. In particular, many constructions of new [[mathematical object]]s from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include [[quotient space (disambiguation)|quotient space]]s, [[direct product]]s, completion, and [[duality (mathematics)|duality]].

Many areas of [[computer science]] also rely on category theory, such as [[functional programming]] and [[Semantics (computer science)|semantics]].

A [[category (mathematics)|category]] is formed by two sorts of [[mathematical object|objects]]: the [[object (category theory)|object]]s of the category, and the [[morphism]]s, which relate two objects called the ''source'' and the ''target'' of the morphism. Metaphorically, a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one. Morphism composition has similar properties as [[function composition]] ([[associativity]] and existence of an [[identity element|identity morphism]] for each object). Morphisms are often some sort of [[function (mathematics)|function]]s, but this is not always the case. For example, a [[monoid]] may be viewed as a category with a single object, whose morphisms are the elements of the monoid.

The second fundamental concept of category theory is the concept of a [[functor]], which plays the role of a morphism between two categories <math>\mathcal{C}_1</math> and <math>\mathcal{C}_2</math>: it maps objects of <math>\mathcal{C}_1</math> to objects of <math>\mathcal{C}_2</math> and morphisms of <math>\mathcal{C}_1</math> to morphisms of <math>\mathcal{C}_2</math> in such a way that sources are mapped to sources, and targets are mapped to targets (or, in the case of a [[contravariant functor]], sources are mapped to targets and ''vice-versa''). A third fundamental concept is a [[natural transformation]] that may be viewed as a morphism of functors.

== Categories, objects, and morphisms ==
{{Main|Category (mathematics)|Morphism}}

=== Categories ===
A ''category'' <math>\mathcal{C}</math> consists of the following three mathematical entities:
* A [[Class (set theory)|class]] <math>\text{ob}(\mathcal{C})</math>, whose elements are called ''objects'';
* A class <math>\text{hom}(\mathcal{C})</math>, whose elements are called [[morphism]]s or [[Map (mathematics)|maps]] or ''arrows''. <p>Each morphism '''<math>f</math>''' has a ''source object '' '''<math>a</math>''' and ''target object'' '''<math>b</math>'''.</p><p>The expression <math>f:a \mapsto b</math> would be verbally stated as "<math>f</math> is a morphism from {{mvar|a}} to {{mvar|b}}".</p><p>The expression <math>\text{hom}(a, b)</math> – alternatively expressed as <math>\text{hom}_\mathcal{C}(a, b)</math>, <math>\text{mor}(a, b)</math>, or <math>\mathcal{C}(a, b)</math> – denotes the ''hom-class'' of all morphisms from <math>a</math> to <math>b</math>.{{efn|The name "hom" derives from the fact that the notion of morphism is a generalisation of the notion of [[homomorphism]]. But even in categories whose objects have no notion of homomorphism or where the morphisms are explicitly not (or not precisely) homomorphisms, the classes <math>\text{hom}(a, b)</math> are still referred to as hom-classes.}}</p>
* A [[binary operation]] <math>\circ</math>, called ''composition of morphisms'', such that for any three objects ''{{mvar|a}}'', ''{{mvar|b}}'', and ''{{mvar|c}}'', we have<math display="block">\circ : \text{hom}(b, c) \times \text{hom}(a, b) \mapsto \text{hom}(a, c)</math>The composition of <math>f : a \mapsto b</math> and <math>g: b \mapsto c</math> is written as <math>g \circ f</math> or <math>gf</math>,{{efn|Some authors compose in the opposite order, writing ''fg'' or {{nowrap|1=''f'' ∘ ''g''}} for {{nowrap|1=''g'' ∘ ''f''}}. Computer scientists using category theory very commonly write {{nowrap|1=''f'' ; ''g''}} for {{nowrap|1=''g'' ∘ ''f''}}}} governed by two axioms:
*# [[Associativity]]: If <math>f: a \mapsto b</math>, <math>g: b \mapsto c</math>, and <math>h: c \mapsto d</math> then <math display="block">h \circ (g \circ f) = (h \circ g) \circ f</math>
*# [[Identity (mathematics)|Identity]]: For every object {{mvar|x}}, there exists a morphism <math>1_x : x \mapsto x</math> (also denoted as <math>\text{id}_x</math>) called the ''[[identity morphism]] for {{mvar|x}}'', such that for every morphism <math>f: a \mapsto b</math>, we have<math display="block">1_b \circ f = f = f \circ 1_a</math><p>From the axioms, it can be proved that there is exactly one [[identity morphism]] for every object.</p>

====Examples====
* The category '''[[category of sets|Set]]'''
** As the class of objects <math>\text{ob} (\text{Set})</math>, we choose the class of all sets.
** As the class of morphisms <math>\text{hom} (\text{Set})</math>, we choose the class of all [[function (mathematics)|function]]s. Therefore, for two objects {{mvar|A}} and {{mvar|B}}, i.e. sets, we have <math>\text{hom} (A,B)</math> to be the class of all functions {{tmath|f}} such that {{tmath|f:A \mapsto B}}.
** The composition of morphisms {{tmath|\circ}} is simply the usual [[function composition]], i.e. for two morphisms {{tmath|f:A \mapsto B}} and {{tmath|g:B \mapsto C}}, we have {{tmath|g \circ f:A \mapsto C}}, <math>(g \circ f)(x) = g(f(x))</math>, which is obviously associative. Furthermore, for every object {{mvar|A}} we have the identity morphism <math>\text{id}_A</math> to be the identity map <math>\text{id}_A : A \mapsto A</math>, <math>\text{id}_A (x) = x</math> on {{mvar|A}}

=== Morphisms ===
Relations among morphisms (such as {{nowrap|1=''fg'' = ''h''}}) are often depicted using [[commutative diagram]]s, with "points" (corners) representing objects and "arrows" representing morphisms.

[[Morphism]]s can have any of the following properties. A morphism {{nowrap|1=''f'' : ''a'' → ''b''}} is:
* a [[monomorphism]] (or ''monic'') if {{nowrap|1=''f'' ∘ ''g''<sub>1</sub> = ''f'' ∘ ''g''<sub>2</sub>}} implies {{nowrap|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms {{nowrap|1=''g''<sub>1</sub>, ''g<sub>2</sub>'' : ''x'' → ''a''}}.
* an [[epimorphism]] (or ''epic'') if {{nowrap|1=''g''<sub>1</sub> ∘ ''f'' = ''g''<sub>2</sub> ∘ ''f''}} implies {{nowrap|1=''g<sub>1</sub>'' = ''g<sub>2</sub>''}} for all morphisms {{nowrap|1=''g''<sub>1</sub>, ''g''<sub>2</sub> : ''b'' → ''x''}}.
* a ''bimorphism'' if ''f'' is both epic and monic.
* an [[isomorphism]] if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} such that {{nowrap|1=''f'' ∘ ''g'' = 1<sub>''b''</sub> and ''g'' ∘ ''f'' = 1<sub>''a''</sub>}}.{{efn|A morphism that is both epic and monic is not necessarily an isomorphism. An elementary counterexample: in the category consisting of two objects ''A'' and ''B'', the identity morphisms, and a single morphism ''f'' from ''A'' to ''B'', ''f'' is both epic and monic but is not an isomorphism.}}
* an [[endomorphism]] if {{nowrap|1=''a'' = ''b''}}. end(''a'') denotes the class of endomorphisms of ''a''.
* an [[automorphism]] if ''f'' is both an endomorphism and an isomorphism. aut(''a'') denotes the class of automorphisms of ''a''.
* a [[retract (category theory)|retraction]] if a right inverse of ''f'' exists, i.e. if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} with {{nowrap|1=''f'' ∘ ''g'' = 1<sub>''b''</sub>}}.
* a [[section (category theory)|section]] if a left inverse of ''f'' exists, i.e. if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} with {{nowrap|1=''g'' ∘ ''f'' = 1<sub>''a''</sub>}}.

Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
* ''f'' is a monomorphism and a retraction;
* ''f'' is an epimorphism and a section;
* ''f'' is an isomorphism.

== Functors ==
{{Main|Functor}}

[[Functor]]s are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories.

A ('''covariant''') functor ''F'' from a category ''C'' to a category ''D'', written {{nowrap|1=''F'' : ''C'' → ''D''}}, consists of:
* for each object ''x'' in ''C'', an object ''F''(''x'') in ''D''; and
* for each morphism {{nowrap|1=''f'' : ''x'' → ''y''}} in ''C'', a morphism {{nowrap|1=''F''(''f'') : ''F''(''x'') → ''F''(''y'')}} in ''D'',

such that the following two properties hold:
* For every object ''x'' in ''C'', {{nowrap|1=''F''(1<sub>''x''</sub>) = 1<sub>''F''(''x'')</sub>}};
* For all morphisms {{nowrap|1=''f'' : ''x'' → ''y''}} and {{nowrap|1=''g'' : ''y'' → ''z''}}, {{nowrap|1=''F''(''g'' ∘ ''f'') = ''F''(''g'') ∘ ''F''(''f'')}}.

A '''contravariant''' functor {{nowrap|1=''F'': ''C'' → ''D''}} is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism {{nowrap|1=''f'' : ''x'' → ''y''}} in ''C'' must be assigned to a morphism  {{nowrap|1=''F''(''f'') : ''F''(''y'') → ''F''(''x'')}} in ''D''. In other words, a contravariant functor acts as a covariant functor from the [[opposite category]] ''C''<sup>op</sup> to ''D''.

== Natural transformations ==
{{Main|Natural transformation}}

A ''natural transformation'' is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.

If ''F'' and ''G'' are (covariant) functors between the categories ''C'' and ''D'', then a natural transformation ''η'' from ''F'' to ''G'' associates to every object ''X'' in ''C'' a morphism {{nowrap|1=''η''<sub>''X''</sub> : ''F''(''X'') → ''G''(''X'')}} in ''D'' such that for every morphism {{nowrap|1=''f'' : ''X'' → ''Y''}} in ''C'', we have {{nowrap|1=''η''<sub>''Y''</sub> ∘ ''F''(''f'') = ''G''(''f'') ∘ ''η''<sub>''X''</sub>}}; this means that the following diagram is [[commutative diagram|commutative]]:

[[Image:Natural transformation.svg|175px|center|Commutative diagram defining natural transformations]]

The two functors ''F'' and ''G'' are called ''naturally isomorphic'' if there exists a natural transformation from ''F'' to ''G'' such that ''η''<sub>''X''</sub> is an isomorphism for every object ''X'' in ''C''.

== Other concepts ==

=== Universal constructions, limits, and colimits ===
{{Main|Universal property|Limit (category theory)}}

Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies.

Each category is distinguished by properties that all its objects have in common, such as the [[empty set]] or the [[product topology|product of two topologies]], yet in the definition of a category, objects are considered atomic, i.e., we ''do not know'' whether an object ''A'' is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find ''[[universal property|universal properties]]'' that uniquely determine the objects of interest.

Numerous important constructions can be described in a purely categorical way if the ''category limit'' can be developed and dualized to yield the notion of a ''colimit''.

=== Equivalent categories ===
{{Main|Equivalence of categories|Isomorphism of categories}}

It is a natural question to ask: under which conditions can two categories be considered ''essentially the same'', in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called ''equivalence of categories'', which is given by appropriate functors between two categories. Categorical equivalence has found [[Equivalence of categories#Examples|numerous applications]] in mathematics.

=== Further concepts and results ===
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
* The [[functor category]] ''D''<sup>''C''</sup> has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The [[Yoneda lemma]] is one of the most famous basic results of category theory; it describes representable functors in functor categories.
* [[Dual (category theory)|Duality]]: Every statement, theorem, or definition in category theory has a ''dual'' which is essentially obtained by "reversing all the arrows". If one statement is true in a category ''C'' then its dual is true in the dual category ''C''<sup>op</sup>. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.
* [[Adjoint functors]]: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.

=== Higher-dimensional categories ===
{{Main|Higher category theory}}
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of ''higher-dimensional categories''. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".

For example, a (strict) [[2-category]] is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is '''Cat''', the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply [[natural transformation]]s of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially [[monoidal category|monoidal categories]]. [[bicategory|Bicategories]] are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.

This process can be extended for all [[natural number]]s ''n'', and these are called [[n-category|''n''-categories]]. There is even a notion of ''[[quasi-category|ω-category]]'' corresponding to the [[ordinal number]] [[ω (ordinal number)|ω]].

Higher-dimensional categories are part of the broader mathematical field of [[higher-dimensional algebra]], a concept introduced by [[Ronald Brown (mathematician)|Ronald Brown]].  For a conversational introduction to these ideas, see [http://math.ucr.edu/home/baez/week73.html John Baez, 'A Tale of ''n''-categories' (1996).]

== 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).

== See also ==
{{Portal|Mathematics}}
{{cols|colwidth=26em}}
* [[Domain theory]]
* [[Enriched category|Enriched category theory]]
* [[Glossary of category theory]]
* [[Group theory]]
* [[Higher category theory]]
* [[Higher-dimensional algebra]]
* [[List of publications in mathematics#Category theory|Important publications in category theory]]
* [[Lambda calculus]]
* [[Outline of category theory]]
* [[Timeline of category theory and related mathematics]]
* [[Applied category theory]]
{{colend}}

== Notes ==
{{notelist|30em}}

== References ==
=== Citations ===
{{reflist|30em}}

=== Sources ===
{{refbegin|30em}}
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* {{cite book |first1=F. William |last1=Lawvere |first2=Stephen Hoel |last2=Schanuel |author2-link=Stephen Schanuel |title = Conceptual Mathematics: A First Introduction to Categories |url = https://archive.org/details/conceptualmathem00lawv |url-access=registration |year=2009 |publisher=Cambridge University Press |isbn = 978-0-521-89485-2 |edition=2nd |orig-year=1997 }}
* {{cite book |last=Leinster |first=Tom |author-link= :de:Tom Leinster|title=Higher Operads, Higher Categories |publisher=[[Cambridge University Press]] |year=2004 |isbn=978-0-521-53215-0 |series=London Math. Society Lecture Note Series |volume=298 |pages=448 |url=http://www.maths.gla.ac.uk/~tl/book.html |bibcode=2004hohc.book.....L |access-date=2006-04-03 |archive-url=https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html |archive-date=2003-10-25 |url-status=dead }}
* {{cite book |last=Leinster |first=Tom |author-link = :de:Tom Leinster |title = Basic Category Theory |publisher=Cambridge University Press |year=2014 |arxiv=1612.09375 |url=https://books.google.com/books?id=Q3vsAwAAQBAJ |isbn=9781107044241 |volume=143 |series=Cambridge Studies in Advanced Mathematics}}
* {{cite book |last=Lurie |first=Jacob |author-link=Jacob Lurie |title = Higher Topos Theory| publisher=Princeton University Press |year=2009 |series=Annals of Mathematics Studies |volume=170 |arxiv=math.CT/0608040 |mr=2522659 |isbn=978-0-691-14049-0 }}
* {{cite book |last=Mac Lane |first=Saunders |title=Categories for the Working Mathematician |publisher=Springer-Verlag |year=1998 |edition=2nd |series=Graduate Texts in Mathematics | volume=5 |author-link=Saunders Mac Lane |isbn=978-0-387-98403-2 | mr=1712872|title-link=Categories for the Working Mathematician }}
* {{cite book |title=Algebra |last1=Mac Lane |first1=Saunders |author1-link=Saunders Mac Lane |first2=Garrett |last2=Birkhoff |author2-link=Garrett Birkhoff |publisher=Chelsea |year=1999| edition=2nd |isbn = 978-0-8218-1646-2 |orig-year=1967 }}
* {{cite journal |year=1996|title=Elements of basic category theory |journal=Technical Report|volume=96 |issue=5 |url = http://citeseer.ist.psu.edu/martini96element.html |first1=A. |last1=Martini |first2=H. |last2=Ehrig |first3=D. |last3=Nunes}}
* {{cite book |first=Guerino |last=Mazzola |author-link=Guerino Mazzola |title=The Topos of Music, Geometric Logic of Concepts, Theory, and Performance |publisher=Birkhäuser |year=2002 |isbn = 978-3-7643-5731-3 }}
* {{cite book |zbl=1034.18001 |editor1-last=Pedicchio |editor1-first = Maria Cristina |editor2-last = Tholen |editor2-first=Walter |title = Categorical foundations. Special topics in order, topology, algebra, and sheaf theory |series=Encyclopedia of Mathematics and Its Applications |volume=97  |publisher=[[Cambridge University Press]] |year=2004 |isbn = 978-0-521-83414-8 }}
* {{cite book |first=Benjamin C. |last=Pierce |author-link=Benjamin C. Pierce |title = Basic Category Theory for Computer Scientists |url = https://books.google.com/books?id=ezdeaHfpYPwC |year=1991 |publisher=MIT Press |isbn = 978-0-262-66071-6 }}
* {{cite book |title=An introduction to Category Theory in four easy movements |year=2005 |url=http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf |last1=Schalk |first1=A. |last2=Simmons |first2=H. |access-date=2007-12-03 |archive-url=https://web.archive.org/web/20170321152109/http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf |archive-date=2017-03-21 |url-status=dead }} Notes for a course offered as part of the MSc. in [[Mathematical Logic]], [[Manchester University]].
* {{citation <!-- Harold Simmons, 2011 CUP book -->
 | last1 = Simmons
 | first1 = Harold
 | title = An Introduction to Category Theory
 | year = 2011
 | isbn = 978-0521283045
}}.
* {{cite book |title = Homotopy theory of higher categories |url = https://archive.org/details/arxiv-1001.4071 |last=Simpson |first=Carlos |author-link=Carlos Simpson |arxiv=1001.4071 |bibcode=2010arXiv1001.4071S |year=2010 }}, draft of a book.
* {{cite book |first=Paul |last=Taylor |title=Practical Foundations of Mathematics |url= https://books.google.com/books?id=iSCqyNgzamcC |year=1999 |publisher= Cambridge University Press |isbn = 978-0-521-63107-5 |series = Cambridge Studies in Advanced Mathematics |volume = 59 }}
* {{cite web |url = http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf |title = Category Theory Lecture Notes |last=Turi |first=Daniele |date = 1996–2001 |access-date = 11 December 2009 }} Based on {{harvnb|Mac Lane|1998}}.
{{refend}}

== Further reading ==
{{refbegin}}
* {{cite book |first=Jean-Pierre |last=Marquis |title=From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory |year=2008 |publisher=Springer |isbn=978-1-4020-9384-5}}
{{refend}}

== External links ==
{{Commons category}}
{{wikiquote}}
{{refbegin}}
* [http://www.tac.mta.ca/tac/ Theory and Application of Categories], an electronic journal of category theory, full text, free, since 1995.
* [https://cahierstgdc.com/ Cahiers de Topologie et Géométrie Différentielle Catégoriques], an electronic journal of category theory, full text, free, funded in 1957.
* [http://ncatlab.org/nlab nLab], a wiki project on mathematics, physics and philosophy with emphasis on the ''n''-categorical point of view.
* [https://golem.ph.utexas.edu/category/ The n-Category Café], essentially a colloquium on topics in category theory.
* [http://www.logicmatters.net/categories/ Category Theory], a web page of links to lecture notes and freely available books on category theory.
* {{citation |first=Chris |last=Hillman |title=A Categorical Primer |year=2001 |citeseerx=10.1.1.24.3264}}, a formal introduction to category theory.
* {{cite web |first1=J. |last1=Adamek |first2=H. |last2=Herrlich |first3=G. |last3=Stecker  |title=Abstract and Concrete Categories-The Joy of Cats |url=http://katmat.math.uni-bremen.de/acc/acc.pdf |archive-url=https://web.archive.org/web/20060610174819/http://katmat.math.uni-bremen.de/acc/acc.pdf |archive-date=2006-06-10 |url-status=live}}
* {{SEP|category-theory|Category Theory|Jean-Pierre Marquis}}, with an extensive bibliography.
* [http://www.mta.ca/~cat-dist/ List of academic conferences on category theory]
* {{cite web |last=Baez |first=John |title=The Tale of ''n''-categories |year=1996 |url=http://math.ucr.edu/home/baez/week73.html}} — An informal introduction to higher order categories.
* [http://wildcatsformma.wordpress.com WildCats] is a category theory package for [[Mathematica]]. Manipulation and visualization of objects, [[morphism]]s, categories, [[functor]]s, [[natural transformation]]s, [[universal properties]].
* {{YouTube|user=TheCatsters|title=The catsters}}, a channel about category theory.
* {{PlanetMath|urlname=9categorytheory|title=Category theory}}.
* [http://categorieslogicphysics.wikidot.com/events Video archive] of recorded talks relevant to categories, logic and the foundations of physics.
* [https://web.archive.org/web/20080916162345/http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page] which generates examples of categorical constructions in the category of finite sets.
* [https://web.archive.org/web/20150109111227/http://category-theory.mitpress.mit.edu/index.html Category Theory for the Sciences], an instruction on category theory as a tool throughout the sciences.
* [https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ Category Theory for Programmers] A book in blog form explaining category theory for computer programmers.
* [http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf Introduction to category theory.]
{{refend}}

{{Areas of mathematics}}
{{Category theory}}
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