Lihat sumber Teori kategori

[[Berkas:Commutative diagram for morphism.svg|ka|jmpl|200px|Teori kategori. Sebuah kategori dengan objek ''X'', ''Y'', ''Z'', dan morfisme ''f'', ''g'', ''g'' ∘ ''f'', dan tiga morfisma identitas (tidak ditunjukkan) 1<sub>''X''</sub>, 1<sub>''Y''</sub>, dan 1<sub>''Z''</sub>.]]

'''Teori kategori''' berhubungan dengan struktur [[matematika]] dan hubungan antar struktur tersebut secara abstrak. Saat ini kategori digunakan dalam matematika, informatika teori, dan [[fisika matematis]]. Kategori diperkenalkan pertama kali oleh [[Samuel Eilenberg]] dan [[Saunders Mac Lane]] pada tahun [[1942]]-[[1945]], dalam hubungannya dengan [[topologi aljabar]].

== Definisi Kategori ==
Suatu kategori <math> \mathcal{C} </math> terdiri atas:
* Suatu kelas Obj(<math> \mathcal{C} </math>) yang berisi objek dari kategori tersebut
* Untuk sebarang <math> A, B \in </math> Obj(<math> \mathcal{C} </math>) terdapat suatu himpunan morfisma Hom(<math>A,B</math>) yang berisi morfisma atau panah dari objek <math> A </math> ke objek <math> B </math> dengan sifat:
** Komposisi Morfisma. Kita bisa menggabungkan dua morfisma <math>f</math> dan <math>g</math>, jika himpunan target dari morfisma pertama sama dengan himpunan sumber dari morfisma kedua, misal <math> f: A \rightarrow B </math> dan <math> g: B \rightarrow C </math> untuk suatu objek <math> A,B, </math> dan <math> C </math>. Komposisi biasanya dilambangkan dengan <math>g \circ f</math> yang berarti morfisma <math>f</math> dilanjut oleh morfisma <math>g</math> (dibaca dari kanan).
** Morfisma Identitas. Untuk setiap <math>A \in</math> Obj(<math>\mathcal{C}</math>), terdapat morfisma identitas <math>1_A : A \rightarrow A</math> yang untuk sebarang morfisma <math>f \in </math> Hom(<math>A,B</math>) memenuhi <math>f1_A = f</math> dan <math>1_B f = f </math>

Suatu kategori <math> \mathcal{C} </math> dikatakan sebagai '''kategori kecil''' jika Obj(<math>\mathcal{C}</math>) adalah suatu [[himpunan]].

== Contoh Kategori ==
Dari pendefinisian kategori, terasa alami untuk mendefinisikan suatu kategori '''SET''' dengan Obj('''SET''') adalah kelas yang berisi semua himpunan dan untuk sebarang himpunan <math>A,B \in </math> Obj('''SET''') kita punyai Hom(<math>A,B</math>) fungsi dari himpunan <math>A</math> ke himpunan <math>B</math>. Perhatikan bahwa kategori '''SET''' bukanlah sebuah kategori kecil dikarenakan koleksi dari seluruh himpunan bukanlah suatu himpunan (untuk menghindari paradoks Russel)

Beberapa contoh lain dari kategori diberikan pada tabel berikut
{| class="wikitable"
!Kategori
!Objek
!Morfisma
|-
|'''Grp'''
|Grup
|Homomorfisma Grup
|-
|'''Man'''<sup>''p''</sup>
|Manifold mulus
|Pemetaan yang terdiferensialkan kontinu ''p''-kali
|-
|'''Met'''
|Ruang Metrik
|Pemetaan metrik
|-
|'''R-Mod'''
|R-Modul, dengan R suatu gelanggang
|Homomorfisma modul
|-
|'''Ring'''
|Gelanggang
|Homomorfisma gelanggang
|-
|'''Top'''
|Ruang Topologi
|Fungsi kontinu
|-
|'''Vect'''<sub>''K''</sub>
|Ruang vektor atas lapangan ''K''
|Pemetaan linier
|}

== Konstruksi Kategori Baru dari Kategori yang Ada ==
Sebarang kategori <math>\mathcal{C}</math> dapat dikonstruksi menjadi kategori baru dengan membalik setiap morfismanya tanpa mengubah objeknya. Kategori yang demikian disebut kategori ''dual'' dan dinotasikan sebagai <math>\mathcal{C}</math><sup>op</sup>

== Lihat pula ==
{{Portal|Matematika}}
{{cols|colwidth=26em}}
* [[Teori domain]]
* [[Kategori yang diperkaya|Teori kategori yang diperkaya]]
* [[Glosarium teori kategori]]
* [[Teori grup]]
* [[Teori kategori yang lebih tinggi]]
* [[Aljabar berdimensi lebih tinggi]]
* [[Daftar publikasi dalam matematika#Teori kategori|Publikasi penting dalam teori kategori]]
* [[Kalkulus Lambda]]
* [[Garis besar teori kategori]]
* [[Garis waktu teori kategori dan matematika terkait]]
{{colend}}

== Referensi ==
=== Kutipan ===
{{Reflist}}

=== Sumber ===
* {{cite book |title = Abstract and Concrete Categories |last1 = Adámek |first1 = Jiří |last2 = Herrlich |first2 = Horst |author2-link = Horst Herrlich |last3 = Strecker |first3 = George E. |publisher = Heldermann Verlag Berlin |year = 2004 |url = http://katmat.math.uni-bremen.de/acc/acc.htm |access-date = 2020-11-26 |archive-date = 2021-02-24 |archive-url = https://web.archive.org/web/20210224031815/http://katmat.math.uni-bremen.de/acc/acc.htm |dead-url = yes }}
* {{citation
 | last1 = Barr
 | first1 = Michael
 | author1-link = Michael Barr (mathematician)
 | last2 = Wells
 | first2 = Charles
 | author2-link = Charles Wells (mathematician)
 | edition = 3rd
 | series = Reprints in Theory and Applications of Categories
 | title = Category Theory for Computing Science
 | url = http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html
 | volume = 22
 | year = 2012 |origyear=1995
 }}.
* {{citation
 | last1 = Barr | first1 = Michael | author1-link = Michael Barr (mathematician)
 | last2 = Wells | first2 = Charles | author2-link = Charles Wells (mathematician)
 | series = Reprints in Theory and Applications of Categories
 | mr = 2178101
 | title = Toposes, Triples and Theories
 | url = http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html
 | volume = 12
 | year = 2005
}}.
* {{cite book
 | title = Handbook of categorical algebra
 | publisher = Cambridge University Press
 | year = 1994
 | series = Encyclopedia of Mathematics and its Applications |pages=50–52
 | last1 = Borceux
 | first1 = Francis 
 | url = https://books.google.com/books?id=YfzImoopB-IC&q=%22Handbook+of+categorical+algebra%22&pg=PP1
| isbn = 9780521441780
 }}
* {{cite book |last=Freyd |first=Peter J. |title = Abelian Categories |origyear=1964 |url = http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html |authorlink=Peter J. Freyd |year=2003 |series=Reprints in Theory and Applications of Categories |volume=3}}
* {{cite book |title = Categories, allegories |publisher=North Holland |year=1990 |series=North Holland Mathematical Library |volume=39 |last1=Freyd  |first1=Peter J. |author1-link=Peter J. Freyd  |last2=Scedrov  |first2=Andre |url = https://books.google.com/books?id=fCSJRegkKdoC |isbn = 978-0-08-088701-2 }}
* {{cite book |first=Robert |last=Goldblatt |authorlink=Robert Goldblatt |title = Topoi: The Categorial Analysis of Logic |url = https://books.google.com/books?id=AwLc-12-7LMC |year=2006 |publisher=Dover |isbn = 978-0-486-45026-1 |volume=94 |series=Studies in logic and the foundations of mathematics |origyear = 1979 }}
* {{cite book |last1= Herrlich |first1= Horst |author1-link=Horst Herrlich |last2=Strecker |first2=George E. |year=2007 |edition=3rd |title = Category Theory |url= https://archive.org/details/categorytheoryin0000herr |publisher = Heldermann Verlag Berlin |isbn = 978-3-88538-001-6}}.
* {{cite book |first1=Masaki |last1=Kashiwara |author1-link=Masaki Kashiwara |first2=Pierre |last2=Schapira |author2-link=Pierre Schapira (mathematician) |title = Categories and Sheaves |url = https://books.google.com/books?id=K-SjOw_2gXwC |year=2006 |publisher=Springer |isbn = 978-3-540-27949-5 |volume=332 |series=Grundlehren der Mathematischen Wissenschaften }}
* {{cite book |first1=F. William |last1=Lawvere |author1-link=William Lawvere |first2=Robert |last2=Rosebrugh |title = Sets for Mathematics |url = https://archive.org/details/setsformathemati0000lawv |url-access=registration |year=2003 |publisher=Cambridge University Press |isbn = 978-0-521-01060-3 }}
* {{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 |origyear=1997 }}
* {{cite book |last=Leinster |first=Tom |title=Higher Operads, Higher Categories |journal=Higher Operads |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 |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 |authorlink=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 |authorlink=Saunders Mac Lane |isbn=978-0-387-98403-2 |ref=harv | 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 |author2link=Garrett Birkhoff |publisher=Chelsea |year=1999| edition=2nd |isbn = 978-0-8218-1646-2 |origyear=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 |last=May |first=Peter |authorlink=J. Peter May |title=A Concise Course in Algebraic Topology|publisher=University of Chicago Press|year=1999 |isbn=978-0-226-51183-2}}
* {{cite book |first=Guerino |last=Mazzola |authorlink=Guerino Mazzola |title=The Topos of Music, Geometric Logic of Concepts, Theory, and Performance |url=https://archive.org/details/toposofmusicgeom0000mazz |publisher=Birkhäuser |location= |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 |authorlink=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]].
* {{cite book |title = Homotopy theory of higher categories |url = https://archive.org/details/arxiv-1001.4071 |last=Simpson |first=Carlos |authorlink=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 |accessdate = 11 December 2009 }} Based on {{harvnb|Mac Lane|1998}}.

== Bacaan lebih lanjut ==
* {{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}}

== Pranala luar ==
{{Commons category}}
{{wikiquote}}
* [http://www.tac.mta.ca/tac/ Theory and Application of Categories], an electronic journal of category theory, full text, free, since 1995.
* [http://ncatlab.org/nlab nLab], sebuah proyek wiki tentang matematika, fisika dan filsafat dengan penekanan pada sudut pandang '' n '' - kategorikal.
* [https://golem.ph.utexas.edu/category/ The n-Category Café], pada dasarnya sebuah kolokium tentang topik-topik dalam teori kategori. 
* [http://www.logicmatters.net/categories/ Category Theory], halaman web yang berisi tautan ke catatan kuliah dan buku-buku tentang teori kategori yang tersedia secara gratis.
* {{citation |first=Chris |last=Hillman |title=A Categorical Primer |citeseerx=10.1.1.24.3264}}, pengantar formal untuk teori kategori.
* {{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 |access-date=2020-11-26 |archive-date=2015-04-21 |archive-url=https://web.archive.org/web/20150421081851/http://katmat.math.uni-bremen.de/acc/acc.pdf |dead-url=yes }}
* {{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 |website= |publisher= |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.]

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