#noindex
[[대상,object]]과 [[사상,morphism]]들로 이루어진 체계같은건데
보통 사상을 [[화살표,arrow]]로 나타내는?

category에서 [[사상,morphism]]의 조건?
 * 사상은 [[합성,composition]]이 가능해야 함 (morphisms must be composable)
 * 사상은 자기 자신으로 가는 [[항등사상,identity_morphism]]이 존재해야 함 // (curr at [[아이덴티티,identity]])
 * associativity

----
Informal definition[* The Language of Categories | Category Theory and Why We Care 1.1 https://youtu.be/5Ykrfqrxc8o?si=cYlD5dDwmODCbMmm&t=58]
 * [[대상,object]]과 $(A,B,C,\ldots)$
 * arrow([[화살,arrow]] or [[화살표,arrow]]) $(f,g,\ldots)$
가 있는데, 다음 법칙들을 만족.
 * composition - [[합성,composition]]
 * associativity - (maybe [[결합성,associativity]]? curr. [[VG:결합법칙,associativity]])
 * identity - (curr at [[아이덴티티,identity]])

----
MKL

Pagename TBD

F-대수,F-algebra ?? 
F대수,F-algebra ?? <- pagename: ,의 왼쪽에는 _를 고유명사(? 인명?) 좌우 외에는 안 쓰기로 했는데, -도 쓰지 말까?
[[F대수,F-algebra]] =F대수,F-algebra =,F-algebra F대수 F-algebra
{
아님 대수 대신 대수학 ?
https://en.wikipedia.org/wiki/F-algebra
https://ja.wikipedia.org/wiki/F代数
} // F-algebra ... NN:F-algebra Ggl:F-algebra

의 [[쌍대,dual]]인 // [[쌍대성,duality]]

F-coalgebra
F-쌍대대수 ??
KMS는 KmsE:coalgebra = 쌍대대수 이긴 한데.
https://en.wikipedia.org/wiki/F-coalgebra
https://ja.wikipedia.org/wiki/F余代数
// F-coalgebra ... NN:F-coalgebra Ggl:F-coalgebra

QQQ F의 정확한 의미... sigma-algebra의 sigma 비슷해 보이는데

암튼 rel [[코,co]] or [[쌍대,dual]] or [[쌍대성,duality]]
{
rel
[[coinduction]] =,coinduction . coinduction
 {
 '''coinduction'''
 쌍대귀납 ? 

 MKL [[귀납,induction]]

 https://en.wikipedia.org/wiki/Coinduction
 KmsE:coinduction 없음. (2023-11) [[귀납,induction]] ... 에 대한 [[쌍대귀납,coinduction]]???
 } // coinduction ... NN:coinduction Ggl:coinduction // 쌍대귀납 ... NN:쌍대귀납 Ggl:쌍대귀납
} 

coalgebra 말고도 dialgebra 라는 게
이건 algebra + coalgebra 모두를 일반화 한. "In abstract algebra, a dialgebra is the generalization of both algebra and coalgebra."
https://en.wikipedia.org/wiki/Dialgebra
https://ncatlab.org/nlab/show/dialgebra

----
[[범주,category]]의 [[동치,equivalence]]: [[category_equivalence]] - w

[[범주론,category_theory]] =범주론,category_theory =,category_theory .
{
WtEn:category_theory 

Sites
Maths - Category Theory - Martin Baker
https://www.euclideanspace.com/maths/discrete/category/index.htm

Videos
Tutorial on Category Theory: Part 1 – Pure and Classical - YouTube
https://www.youtube.com/watch?v=6eWn9nG5d7o


Topics
[[함수,function]]
[[대상,object]]
[[모노이드,monoid]]
[[모노이드대상,monoid_object]]
[[모나드,monad]] ? =모나드, =,monad . monad
{
KmsE:monad

https://www.euclideanspace.com/maths/discrete/category/higher/monad/index.htm


[[WtEn:monad]]
= https://en.wiktionary.org/wiki/monad
 (4.) "A monoid object in the category of endofunctors of a fixed category."

[[WpEn:Monad]] = https://en.wikipedia.org/wiki/Monad
esp
[[WpEn:Monad_(category_theory)]] = https://en.wikipedia.org/wiki/Monad_(category_theory)

[[WpKo:모나드_(범주론)]] = https://ko.wikipedia.org/wiki/모나드_(범주론)
[[WpJa:モナド_(圏論)]] = https://ja.wikipedia.org/wiki/モナド_(圏論)
Cmp [[모노이드,monoid]]
} // monad Ggl:"monad category theory" Ggl:"범주론 monad"
[[comonad]] =,comonad =,comonad . comonad
{
[[WtEn:comonad]] = https://en.wiktionary.org/wiki/comonad
 "A monad of the opposite category."
KmsE:comonad ?
Ggl:comonad
} // comonad Ggl:"comonad category theory" Ggl:"범주론 comonad"

[[범주,category]]의 [[동치,equivalence]]: [[category_equivalence]] - w


[[map]] / [[mapping]] ... [[사상,map]](vg)


[[functor]] =,functor . [[펑터,functor]] or [[함자,functor]] ....
{
'''functor'''
KmsE:functor
WtEn:functor
함자
펑터
Cmp [[함수,function]]
Cmp [[pseudofunctor]] =,pseudofunctor =,pseudofunctor . pseudofunctor { pseudofunctor Ggl:pseudofunctor pseudofunctor }
Bing:펑터 Ggl:펑터
} functor Ggl:"functor"

----
[[morphism]] =,morphism =,morphism . 사상 morphism ... [[모피즘,morphism]]? [[사상,morphism]]? 
{
KmsE:morphism
[[WtEn:morphism]] = 
 * 형태([[셰이프,shape]] [[피겨,figure]] ....)에 대한 것. 이건 몇 개 안 됨.
  * ex. [[다형성,polymorphism]] [[단형성,monomorphism]]
 * 수학의 일반적인 뜻(esp 범주, 함수, 사상) - 보통 [[사상,morphism]]으로 번역되는 - [[사상,map]] [[사상,mapping]] [[함수,function]] [[연산,operation]] [[연산자,operator]] [[변환,transformation]](curr [[변환]])과 거의 같은 그거
  * ex. (수없이 많이 추가될것이므로 아래에)
로 나눌 수 있음


사상
모피즘 ? // transliteration: morphism-모피즘 맞는듯, via '이소모피즘-isomorphism' via kornorms. / 근데 저 예 하나는 아이소에 가깝지 않나. 영어로는? - NdEn:isomorphism : "/àisəmɔ́:rfizm/". Yes, 아이소에 가까움.


MKL
[[동형사상,isomorphism]]
[[자기동형사상,automorphism]]
[[자기사상,endomorphism]]
[[준동형사상,homomorphism]] // [[WtEn:homomorphism]] = https://en.wiktionary.org/wiki/homomorphism
[[monomorphism]] =,monomorphism =,monomorphism . monomorphism // [[단형성,monomorphism]] 말고
 {
 '''monomorphism'''
 [[WtEn:monomorphism]] = https://en.wiktionary.org/wiki/monomorphism
  1. 수학에선 " an injective homomorphism"
  2. 생물학에선 sexual_dimorphism (sexual WtEn:dimorphism )의 absense. 저건뭐냐?
   Ndict:"sexual dimorphism" = https://terms.naver.com/search.naver?query=sexual+dimorphism
    즉 암수의 뚜렷한 형태 차이점? chk

 WpEn:Monomorphism = 666666666666666666666777777777777 ??
 ...
 Naver:monomorphism
 Ggl:monomorphism
 Bing:"definition of monomorphism"
 } // monomorphism (단형성 말고)

[[bimorphism]] =,bimorphism =,bimorphism . bimorphism
 {
 KmsE:bimorphism = https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=bimorphism x [[Date(2023-11-14T09:16:39)]]
 WtEn:bimorphism = https://en.wiktionary.org/wiki/bimorphism
  "A morphism which is both a monomorphism and an epimorphism." 
 Ndict:bimorphism ?
 Ggl:bimorphism
 } // bimorphism

(그럼 mono-bi 다음 multimorphism ? Ggl:multimorphism )
 
ADDMORPHISMHERE
ADDMORPHISMHERE
ADDMORPHISMHERE
ADD_MORPHISM_HERE
ADD_MORPHISM_HERE
ADD_MORPHISM_HERE


RANDOMLINKS TOCLEANUP
[[WtEn:classifying_morphism]]
= https://en.wiktionary.org/wiki/classifying_morphism
 ... [[분류,classification]]?

...
Ndict:morphism
Ggl:morphism
} // morphism

----
//이하 [[연산,operation]] esp [[이항연산,binary_operator]]
//and [[연산자,operator]](=[[작용소,operator]]) esp [[이항연산자,binary_operator]](=[[이항작용소,binary_operator]]) 와 mkl
[[결합법칙,associativity]] (vg) =,associativity =결합 .... [[결합성,associativity]] [[결합법칙,commutative_law]] ? or rule?
{
'''associativity'''
'''결합성?'''

KmsE:associativ
WtEn:associativity


https://ko.wikipedia.org/wiki/결합법칙
https://en.wikipedia.org/wiki/Associative_property
} // associativity 

[[교환법칙,commutativity]] (vg) =,commutativity =교환 .... [[교환성,commutativity]] [[교환법칙,commutative_law]] or rule?
{
'''commutativity'''
'''교환성'''? 가환성?

KmsE:commutativ
WtEn:commutativity
mkl
가환
 KmsK:가환 

Sub:
[[commmutative_diagram]] - isa [[다이어그램,diagram]] or [[그림,diagram]]

https://en.wikipedia.org/wiki/Commutative_property
} // commutativity ... NN:commutativity Ggl:commutativity

----
<<tableofcontents>>

= bmks ko =

tmp bmks ko
GitHub - pilgwon/CategoryTheory: ![번역] 프로그래머를 위한 카테고리 이론 (Category Theory for Programmers)
https://github.com/pilgwon/CategoryTheory

Bartosz Milewski's "Category Theory for Programmers" Korean translation
프로그래머를 위한 범주론
"본 레파지토리는 Bartosz Milewsk의 Category Theory for Programmers을 번역하며 학습한 레파지토리입니다."
https://github.com/alstn2468/category-theory-for-programmers

https://theworldaswillandidea.tistory.com/category/Series/함수적인%2C%20너무나%20함수적인

= bmks en =
Category theory: online lecture notes, etc. - Logic Matters
https://www.logicmatters.net/categories/


= books? =
Naver:범주론+교재
Naver:범주론+서적
Naver:카테고리이론+교재
Naver:카테고리이론+서적

Ggl:범주론+교재
Ggl:범주론+서적
Ggl:카테고리이론+교재
Ggl:카테고리이론+서적
Ggl:category+theory+book+recommendation

Bing:범주론+교재
Bing:범주론+서적
Bing:카테고리이론+교재
Bing:카테고리이론+서적
Bing:category+theory+book+recommendation

= tmp videos en / category theory =
Category Theory in Life - Eugenia Cheng
https://youtu.be/ho7oagHeqNc?si=YQUeKoKJJzgLQp-H

The Language of Categories | Category Theory and Why We Care 1.1 - YouTube
https://www.youtube.com/watch?v=5Ykrfqrxc8o


...
YouTube:범주론
YouTube:카테고리이론

----
[[NoSmoke:CategoryTheory]]
= pppppppppp

[[Wiki:CategoryTheory]]
= http://wiki.c2.com/?CategoryTheory

https://wiki.haskell.org/Category_theory
} // category theory

[[mathematical_structure]] - 수학적 구조(mathematical_structure), [[수학,math]] curr at [[구조,structure?action=highlight&value=mathematical_structure]]

Sub:
[[opposite_category]] = [[dual_category]] ? // =,opposite_category opposite_category / =,dual_category dual_category
{
'''opposite category'''
(tmp) kms opposite => https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=opposite

WtEn:dual_category ?
WtEn:opposite_category x [[Date(2024-01-12T13:57:33)]]

https://artofproblemsolving.com/wiki/index.php/Opposite_category
[[WpEn:Opposite_category]]
 = https://en.wikipedia.org/wiki/Opposite_category
 보면 aka [[dual_category]] ... [[쌍대,dual]] [[쌍대성,duality]]
... Google:opposite_category Naver:opposite_category

https://encyclopediaofmath.org/wiki/Dual_category

} // opposite category Ggl:"opposite category"

[[small_category]] =,small_category . small_category
{
'''small category'''
[[WtEn:small_category]] = https://en.wiktionary.org/wiki/small_category
https://encyclopediaofmath.org/wiki/Small_category
} // small category Ggl:"small category"

[[closed_category]] =,closed_category . closed_category
{
'''closed category'''
WtEn:closed_category ? x [[Date(2024-01-12T13:57:33)]]

https://encyclopediaofmath.org/wiki/Closed_category
https://ko.wikipedia.org/wiki/데카르트_닫힌_범주 "Cartesian closed category, 약자 CCC"

} // closed category Ggl:"closed category" Ggl:"닫힌 범주" Naver:"closed category" Naver:"닫힌 범주"

[[bicategory]] =,bicategory =,bicategory . bicategory
{
https://ko.wikipedia.org/wiki/이차_범주
https://en.wikipedia.org/wiki/Bicategory
https://encyclopediaofmath.org/wiki/Bicategory
https://ncatlab.org/nlab/show/bicategory
} // bicategory ... Google:bicategory

[[quotient_category]]
 {
 '''quotient category'''
 MKLINK [[quotient_object]] [[몫,quotient]] https://encyclopediaofmath.org/wiki/Quotient_category
 } // quotient category Ggl:"quotient category"
[[topologized_category]] => https://encyclopediaofmath.org/wiki/Site
[[Abelian_category]] { https://mathworld.wolfram.com/AbelianCategory.html https://encyclopediaofmath.org/wiki/Abelian_category }
[[Grothendieck_category]] { https://encyclopediaofmath.org/wiki/Grothendieck_category }
[[derived_category]] =,derived_category . derived_category { '''derived category''' https://encyclopediaofmath.org/wiki/Derived_category } // derived category Ggl:"derived category"
[[additive_category]] =,additive_category . additive_category
{
'''additive category'''
[[MathWorld:AdditiveCategory]] = https://mathworld.wolfram.com/AdditiveCategory.html
https://encyclopediaofmath.org/wiki/Additive_category
} // additive category Ggl:"additive category"
[[subcategory]] =,subcategory . subcategory - w
{
'''subcategory'''
부범주? 부분범주?
https://mathworld.wolfram.com/Subcategory.html
[[WtEn:subcategory]] = https://en.wiktionary.org/wiki/subcategory
https://ncatlab.org/nlab/show/subcategory
} // subcategory Ggl:subcategory
[[supercategory]] =,supercategory . supercategory - w
{
'''supercategory'''
[[WtEn:supercategory]] = https://en.wiktionary.org/wiki/supercategory
} // supercategory Ggl:supercategory

[[metacategory]] =,metacategory =,metacategory . metacategory
{
'''metacategory'''
메타범주?

WtEn:metacategory
Ndict:metacategory

MKLINK [[metagraph]] =,metagraph =,metagraph . metagraph { https://proofwiki.org/wiki/Definition:Metagraph } [[메타,meta]]
https://proofwiki.org/wiki/Definition:Metacategory
} // metacategory Ggl:metacategory
[[distributive_lattice]] =,distributive_lattice . distributive_lattice
 {
 '''distributive lattice'''
 WtEn:distributive_lattice
 https://ncatlab.org/nlab/show/distributive+lattice
 KmsE:"distributive lattice": 분배격자
 } // distributive lattice ... Naver:distributive+lattice Ggl:distributive+lattice

= MKLINK =
[[대상,object]]

= Topics =
== categorification ==
[[WtEn:categorification]]
= https://en.wiktionary.org/wiki/categorification
 "A procedure that defines theorems,,[[정리,theorem]],, in terms of category theory by mapping concepts from set theory,,[[집합론,set_theory]],, to category theory^^[[범주론,category_theory]]^^."

== decategorification ==
decategorification
[[decategorification]] =,decategorification =,decategorification . decategorification
{

https://ncatlab.org/nlab/show/decategorification
 "process which turns a category into a set "

WtEn:decategorification
Ggl:decategorification
decategorification
}

= 이상엽 =
// from https://www.youtube.com/watch?v=aggoIxEkr6Q 범주론 category theory 이란?

(0:29)

범주란?

범주의 두 요소
 1. 대상의 모임
 1. 사상의 모임
정확히 얘기하면
 1. [[대상,object]]의 모임 ob(C)
 1. 임의의 두 대상 X,Y∈ob(C)에 대해 X를 [[정의역,domain]], Y를 [[공역,codomain]]으로 하는 [[사상,map]] f:X→Y의 모임 hom(X,Y)

두 범주 사이에서 정의되는 사상은 [[함자,functor]]라고 부른다.

(6:50)

집합론적 함수와의 비교 // [[집합론,set_theory]]
 * 집합론이 [[집합,set]]을 주 대상으로 하고 이로부터 함수가 파생되는 구조를 가지고 있다면,
 범주론은 함수가 주 대상이고 이로부터 파생된 대상의 성질을 연구한다.
 * 범주의 대상은 집합일 필요가 없으며, [[사상,morphism]]도 [[함수,function]]일 필요가 없다.
 ex. [[자연수,natural_number]]([[대상,object]])에 대해 부등호 ≤는 사상일 수 있다. 즉 1≤2를 1→2로 볼 수 있다.

''(즉 사상(함수)이 집합론적 함수에서 벗어났다)''
''1≤2에서 ≤는 집합론에서 말하는 함수는 아니다. 근데 범주론에선 사상이다.''

= free/open textbooks =
http://xahlee.info/math/category_theory_tom_leinster.html

= files =
Category Theory / Randall R. Holmes / October 8, 2019
99p
https://web.auburn.edu/holmerr/8970/Textbook/CategoryTheory.pdf

Notes on Category Theory / with examples from basic mathematics / Paolo Perrone http://www.paoloperrone.org / Last update: February 2021
181p
https://arxiv.org/pdf/1912.10642.pdf


= 같은영단어 =
[[분류,category]]??
rel [[분류,classification]]

[[카테고리,category]] - via kornorms.

----
Twins:
[[https://artofproblemsolving.com/wiki/index.php/Category_(category_theory)]]

https://encyclopediaofmath.org/wiki/Category
https://mathworld.wolfram.com/Category.html

[[WpKo:범주_(수학)]]
= https://ko.wikipedia.org/wiki/범주_(수학)
= https://ko.wikipedia.org/wiki/범주_%28수학%29
 "는 추상적인 [[구조,structure]](와 이를 보존하는 변환의 개념을 형식화한 것이다"

----
Sub:

https://en.wiktionary.org/wiki/Kleisli_category