범주,category

Difference between r1.33 and the current

@@ -4,32 +4,57 @@

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
* ...tbw
* 사상은 자기 자신으로 가는 [[항등사상,identity_morphism]]이 존재해야 함 // (curr at [[아이덴티티,identity]])
* associativity
----
mkl
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
https://en.wikipedia.org/wiki/Coinduction
[[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."
@@ -43,22 +68,22 @@
{
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]] =모노이드대상,monoid_object 모노이드대상,monoid_object 모노이드대상,monoid_object
[[모노이드대상,monoid_object]] 
[[모나드,monad]] ? =모나드, =,monad . monad
{
[[모노이드,monoid]]
[[대상,object]]
 
https://ko.wikipedia.org/wiki/모노이드_대상
https://ja.wikipedia.org/wiki/モノイド対象
}
[[모나드,monad]] ? =모나드, =,monad .
KmsE:monad

https://www.euclideanspace.com/maths/discrete/category/higher/monad/index.htm
@@ -67,20 +92,22 @@
[[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

@@ -90,15 +117,15 @@

[[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
}
Bing:펑터 Ggl:펑터
} functor Ggl:"functor"

----
[[morphism]] =,morphism =,morphism . 사상 morphism ... [[모피즘,morphism]]? [[사상,morphism]]?
@@ -136,6 +163,7 @@
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)]]
@@ -144,6 +172,8 @@
Ndict:bimorphism ?
Ggl:bimorphism
} // bimorphism
 
(그럼 mono-bi 다음 multimorphism ? Ggl:multimorphism )

ADDMORPHISMHERE
ADDMORPHISMHERE
@@ -156,6 +186,7 @@
RANDOMLINKS TOCLEANUP
[[WtEn:classifying_morphism]]
= https://en.wiktionary.org/wiki/classifying_morphism
... [[분류,classification]]?

...
Ndict:morphism
@@ -167,16 +198,22 @@
//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
@@ -187,7 +224,10 @@
[[commmutative_diagram]] - isa [[다이어그램,diagram]] or [[그림,diagram]]

https://en.wikipedia.org/wiki/Commutative_property
}
} // commutativity ... NN:commutativity Ggl:commutativity
 
----
<<tableofcontents>>

= bmks ko =

@@ -225,10 +265,14 @@
Bing:카테고리이론+서적
Bing:category+theory+book+recommendation

= watch yt =
= 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
@@ -241,17 +285,18 @@
= http://wiki.c2.com/?CategoryTheory

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

수학적 구조(mathematical_structure), [[수학,math]] [[구조,structure]]
[[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 ??
WtEn:opposite_category x [[Date(2024-01-12T13:57:33)]]

https://artofproblemsolving.com/wiki/index.php/Opposite_category
[[WpEn:Opposite_category]]
@@ -261,71 +306,81 @@

https://encyclopediaofmath.org/wiki/Dual_category

}
} // opposite category Ggl:"opposite category"
[[small_category]] =,small_category . small_category
{
WtEn:small_category ? qqqqqqqqqqq
'''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 ? ffffffff
'''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"
}
} // 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
... Google:bicategory
}
[[quotient_category]] { MKLINK [[quotient_object]] [[몫,quotient]] https://encyclopediaofmath.org/wiki/Quotient_category }
} // 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 
{ https://encyclopediaofmath.org/wiki/Derived_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
'''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 }
MKLINK [[metagraph]] =,metagraph =,metagraph . metagraph { https://proofwiki.org/wiki/Definition:Metagraph } [[메타,meta]]
https://proofwiki.org/wiki/Definition:Metacategory
}
distributive_lattice { https://ncatlab.org/nlab/show/distributive+lattice
KmsE:"distributive lattice": 분배격자
... 
Naver:distributive+lattice 
Ggl:distributive+lattice
}
} // 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]]


대상,object과 사상,morphism들로 이루어진 체계같은건데
보통 사상을 화살표,arrow로 나타내는?

category에서 사상,morphism의 조건?
Informal definition[1]
가 있는데, 다음 법칙들을 만족.
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


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:coinduction없음. (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


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











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 =
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 2023-11-14
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







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




[edit]

1. 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


[edit]

2. bmks en

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


[edit]

3. books?

[edit]

4. 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



NoSmoke:CategoryTheory
= pppppppppp




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




} // opposite category Ggl:opposite category


closed_category =,closed_category . closed_category
{
closed category
WtEn:closed_category ? x 2024-01-12





metacategory =,metacategory =,metacategory . metacategory
{
metacategory
메타범주?


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

[edit]

5. MKLINK

[edit]

6. Topics

[edit]

6.1. categorification

WtEn:categorification
= https://en.wiktionary.org/wiki/categorification
"A procedure that defines theorems정리,theorem in terms of category theory by mapping concepts from set theory집합론,set_theory to category theory범주론,category_theory."

[edit]

6.2. 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
}

[edit]

7. 이상엽

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

(0:29)

범주란?

범주의 두 요소
  1. 대상의 모임
  2. 사상의 모임
정확히 얘기하면
  1. 대상,object의 모임 ob(C)
  2. 임의의 두 대상 X,Y∈ob(C)에 대해 X를 정의역,domain, Y를 공역,codomain으로 하는 사상,map f:X→Y의 모임 hom(X,Y)

두 범주 사이에서 정의되는 사상은 함자,functor라고 부른다.

(6:50)

집합론적 함수와의 비교 // 집합론,set_theory
  • 집합론이 집합,set을 주 대상으로 하고 이로부터 함수가 파생되는 구조를 가지고 있다면,
    범주론은 함수가 주 대상이고 이로부터 파생된 대상의 성질을 연구한다.
  • 범주의 대상은 집합일 필요가 없으며, 사상,morphism도 함수,function일 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.

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8. free/open textbooks

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9. 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


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10. 같은영단어

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범주,category Modified on 9:30 pm, March 17, 2024.
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