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

category에서 사상,morphism의 조건?

사상은 합성,composition이 가능해야 함 (morphisms must be composable)
사상은 자기 자신으로 가는 항등사상,identity_morphism이 존재해야 함 // (curr at 아이덴티티,identity)
associativity

Informal definition[1]

대상,object과 $\displaystyle (A,B,C,\ldots)$
arrow(애로,arrow - 화살,arrow or 화살표,arrow) $\displaystyle (f,g,\ldots)$

가 있는데, 다음 법칙들을 만족.

composition - 합성,composition
associativity - (maybe 결합성,associativity? curr. 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 ...

F-algebra

의 쌍대,dual인 // 쌍대성,duality

F-coalgebra
F-쌍대대수 ??
KMS는

coalgebra = 쌍대대수 이긴 한데.
https://en.wikipedia.org/wiki/F-coalgebra
https://ja.wikipedia.org/wiki/F余代数
// F-coalgebra ...

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

coinduction없음. (2023-11) 귀납,induction ... 에 대한 쌍대귀납,coinduction???
} // coinduction ...

coinduction

coinduction // 쌍대귀납 ...

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,category의 동치,equivalence: category_equivalence - w

범주론,category_theory =범주론,category_theory =,category_theory .
{

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
{

monad x 2014-07

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

monad
= https://en.wiktionary.org/wiki/monad

(4.) "A monoid object in the category of endofunctors of a fixed category."

Monad = https://en.wikipedia.org/wiki/Monad
esp

Monad_(category_theory) = https://en.wikipedia.org/wiki/Monad_(category_theory)

모나드_(범주론) = https://ko.wikipedia.org/wiki/모나드_(범주론)

モナド_(圏論) = https://ja.wikipedia.org/wiki/モナド_(圏論)

https://ncatlab.org/nlab/show/monad
https://ncatlab.org/nlab/show/monad (disambiguation)

하나,one

https://ncatlab.org/nlab/show/monad (in computer science)
입출력,IO - https://ncatlab.org/nlab/show/IO-monad

Cmp 모노이드,monoid

} // monad

monad category theory

범주론 monad
comonad =,comonad =,comonad . comonad
{

comonad = https://en.wiktionary.org/wiki/comonad

"A monad of the opposite category."

comonad ?

comonad
} // comonad

comonad category theory

범주론 comonad

범주,category의 동치,equivalence: category_equivalence - w

map / mapping ... 사상,map(vg)

functor =,functor . 펑터,functor or 함자,functor ....
{
functor

functor

functor
함자
펑터
Cmp 함수,function
Cmp pseudofunctor =,pseudofunctor =,pseudofunctor . pseudofunctor { pseudofunctor

pseudofunctor pseudofunctor }

펑터

펑터
} functor

functor

morphism =,morphism =,morphism . 사상 morphism ... 모피즘,morphism? 사상,morphism?
{

morphism

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,morphism으로 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? -

isomorphism : "/àisəmɔ́:rfizm/". Yes, 아이소에 가까움.

MKL
동형사상,isomorphism
자기동형사상,automorphism
자기사상,endomorphism
준동형사상,homomorphism //

homomorphism = https://en.wiktionary.org/wiki/homomorphism
monomorphism =,monomorphism =,monomorphism . monomorphism // 단형성,monomorphism 말고

{
monomorphism

monomorphism = https://en.wiktionary.org/wiki/monomorphism

수학에선 " an injective homomorphism"
생물학에선 sexual_dimorphism (sexual dimorphism )의 absense. 저건뭐냐?

sexual dimorphism = https://terms.naver.com/search.naver?query=sexual dimorphism

즉 암수의 뚜렷한 형태 차이점? chk

Monomorphism = 666666666666666666666777777777777 ??
...

monomorphism

definition of monomorphism
} // monomorphism (단형성 말고)

bimorphism =,bimorphism =,bimorphism . bimorphism

{

bimorphism = https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=bimorphism x 2023-11-14

bimorphism = https://en.wiktionary.org/wiki/bimorphism

"A morphism which is both a monomorphism and an epimorphism."

bimorphism ?

bimorphism
} // bimorphism

(그럼 mono-bi 다음 multimorphism ?

multimorphism )

ADDMORPHISMHERE
ADDMORPHISMHERE
ADDMORPHISMHERE
ADD_MORPHISM_HERE
ADD_MORPHISM_HERE
ADD_MORPHISM_HERE

RANDOMLINKS TOCLEANUP

classifying_morphism
= https://en.wiktionary.org/wiki/classifying_morphism

... 분류,classification?

...

morphism

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?

associativ

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?

commutativ

commutativity
mkl
가환

가환

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

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

4. tmp videos en / category theory

5. MKLINK

6. Topics

6.1. categorification
6.2. decategorification

7. 이상엽

8. free/open textbooks

9. files

10. 같은영단어

[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

https://theworldaswillandidea.tistory.com/category/Series/함수적인, 너무나 함수적인

[edit]

2. bmks en ¶

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

[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

...

범주론

카테고리이론

CategoryTheory
= pppppppppp

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

dual_category ?

opposite_category x 2024-01-12

https://artofproblemsolving.com/wiki/index.php/Opposite_category

Opposite_category

= https://en.wikipedia.org/wiki/Opposite_category
보면 aka dual_category ... 쌍대,dual 쌍대성,duality

...

opposite_category

https://encyclopediaofmath.org/wiki/Dual_category

} // opposite category

opposite category

small_category =,small_category . small_category
{
small category

small_category = https://en.wiktionary.org/wiki/small_category
https://encyclopediaofmath.org/wiki/Small_category
} // small category

small category

closed_category =,closed_category . closed_category
{
closed category

closed_category ? x 2024-01-12

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

} // closed category

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 ...

bicategory

완비범주,complete_category =완비범주,complete_category =,complete_category 완비범주 complete_category
{
Sub:
complete_small_category = small_complete_category ?
}

몫범주,quotient_category =몫범주,quotient_category =,quotient_category 몫범주 quotient_category

{
quotient category
몫범주 - 가 최선일 듯, determined 2024-07-05.
MKLINK quotient_object 몫,quotient https://encyclopediaofmath.org/wiki/Quotient_category
Up: 몫,quotient 범주,category
} // quotient category

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

derived category
additive_category =,additive_category =,additive_category . additive_category
{
additive category

AdditiveCategory = https://mathworld.wolfram.com/AdditiveCategory.html
https://encyclopediaofmath.org/wiki/Additive_category
} // additive category ....

additive category

additive category
subcategory =,subcategory =,subcategory . subcategory - w
{
subcategory
부범주? 부분범주?

https://mathworld.wolfram.com/Subcategory.html

subcategory = https://en.wiktionary.org/wiki/subcategory
https://ncatlab.org/nlab/show/subcategory
} // subcategory

subcategory
supercategory =,supercategory . supercategory - w
{
supercategory

supercategory = https://en.wiktionary.org/wiki/supercategory
} // supercategory

supercategory

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

metacategory

MKLINK metagraph =,metagraph =,metagraph . metagraph { https://proofwiki.org/wiki/Definition:Metagraph } 메타,meta
https://proofwiki.org/wiki/Definition:Metacategory
} // metacategory

metacategory
distributive_lattice =,distributive_lattice . distributive_lattice

{
distributive lattice

distributive_lattice
https://ncatlab.org/nlab/show/distributive lattice

distributive lattice: 분배격자
} // distributive lattice ...

distributive lattice

[edit]

5. MKLINK ¶

대상,object

[edit]

6. Topics ¶

[edit]

6.1. categorification ¶

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}."

[edit]

6.2. decategorification ¶

decategorification
decategorification =,decategorification =,decategorification . decategorification
{

https://ncatlab.org/nlab/show/decategorification

"process which turns a category into a set "

decategorification

decategorification
decategorification
}

[edit]

7. 이상엽 ¶

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

(0:29)

범주란?

범주의 두 요소

대상의 모임
사상의 모임

정확히 얘기하면

대상,object의 모임 ob(C)
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.

[edit]

8. free/open textbooks ¶

http://xahlee.info/math/category_theory_tom_leinster.html

[edit]

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

[edit]