범주의 동치

범주론에서, 두 범주 사이의 동치(同値, 영어: equivalence (of categories))는 두 범주가 사실상 같은 구조를 지니게 하는 함자이다. 범주의 동형보다 더 약한 개념이며, 범주의 동형보다 더 널리 쓰인다.

정의

두 범주 ${\mathcal {C}}$ 와 ${\mathcal {D}}$ 사이의 함자 $F\colon {\mathcal {C}}\to {\mathcal {D}}$ 에 대하여, 다음 두 조건이 서로 필요충분조건이며, 이 조건을 만족시키는 $F$ 를 ${\mathcal {C}}$ 와 ${\mathcal {D}}$ 사이의 동치라고 한다.

$F^{-1}F$ 와 $FF^{-1}$ 가 항등 함자와 자연 동형인 함자 $F^{-1}\colon {\mathcal {D}}\to {\mathcal {C}}$ 가 존재한다 (이러한 함자는 유일하지 않을 수 있다).
충실충만한 함자이며, 본질적 전사 함자(영어: essentially surjective functor)이다 (즉, ${\mathcal {D}}$ 의 모든 원소는 적어도 하나의 $F$ 의 상의 원소와 동형이다).

성질

서로 동치인 두 범주는 범주론에서 다루는 거의 모든 성질이 같다. 예를 들어, 동치는 모든 극한과 쌍대극한을 보존하며, 또한 전사 사상 및 단사 사상을 보존한다.

작은 범주

범주 ${\mathcal {C}}$ 의 부분 범주 ${\mathcal {D}}$ 에 대하여, 다음 두 조건이 서로 필요충분조건이며, 이를 만족시키는 ${\mathcal {D}}$ 를 ${\mathcal {C}}$ 의 뼈대(영어: skeleton)라고 한다.

충만한 부분 범주이며, ${\mathcal {C}}$ 의 임의의 대상은 정확히 하나의 ${\mathcal {D}}$ 의 대상과 동형이다.
포함 함자 ${\mathcal {D}}\to {\mathcal {C}}$ 는 동치이며, 뼈대 범주(영어: skeletal category)이다 (즉, ${\mathcal {D}}$ 의 임의의 서로 다른 두 대상은 동형이 아니다).

모든 작은 범주는 뼈대를 가지며, 이는 범주의 동형 아래 유일하다. 이는 선택 공리와 동치이다. 또한, 두 작은 범주 ${\mathcal {C}}$ 와 ${\mathcal {D}}$ 에 대하여, 다음 두 조건이 서로 필요충분조건이다.

${\mathcal {C}}$ 와 ${\mathcal {D}}$ 는 서로 동치이다.
${\mathcal {C}}$ 와 ${\mathcal {D}}$ 의 뼈대는 서로 동형이다.

참고 문헌

Mac Lane, Saunders (1998). 《Categories for the Working Mathematician》. Graduate Texts in Mathematics 5 2판. Springer. ISBN 0-387-98403-8. CS1 관리 - 추가 문구 (링크)

외부 링크

“Equivalence of categories”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. ISBN 978-1-55608-010-4.
“Skeleton of a category”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. ISBN 978-1-55608-010-4.
“Equivalence of categories”. 《nLab》 (영어).
“Equivalence”. 《nLab》 (영어).
“Skeleton”. 《nLab》 (영어).

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