Categoria vírgula

Na teoria das categorias, uma categoria vírgula (em inglês, comma category) é uma categoria cujos objetos correspondem a certos morfismos de outra categoria. Sua definição foi introduzida por William Lawvere, em 1963; o nome provém de uma de suas notações, que usa o sinal de pontuação vírgula.[1][2]

Definição

Para quaisquer functores F : A → C e G : B → C, pode-se formar a categoria de vírgula F ↓ G,[3] também denotada por (F ↓ G) e por (F, G),[4] para a qual:

  • a coleção de objetos consiste nas triplas (a, b, f), em que a é objeto de A, b é objeto de B, e f : F(a) → G(b) é morfismo em C;
  • a coleção de morfismos do objeto (a, b, f) ao objeto (a′, b′, f′) consiste nas duplas (h, k), em que h : a → a′ é morfismo em A e k : b → b′ é morfismo em B, satisfazendo f′ ∘ F(h) = G(k) ∘ f, condição representada no diagrama comutativo:
  • as identidades e a operação de composição são dadas por:

Exemplos

  • Se F : 1 → Set for o functor levando o único objeto da categoria com um objeto e sem morfismos além da identidade ao conjunto de um elemento, e se G : B → Set é qualquer functor, então F ↓ G é isomorfa à categoria de elementos ∫ G.[5]
  • Generalizando o exemplo anterior, se F : 1 → C e G : B → C são functores quaisquer, F ↓ G é chamada categoria de setas do objeto a ao functor G, onde a é a imagem de F no único objeto de 1, e é denotada habitualmente por a ↓ G.[5]
  • Denotando-se por AnelC a categoria dos anéis comutativos, para cada K ∈ AnelC, F ↓ G, em que F : 1 → AnelC é o único functor com imagem K e G : AnelC → AnelC é o functor identidade, é isomorfa à categoria de álgebras comutativas sobre K. Em vez de F ↓ G, pode-se escrever K ↓ AnelC.[4]

Limites e colimites

Dados functores F : A → C e G : B → C, a categoria vírgula F ↓ G:

  • é completa desde que A e B sejam completas e G seja functor contínuo;[6]
  • é cocompleta desde que A e B sejam cocompletas e F seja functor cocontínuo.[6][7]

Referências

  1. ↑ (Mac Lane, §II.notas)
  2. ↑ (Lawvere, §A.3.1, pgs. 12–13): "Unfortunately, I did not suggest a name for the operation, so due to the need for reading it somehow or other, it rather distressingly came to be known by the subjective name 'comma category', even when it came to be also denoted by a vertical arrow in place of the comma."
  3. ↑ (Riehl, Exercício 1.3.vi)
  4. ↑ a b (Mac Lane, §II.6)
  5. ↑ a b (Mac Lane, §III.1)
  6. ↑ a b «Comma category – nLab». Consultado em 3 de dezembro de 2020 
  7. ↑ (Rydeheard, Burstall 1988, §5.2)

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