Obiekty początkowy i końcowy

Obiekt początkowy (końcowy) – dla ustalonej kategorii obiekt o tej własności, że dla każdego obiektu tej kategorii istnieje dokładnie jeden morfizm (odpowiednio ). Obiekty początkowy i końcowy danej kategorii, o ile tylko istnieją, są wyznaczone jednoznacznie z dokładnością do (jedynego) izomorfizmu. Obiekt, który jest jednocześnie początkowy i końcowy, nazywany jest obiektem zerowym kategorii

Przykłady

  • Zbiór pusty jest obiektem początkowym w kategorii wszystkich zbiorów. Każdy zbiór jednoelementowy jest obiektem końcowym tej kategorii.
  • W kategorii wszystkich grup obiektem początkowym, a zarazem końcowym (a więc zerowym), jest grupa jednoelementowa.
  • W kategorii punktowanych przestrzeni topologicznych (tj. przestrzeni z wyróżnionym punktem, w której od morfizmów wymagamy, by przeprowadzały wyróżnione punkty na wyróżnione punkty), obiektem zerowym jest przestrzeń jednopunktowa.
  • W kategorii wszystkich pierścieni z jedynką obiektem początkowym jest pierścień liczb całkowitych, obiektem końcowym natomiast pierścień zerowy.
  • Każdy zbiór częściowo uporządkowany może być rozpatrywany jako kategoria, której obiektami są elementy zbioru Powiemy, że istnieje morfizm między elementami wtedy i tylko wtedy, gdy Kategoria ta ma obiekt początkowy (końcowy) wtedy i tylko wtedy, gdy w zbiorze istnieje element najmniejszy (odpowiednio największy).

Bibliografia

Linki zewnętrzne

  • publikacja w otwartym dostępie – możesz ją przeczytać Final object (ang.), Encyclopedia of Mathematics, encyclopediaofmath.org [dostęp 2024-04-05].

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