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filtered in

Filtered category

In category theory, filtered categories generalize the notion of directed set.

A category J is filtered when

  • it is not empty,
  • for every two objects j and j' in J there exists an object k and two arrows f:jto k and f':j'to k in J,
  • for every two parallel arrows u,v:ito j in J, there exists an object k and an arrow w:jto k such that wu=wv.

A filtered colimit is a colimit of a functor F:Jto C where J is a filtered category.

Cofiltered categories

There is a dual notion of cofiltered category. A category J is cofiltered if the opposite category J^{mathrm{op}} is filtered. In detail, a category is cofiltered when

  • it is not empty
  • for every two objects j and j' in J there exists an object k and two arrows f:kto j and f':k to j' in J,
  • for every two parallel arrows u,v:jto i in J, there exists an object k and an arrow w:kto j such that uw=vw.

A (co)filtered limit is a limit of a functor F:J to C where J is a cofiltered category.

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