Filtered category


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:j o k and f':j' o k in J,
* for every two parallel arrows u,v:i o j in J, there exists an object k and an arrow w:j o k such that wu=wv.

A filtered colimit is a colimit of a functor F:J o 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:k o j and f':k o j' in J,
* for every two parallel arrows u,v:j o i in J, there exists an object k and an arrow w:k o j such that uw=vw.

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


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