Filtered category
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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
and 
in J, - for every two parallel arrows
in J, there exists an object k and an arrow 
such that wu = wv.
A filtered colimit is a colimit of a functor 
where J is a filtered category.
[edit] Cofiltered categories
There is a dual notion of cofiltered category. A category J is cofiltered if the opposite category Jop 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
and 
in J, - for every two parallel arrows
in J, there exists an object k and an arrow 
such that uw = vw.
A (co)filtered limit is a limit of a functor where J is a cofiltered category.
