The concept of filtered category is a categorification of the concept of directed set: In addition to having an upper bound (but not necessarily a coproduct) for every pair of objects, there must also be an upper bound (but not necessarily a coequaliser) for every pair of parallel morphisms.
A diagram where is a filtered category is called a filtered diagram. A colimit of a filtered diagram is called a filtered colimit.
The dual notion of filtered category is that of cofiltered category: a category whose opposite is filtered.
More in details, this requirement is that: For any finite category and any functor , there exists an object and a natural transformation where is the constant diagram at . If is the result of freely adjoining a terminal object to a category , then the condition is the same as that any functor with finite domain admits an extension .
Equivalently, filtered categories can be characterized as those categories where, for every finite diagram , the diagonal functor is final. This point of view can be generalized to other kinds of categories whose colimits are well-behaved with respect to a type of limit, such as sifted categories.
All this may be rephrased in more elementary terms by saying that:
There exists an object of (the case when )
For any two objects , there exists an object and morphisms and .
For any two parallel morphisms in , there exists a morphism such that .
Just as all finite colimits can be constructed from initial objects, binary coproducts, and coequalizers, so a cocone on any finite diagram can be constructed from these three.
In constructive mathematics, the elementary rephrasing above is equivalent to every Bishop-finite diagram admitting a cocone.
More generally, if is an infinite regular cardinal (or an arity class), then a -filtered category is one such that any diagram has a cocone when has arrows, or equivalently that any functor whose domain has fewer than morphisms admits an extension . The usual filtered categories are then the case , i.e., where the have fewer than morphisms (in other words are finite). (We could also say in this case “-filtered”, but -filtered is more usual in the literature.)
Note that a preorder is -filtered as a category just when it is -directed as a preorder.
In ABLR, they use the term -filtered for a category that is -filtered for every cardinal . Thus, an -filtered category is equivalently one in which every small diagram admits a cocone.
Even more generally, if is a class of small categories, a category is called -filtered if -colimits commute with -limits in Set. When is the class of all -small categories for an infinite regular cardinal , then -filteredness is the same as -filteredness as defined above. See ABLR.
If is the class consisting of the terminal category and the empty category — which is to say, the class of -small categories when is the finite regular cardinal — then being -filtered in this sense is equivalent to being connected. Note that this is not what the explicit definition given above for infinite regular cardinals would specialize to by simply setting (that would be simply inhabitation).
A filtered preorder is the same as a directed one: a filtered (0,1)-category.
Every category with a terminal object is filtered.
Every category which has finite colimits is filtered.
A product of filtered categories is filtered.
directed set, filtered category, filtered (∞,1)-category
Last revised on December 9, 2023 at 17:52:30. See the history of this page for a list of all contributions to it.