Filtered category
Encyclopedia
In category theory
, filtered categories generalize the notion of directed set
understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category).
A category
is filtered when
A diagram is said to be of cardinality
if the morphism set of its domain is of cardinality 
. A category 
is filtered if and only if there is a cone over any finite diagram 
; more generally, for a regular cardinal 
, a category 
is said to be 
-filtered if for every diagram 
in 
of cardinality smaller than 
there is a cone over 
.
A filtered colimit is a colimit of a functor
where 
is a filtered category. This readily generalizes to 
-filtered limits. An ind-object in a category 
is a presheaf of sets 
which is a small filtered colimit of representable presheaves. Ind-objects in a category 
form a full subcategory 
in the category of functors 
. The category 
of pro-objects in 
is the opposite of the category of ind-objects in the opposite category 
.
is cofiltered if the opposite category
is filtered. In detail, a category is cofiltered when
A cofiltered limit is a limit
of a functor
where 
is a cofiltered category.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, filtered categories generalize the notion of directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category).
A category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
is filtered when
- it is not empty,
- for every two objects
and 
in 
there exists an object 
and two arrows 
and 
in 
, - for every two parallel arrows
in 
, there exists an object 
and an arrow 
such that 
.
A diagram is said to be of cardinality
if the morphism set of its domain is of cardinality . A category is filtered if and only if there is a cone over any finite diagram ; more generally, for a regular cardinal , a category is said to be -filtered if for every diagram in of cardinality smaller than there is a cone over .A filtered colimit is a colimit of a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
where is a filtered category. This readily generalizes to -filtered limits. An ind-object in a category is a presheaf of sets which is a small filtered colimit of representable presheaves. Ind-objects in a category form a full subcategory in the category of functors . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category .Cofiltered categories
There is a dual notion of cofiltered category. A category is cofiltered if the opposite categoryOpposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite...
is filtered. In detail, a category is cofiltered when
- it is not empty
- for every two objects
and 
in 
there exists an object 
and two arrows 
and 
in 
, - for every two parallel arrows
in 
, there exists an object 
and an arrow 
such that 
.
A cofiltered limit is a limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
where is a cofiltered category.