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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...
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- 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
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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....
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Cofiltered categories
There is a dual notion of cofiltered category. A category![](http://image.absoluteastronomy.com/images/formulas/6/6/2660335-36.gif)
Opposite 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...
![](http://image.absoluteastronomy.com/images/formulas/6/6/2660335-37.gif)
- 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....
![](http://image.absoluteastronomy.com/images/formulas/6/6/2660335-50.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/6/2660335-51.gif)