nLab
category of filters

The category of filters

The category of filters

Definition

A filter 𝔉\mathfrak{F} on a set XX is a set of of subsets of X which has the following properties:

(F IF_I) Every subset of X which contains a set of 𝔉\mathfrak{F} belongs to 𝔉\mathfrak{F}.

(F IIF_{II}) Every finite intersection of sets of 𝔉\mathfrak{F} belongs to 𝔉\mathfrak{F}.

Subsets in 𝔉\mathfrak{F} are called neighbourhoods or 𝔉\mathfrak{F}-big.

A mapping of underlying sets of filters is continuous iff the preimage of a neighbourhood is necessarily a neighbourhood. Note that 𝔉βˆͺ{βˆ…}\mathfrak{F}\cup \{\emptyset\} is a topology on XX, but this notion of continuity is stronger: it means continuous with dense image.

Let α‹‹ denote the category of filters and the continuous mappings of underlying sets.

Idea

It is convenient to talk about a filter on a set as a filter structure on a set and think of it as a space with a notion of smallness, in analogy to a topological structure on a set and thinking of it as a (topological) space with a notion of nearness of points.

A filter 𝔉\mathfrak{F} on a set XX gives a precise meaning to phrases β€œproperty P(x)P(x) holds for all points small enoughβ€™β€˜ and β€œholds for almost all elements of XX’’, namely that
{x:P(x)}βˆˆπ”‰ \{ x : P(x) \} \in \mathfrak{F} , in the same way that the topology (topological structure) on a set gives a precise meaning to the phrase β€œproperty P(x)P(x) holds for all points sufficiently near a given point x 0x_0β€˜β€™, namely {x:P(x)}\{ x : P(x) \} is a neighbourhood of x 0x_0.

Thus, for example, in the neighbourhood filter of a point x 0x_0 in a topological space, a point xx is small iff xx is near x 0x_0: we think of xx near x 0x_0 as an approximation to x 0x_0 up to a small error.

A number of elementary notions can be formulated using the category of simplicial objects of the category of filters.

Application to analysis and topology

In a topological space XX, a filter 𝔉\mathfrak{F} on XX converges to a point xx of XX if every neighbourhood of xx belongs to 𝔉\mathfrak{F}, i.e. the map from 𝔉\mathfrak{F} to the neighbourhood filter of xx is continuous. The filter on the empty set is the initial object; the proper filter on a single point is the terminal object.

The concepts of continuous function and uniformly continuous function may be defined quite nicely in terms of the continuity of maps of filters. A map of topological spaces is continuous iff it induces continuous maps of neighbourhood filters. A map of uniform spaces is continuous iff it induces a continuous map of the uniform structures (filters on MΓ—MM\times M).

In a metric space MM, a filter 𝔉\mathfrak{F} on MM is Cauchy if it has elements of arbitrarily small diameter, i.e. the map 𝔉×𝔉→SΓ—S\mathfrak{F}\times \mathfrak{F}\to S\times S is continuous with respect to the uniformity (filter) on SΓ—SS\times S. Then a sequence is a Cauchy sequence iff its eventuality filter is Cauchy, i.e. it induces a continuous map of certain filters. This gives a diagram chasing meaning to the concept of completion of a metric space.

Created on March 31, 2021 at 20:23:25. See the history of this page for a list of all contributions to it.