nLab category of filters

Definition

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

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

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

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

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

Let ዋ 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 $\mathfrak{F}$ on a set $X$ gives a precise meaning to phrases “property $P(x)$ holds for all points small enough’‘ and “holds for almost all elements of $X$’’, namely that
$\{ x : P(x) \} \in \mathfrak{F}$, in the same way that the topology (topological structure) on a set gives a precise meaning to the phrase “property $P(x)$ holds for all points sufficiently near a given point $x_0$‘’, namely $\{ x : P(x) \}$ is a neighbourhood of $x_0$.

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

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

The simplicial category of the category of filters provides a somewhat more formalisation of the intuition of “nearness” than the usual topological one; in particular, it contains the categories of topological and of uniform spaces, of simplicial sets, and of filters themselves, allowing to reformulate in terms of this category notions such as limit, equicontinuity, locally trivial, and geometric realization.

Application to analysis and topology

In a topological space $X$, a filter $\mathfrak{F}$ on $X$ converges to a point $x$ of $X$ if every neighbourhood of $x$ belongs to $\mathfrak{F}$, i.e. the map from $\mathfrak{F}$ to the neighbourhood filter of $x$ 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 $M\times M$).

In a metric space $M$, a filter $\mathfrak{F}$ on $M$ is Cauchy if it has elements of arbitrarily small diameter, i.e. the map $\mathfrak{F}\times \mathfrak{F}\to S\times S$ is continuous with respect to the uniformity (filter) on $S\times S$. 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.

 See also

• filter