nLab gauge space

Gauge spaces

Gauge spaces

Idea

A gauge space is a topological space (necessary completely regular) whose topology is given by a family of pseudometrics. More generally, a quasigauge space is a space (not necessarily completely regular) whose topology is given by a family of quasipseudometrics.

Actually, a gauge space has additional structure, so that it can be seen as giving a (completely regular) Cauchy space, a uniform space, or even a generalisation of a metric space in which the category MetMet of metric spaces and short maps is a full subcategory.

Please note that, while this is based on the presentation in HAF, the precise definitions of the objects and morphisms of the category of gauge spaces below constitute original research. (In particular, HAF really considers the category of pregauge spaces and uniformly continuous maps, which is equivalent to the category of uniform spaces, since it uses them only to study that category.)

Definitions

Given a set XX, a pregauge on XX is simply a family of pseudometrics on XX. A gauge is a \geq-filter of pseudometrics on XX, that is a collection GG of gauging distances such that

  1. There is an element of GG; in the light of (3), the zero pseudometric (x,y0)( x, y \mapsto 0 ) is a gauging distance.
  2. Given d,eGd, e \in G, some fGf \in G satisfies
    d(x,y),e(x,y)f(x,y) d(x,y), e(x,y) \leq f(x,y)

    for all x,yx, y in XX; in the light of (3), the pseudometric (x,ymax(d(x,y),e(x,y)))( x, y \mapsto max(d(x,y), e(x,y)) ) is a gauging distance.

  3. Given dGd \in G and any pseudometric ee on XX, if
    e(x,y)d(x,y) e(x,y) \leq d(x,y)

    for all x,yx, y in XX, then eGe \in G.

A pregauge satisfying axioms (1&2) is a base for a gauge; a base is precisely what generates a gauge by taking the downward closure. Any pregauge whatsoever is a subbase for a gauge; a subbase is precisely what generates a base by closing under finitary joins.

A gauge space is a set equipped with a gauge. A quasigauge is a collection of quasipseudometrics satisfying (1–3); a quasigauge space is a set equipped with a quasigauge.

Given (quasi)gauge spaces XX and XX', a short map from XX to XX' is a function FF (on their underlying sets) such that the composite with FF (or with F×FF \times F, to be precise) of any gauging distance on XX' is a gauging distance on XX. That is,

  • for every eGe \in G', there is a dGd \in G such that
    e(F(x),F(y))d(x,y) e(F(x),F(y)) \leq d(x,y)

    for all x,yx, y in XX; in the light of (3), the pseudometric (x,ye(F(x),F(y)))( x, y \mapsto e(F(x),F(y)) ) (this is the composite) is a gauging distance.

(Warning: this definition of short map is probably the most significant original research on this page.)

Gauge spaces and short maps between them form the category GauGau of gauge spaces; quasigauge spaces and short maps between them form the category QGauQGau of quasigauge spaces. Note that any gauge is a base for a quasigauge; in this way, GauGau is (equivalent to) a full subcategory of QGauQGau.

Examples

The categories GauGau and QGauQGau are not well known, but some of their subcategories are.

  • A metric space (or pseudometric space) defines a gauge space, taking its single (pseudo)metric as a base; similarly, a quasi(psuedo)metric space defines a quasigauge space. Taking short maps (in the usual sense, that is distance-nonincreasing functions) as morphisms, the categories MetMet and PsMetPsMet are full subcategories of GauGau; similarly, QMetQMet and QPsMetQPsMet are full subcategories of QGauQGau.

  • A uniform space XX defines a gauge space, consisting of all of the pseudometrics on XX that are uniformly continuous as maps from X×XX \times X to the real line; similarly, a quasiuniform space defines a quasigauge space. Taking uniformly continuous maps as morphisms, the category UnifUnif is a full subcategory of GauGau; similarly, QUnifQUnif is a full subcategory of QGauQGau. Thus, uniform spaces can be viewed as gauge spaces whose collection of gauging distances is “saturated” under uniform continuity.

  • A completely regular topological space XX defines a gauge space, consisting of all the pseudometrics on XX that are continuous as maps from X×XX \times X to the real line. In this way the category CRegTopCReg Top of completely regular spaces and continuous maps is a full subcategory of GauGau; it is in fact contained in UnifUnif. (In general, a completely regular space can be uniformized in many ways; this inclusion corresponds to the “initial” uniformity.)

  • An arbitrary topological space defines a quasigauge space in a more complicated way. Given any open set UU in a space XX, define the pseudometric (in fact a pseudoultrametric) d Ud_U as follows:

    d U(x,y)={0 ifxUyU, 1 ifxUyU; d_U(x,y) = \begin{cases} 0 & if\; x \in U \;\Rightarrow\; y \in U ,\\ 1 & if\; x \in U \;\wedge\; y \notin U ;\end{cases}

    then the d Ud_U form a base for a quasigauge, which induces the original topology on XX. In other words, every space is “quasigaugeable.” In this way TopTop also becomes a full subcategory of QGauQGau. Note, though that replacing 11 by any other positive real number gives a different embedding of TopTop in QGauQGau.

  • Another way to get a quasigauge space from a topological space is to take as a base the set of all quasi-pseudometrics dd on XX such that for each xXx\in X, the function d(x,):Xd(x,-):X\to \mathbb{R} is upper semicontinuous. This is also a full embedding of TopTop in QGauQGau, which is perhaps more “canonical.”

Reflections

Many of these full subcategories of GauGau and QGauQGau are reflective.

Details to come

References

Last revised on October 24, 2023 at 19:02:27. See the history of this page for a list of all contributions to it.