nLab
gauge space

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,

(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.

Mike: Back atcha… do you have any good examples of gauge spaces that are not of one of these types? And is there any value in embedding metric spaces, uniform spaces, and topological spaces into this mysterious larger category QGauQGau, in ways so that their images are essentially disjoint? Do we ever, for instance, want to talk about a short map from a metric space to a topological space, or vice versa? I would like the answer to be “yes,” but I haven’t managed to make it come out that way myself yet.

Toby: I don't know any examples of gauge spaces that arise naturally (you can make them artificially, of course, say as disjoint unions) and don't correspond to some other more familiar type of space. However, I can give examples that don't belong to MetMet, UnifUnif, or TopTop … because they're Cauchy spaces, which aren't listed above yet!

Incidentally, I didn't list Cauchy spaces yet (and didn't finish describing general topological spaces), since I haven't checked yet that things behave correctly. (In particular, I haven't checked that TopTop becomes a full subcategory of QGauQGau —did you?—, although I hope that it will.)

I think that it's nice to be able to see these all as special cases of one kind of thing; then the usual ways of finding the ‘underlying’ foo of a bar become reflections (and sometimes coreflections). This is basically how Lowen-Colebunder's book (which I've referenced, for example, on convergence space) works (although her big category of everything is the category of mereological spaces, which includes ConvConv, rather than QGauQGau).

Seeing this as inclusions and (co)reflections prevents any expectation that the diagrams above should commute, since they mix different things in the wrong way. On the other hand, it also shows us that (for example) any topological space has an underlying uniform space … if you want it.

Mike: I believe I did check that TopQGauTop\to QGau is full, but you should verify it.

I removed my comments about triangles commuting; go ahead and write about the reflections and I’ll see whether I still want to make that comment. I can see thinking of uniform spaces as “saturated” gauge spaces, so that the uniform space underlying a metric space is its “saturation” (reflection) in the larger category GauGau. Perhaps Cauchy spaces can also be thought of this way. I will be kind of surprised if TopTop turns out to be reflective in QGauQGau, but if it were that would also be pretty neat.

Mike: I added a new embedding of TopTop in QGauQGau, which seems to me more likely to be (co)reflective.

Toby: Ah yes, I think that that's the good one that I wasn't finding.

Reflections

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

Details to come

References