^{1}

Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note, we consider the analytical properties of diversities, in particular the generalizations of uniform continuity, uniform convergence, Cauchy sequences, and completeness to diversities. We develop conformities, a diversity analogue of uniform spaces, which abstract these concepts in the metric case. We show that much of the theory of uniform spaces admits a natural analogue in this new structure; for example, conformities can be defined either axiomatically or in terms of uniformly continuous pseudodiversities. Just as diversities can be restricted to metrics, conformities can be restricted to uniformities. We find that these two notions of restriction, which are functors in the appropriate categories, are related by a natural transformation.

The theory of metric spaces is well-understood and forms the basis of much of, modern analysis. In 1956, Aronszajn and Panitchpakdi developed the notion of hyperconvex metric spaces [

These minimal hyperconvex spaces, or tight spans, proved to be powerful tools for the analysis of finite metric spaces. The theory of tight spans, or T-theory, is overviewed in [

In light of these applications of T-theory, Bryant and Tupper developed the theory of diversities alongside an associated tight span theory in [

A classic paper by Weil [

In this note, we develop conformities, which generalize diversities in analogy to Weil's uniform space generalization of metrics. We will describe uniform continuity, uniform convergence, Cauchy sequences, and completeness for diversities, and show that these can be characterized in terms of conformities, giving an abstract framework in which to analyze the uniform structure of diversities. This is motivated by the observation that while diversities generalize metric spaces in a straightforward way (in fact they restrict to metric spaces), they can exhibit very nonsmooth behavior with respect to these spaces (cf. Theorem

Throughout this paper, we will denote the finite power set of a given set

We begin with the Bryant-Tupper definition from [

if

if

For a metric space

the diameter diversity

The Steiner tree diversity

(Recall that a Steiner tree on

In fact, these examples are the extremes of diversity behavior relative to their induced metrics, in the sense that for any diversity

To demonstrate the difference between the diameter and Steiner tree diversities, consider the Euclidean metric

The Steiner tree diversity function

Without loss of generality, we show that the result for

A similar construction for the Steiner tree diversity on

In Section

With this goal in mind, we start with the following definitions: let

Finally, if

It is not hard to see that for diameter diversities, these definitions coincide exactly with the standard ones on the induced metric.

For the second half of the paper, we will work extensively with filters, so we state the definition here: given a ground set

In this paper, we additionally require that

In this section, we contrast the convergence of sequences with respect to diversities and their induced metrics. In particular, we show that although the Cauchy property for sequences is much stronger for diversities (we demonstrate a sequence which is not Cauchy with respect to a diversity, even though it is Cauchy with respect to the induced metric), completeness of a diversity is equivalent to completeness of its induced metric. This tells us that every diversity which induces a Euclidean metric (e.g., the Steiner tree diversity on

Since the set of Cauchy sequences in a diversity may be smaller than the set of Cauchy sequences of its induced metric, this may provide a simpler way to determine completeness of metric spaces.

At the end of the section, we construct the analogue of completion for diversities.

Let

Suppose that

Therefore, for all

that is,

As mentioned, the set of Cauchy sequences in a diversity may be strictly smaller than the set of Cauchy sequences in the induced metric. For example, let

Order each set

In light of this example, it is interesting to know that every complete diversity has a complete induced metric, which is proved with the following lemma.

Let

Define the subsequence

Let

Let

Then,

In light of the equivalence between metric completeness and diversity completeness, it is perhaps not so surprising that every diversity can be completed in a canonical way. To do so, we require two more definitions from [

Every diversity

Let

It can then be shown that

This completion is dense in the sense that every member

Let

Let

To show uniqueness of

This is a universal property in the sense that for every complete diversity

In this section we introduce a generalization of diversities analogous to uniformities, which generalize metric spaces. Uniformities lie between metric spaces and topologies, in the sense that every metric space defines a uniformity, and every uniformity defines a topology (which coincides with the metric topology when the uniformity came from a metric). Uniformities characterize uniform continuity, uniform convergence, and Cauchy sequences, which are not topological concepts.

The carry-over from the metric case is natural but nontrivial, since diversities can behave differently on sets of different cardinality. Since this construction is qualitatively different from metric uniformities, it requires a different name. We asked ourselves “what would you call a uniformity that came from a diversity?”, and the answer was clear, a conformity.

Throughout this section, we will give the analogous definitions and results for uniformities, using the standard treatment from Kelley [

We then briefly touch on the problem of completion for conformities.

Finally, we define power conformities; from a conformity defined on a set

Recall that for

Similarly, let

A similar characterization of uniform convergence of sequences of functions can be given in terms of pairs of points. From these observations arises the theory of uniformities, which is described in any standard text on analysis (cf. [

If

For every

In particular, for any pseudometric space

Uniform structure can be defined entirely with respect to uniformities. For example, given sets

To abstract the uniform structure of diversities, uniformities are clearly insufficient. For one thing, since diversities map finite sets rather than pairs, we should seek a filter on

Putting all this together, we define a

For every

For every

An observation that will be necessary later (one which also holds for uniformities) is that for any

As in the metric case, there is a canonical way to generate a conformity from a diversity; if

As in the metric case, uniform structure can be defined on conformities in a way that generalizes that of diversities, let

More generally, given a collection of pseudodiversities

A uniformity is generated by a single pseudometric if and only if it has a countable base.

The standard proof of this theorem goes as follows: it is obvious that any uniformity generated by a pseudometric has a countable base. Conversely, if there exists a countable base for a uniformity on

Given a conformity with a countable base

Nonetheless, the result is true, which is the content of the next theorem.

Let

Let

Let

If

Conversely, let

Define a chain as a sequence

Notice that

We claim that

First of all,

The triangle equality also holds; let

Then,

Next, we notice that

every cycle is a chain, so

If

Finally, we claim that

Trivially,

The case

If

This characterizes the conformities generated by single pseudodiversities. Later, we will describe every conformity in terms of the pseudodiversities that generate them.

Given a conformity

Let

Denote by

Let

By Theorem

Next, we give some standard definitions. For a uniform space

With the same space

The analogous definitions for conformities are as follows.

Let

A pseudodiversity

Suppose

Choose

Conversely, suppose that every Cauchy filter converges in

For any conformity

Let

Suppose that

We end this section with two open questions as follows.

Does the converse to Theorem

We saw in Section

Is there a notion of universal completion for conformities?

Not every conformity has a countable base. For example, let

In this section, we will show that every conformity is generated by the collection of pseudodiversities which are uniformly continuous with respect to it, in an appropriate sense. In the case of uniformities, this is done by constructing a so-called product uniformity; given a uniformity on a set

Since pseudodiversities are functions on finite sets rather than pairs, given a conformity on a set

In fact, such a conformity exists for which we can prove the same result; given a conformity

A power conformity is a conformity.

First, the

Finally, every

Let

First, suppose that every

is in

Conversely, suppose that

Every conformity is generated by the pseudodiversities which are uniformly continuous from its power conformity to

Let

Then, by Theorem

We saw at the beginning of this section that some conformities can be generated by sets of the form

In [

It is not hard to see that for both metric spaces and diversities, nonexpansive maps are uniformly continuous. In the metric case, they are also continuous.

We introduce the category Conf, whose objects are conformities and morphisms uniformly continuous functions. This compares with Unif [

We also recall Top, whose objects are topological spaces and morphisms continuous maps, and CAT, whose objects are categories and morphisms are functors (maps between categories which preserve composition).

With these categories in hand, we can summarize the relationships between diversities, conformities and metric spaces by observing that the maps in the following diagram in CAT are functors, and that the diagram as a whole commutes

and

Notice that each functor leaves the underlying sets unchanged for example,

The author declares that there is no conflict of interests regarding the publication of this paper.

Research funded in part by NSERC.