3
votes
0answers
100 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
3
votes
1answer
49 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
4
votes
0answers
81 views

Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
4
votes
1answer
84 views

What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
4
votes
1answer
63 views

Free Metric Space?

Do free metric spaces exist? Ie.: An object in the category of metric spaces and lipschitzian maps. If so would these be the complete metric spaces, since they satisfy a similar universal property? ...
3
votes
0answers
125 views

Is there a category such that if $\mathbb{R}$ is viewed as an object, we have that $\mathbb{R}^2$ is equipped with the Euclidean distance function?

Viewing $\mathbb{R}$ as an object of the category of metric spaces and metric maps, we have that $\mathbb{R}^2$ is equipped with the distance function $$d(x,y) = ...
5
votes
1answer
97 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
6
votes
1answer
199 views

Metric vs metrizable spaces

A. Helemskii in the book "Lectures on functional analysis" write (in my horrible translation): The category of Hausdorff topological spaces (morphisms are continuous maps) contain the full ...
7
votes
2answers
202 views

Completion of a metric space in categorical terms

Is it possible to define the completion of a metric space using categorical terms?
5
votes
0answers
181 views

Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
7
votes
2answers
106 views

What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?

When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow ...
10
votes
2answers
251 views

Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?

Every metric space is associated with a topological space in a canonical way. According to this source, this amounts to a full functor from the category of metric spaces with continuous maps to the ...
11
votes
2answers
149 views

pseudo-inverse to the operation of turning a metric space into a topological space

The construction of turning a metric space $(X,d)$ into a topological space by inducing the topology generated by the open balls gives rise to a functor $Met\to Top$ for any reasonable category $Met$ ...
38
votes
0answers
631 views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
6
votes
1answer
221 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
25
votes
1answer
644 views

The category of compact metric spaces

Let us denote by $(\mathrm{CompMet})$ the category of compact metric spaces with Lipschitz maps as morphisms. I'm interested in properties of this category. It seems to me that it has finite products ...