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A discussion in chatroom about tag draw attention to tag again.

Here's some data about how tag is used questions about topology and algebra.

My question:

  1. Should we divide into 2, one for topology, and one for algebra?
  2. Are there other tags having the same situation?

Thanks for your kindly discussion.

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    $\begingroup$ I will add links to Wikipedia articles Dimension theory and Dimension theory (algebra) - I suppose that the suggested split is approximately along the lines of the topics of those two articles. $\endgroup$ – Martin Sleziak Apr 18 at 10:09
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    $\begingroup$ As someone who follows the dimension theory tag for the analysis, I would be more than happy to see the tag split. $\endgroup$ – Xander Henderson Apr 19 at 3:12
  • $\begingroup$ @Andrews: The most voted answer has, so far, only half of the votes cast upon to question; this is why I invite you to leave the question longer than usual. Otherwise, a too small number of votes cast upon the answers might not be representative of what the users think. $\endgroup$ – Alex M. Apr 22 at 8:45
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So far it seems from the votes and comments lean towards separating dimension theory in analysis/topology from dimension theory in algebra.

Let me just add one additional suggestion.

Suggestion. Do not put questions about dimension theory in linear algebra under the same tag as dimension of rings/modules.

Whether or not a tag for dimension in the sense of linear algebra (dimension of vector spaces) needs to have its own tag is probably for a separate discussion - but I don't think it is good to use the same tag for dimension of vector spaces and dimension of rings.


Xander Henderson's answer (current revision) suggests that the new tag would be "for questions about algebraic notions of dimension (e.g. dimension of a vector space, Krull dimension)" (emphasis mine). At the same time it says: "Finally, there are a lot of very elementary questions which show up in the tag which really don't belong. These tend to be of the type "What is the dimension of this subspace of a vector space?" These questions should probably be tagged (and nothing else, probably)." A clarification given in chat confirms that level of the question should be considered to decide whether it belongs in the proposed tag.

Some problems I see with this:

  • Generally, I think using level of the question as a basis of choosing tag is not ideal.1
  • The distinction might became vague if we want to decide tag based on division between elementary and advanced questions.
  • I think that the problems about dimension in linear algebra and dimension in rings (or modules) are rather of a different nature and they are likely to be interesting for different groups of users.

1 This was discussed a bit in the past: Sophistication level tags? and some of the questions linked there. It is true that we have elementary-set-theory/set-theory and elementary-number-theory/number-theory. However, there is disagreement whether the latter should really be used in this way, as you can see in some past discussions: How to differentiate between (elementary-number-theory) and (number-theory) and There are 1,732 questions tagged both elementary-number-theory and number-theory.

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My own area of research is fractal geometry. In fractal geometry, there are various notions of dimension (Hausdorff, box, Minkowski, Assouad, packing, complex etc.). Very roughly speaking, the dimension of a set in a metric space describes how the "volume" of a set scales when the set is scaled, where "volume" may be in terms of a measure (e.g. in the case of the Minkowski dimension), in terms of ball-counting functions (e.g. the box and Assouad dimensions), or in terms of some other generalized notion of volume. These analytic notions of dimension help to characterize "fractals", are related to certain embedding problems in the study of metric spaces, and appear in the study of dynamics.

I follow the tag, and am interested in it from the point of view of fractal geometry.

On the other hand, there seems to be some notion of algebraic dimension theory. This is an area where I am far from expert---I got through my graduate algebra class, but the material never really clicked for me (in the sense that I never found it all that engaging). As far as I can tell, algebraic notions of dimension seek to generalize the idea of a basis or generating set for a vector space. I don't really grok what these algebraic notions of dimension are all about, though I can recognize that questions in algebraic dimension theory are very different from the kinds of questions that are in my wheelhouse.

Finally, there are a lot of very elementary questions which show up in the tag which really don't belong. These tend to be of the type "What is the dimension of this subspace of a vector space?" These questions should probably be tagged (and nothing else, probably). I think that they end up getting tagged with because the askers start typing "dimen..." and see a tag which they think is appropriate, without actually reading the tag description.

Proposal

I would be more than happy to divide the tag. I might suggest the creation of two new tags:

  • , for questions about topological and metric notions of dimension (e.g. Hausdorff and box-counting dimension), and
  • , for questions about algebraic notions of dimension (e.g. dimension of a vector space, Krull dimension).

This split makes the roles of the tags more clear, and may help to prevent novice users from using inappropriate tags.

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    $\begingroup$ The idea is great, but I suppose "analytic" is a bit misleading to questions about topology. $\endgroup$ – Andrews Apr 19 at 13:56
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    $\begingroup$ I am not sure that I have ever actually seen a question about topological dimension theory. That is not to say that such a thing doesn't exist---I just don't think that it comes up all that often. $\endgroup$ – Xander Henderson Apr 19 at 15:57
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    $\begingroup$ Krull dimension for commutative rings is the supremum of lengths of chains of prime ideals (by inclusion). It is unrelated to the dimension of a vector space. $\endgroup$ – hardmath Apr 19 at 21:34
  • $\begingroup$ @XanderHenderson I mean the questions about topological and metric notions of dimension, "analytic" might be considered as "analytic" in "analytic geometry". $\endgroup$ – Andrews Apr 20 at 3:52
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    $\begingroup$ Re: I am not sure that I have ever actually seen a question about topological dimension theory. Depending on the viewpoint, some people might count there also questions about Hausdorff dimension or some other types of dimension that you have mentioned. But definitely these ones: Lebesgue covering dimension, small and large inductive dimension $\endgroup$ – Martin Sleziak Apr 20 at 4:40
  • $\begingroup$ @Andrews Yeah, but these notions of dimension are not simply metric; they also typically involve measures, and are properly contained in the the field of analysis. "Analytic dimension theory" seems like the right terminology to contrast with "algebraic number theory." The adjective "topological" doesn't seem quote right to me, though if it came down to it, I suppose that "topological" might be an acceptable alternative adjective. $\endgroup$ – Xander Henderson Apr 20 at 5:05
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    $\begingroup$ Wouldn't names such as (dimension-theory-algebra), (dimension-theory-analysis) or (dimension-theory-topology) make it clearer that the additional word is just a modifier included to distinguish the tags rather than a name of some area of research? Or are the names algebraic dimension theory, analytics dimension theory, topological dimension theory commonly used? $\endgroup$ – Martin Sleziak Apr 20 at 5:32
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    $\begingroup$ Just realized that GEdgar wrote a book which includes both the topological dimension and the Hausdorff dimension. It would be great if he can say something about this current issues. $\endgroup$ – Arctic Char Apr 20 at 7:11
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    $\begingroup$ @ArcticChar I have pinged some users who are active in the (fractals) tag - and mentioned this discussion, let's hope they will come here if they have something to contribute to this. $\endgroup$ – Martin Sleziak Apr 20 at 8:14
  • $\begingroup$ @MartinSleziak I am open to the idea of dimension-theory-analysis, etc. I don't think that dimension-theory-topology is necessary, however. Looking through the list of questions tagged both dimension-theory and general-topology, it seems that nearly all of them are about metric spaces, and fall within the scope of analysis. $\endgroup$ – Xander Henderson Apr 20 at 13:15
  • $\begingroup$ Xander - One way to find some questions on topological dimension is to do a Google searchs on "inductive dimensin" or "topological dimension". I agree that we get far fewer than for Hausdorff dimension but there are some. $\endgroup$ – Mark McClure Apr 21 at 11:48
  • $\begingroup$ BTW if the new tag (dimension-theory-algebra) or (algebraic-dimension-theory) is created, should it be added to the questions which already have tags (krull-dimension), (global-dimension)? $\endgroup$ – Martin Sleziak Apr 22 at 9:13
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    $\begingroup$ The thing that worries me about this proposal the most is that it suggests to put under the same tag questions about dimension of vector spaces and dimension of rings. I am not sure whether a tag for dimension in linear algebra is needed, but if it is, I think it should be separate from the tag intended mostly for rings and modules. $\endgroup$ – Martin Sleziak Apr 22 at 9:15
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    $\begingroup$ That's done now. $\endgroup$ – quid yesterday
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Let me suggest and alternative proposal. (Partly also to make it possible for users to vote/comment also on this suggestion).

Suggestion. Let us keep only one tag. However, we should clearly indicate in the tag-info that:

  • The tag includes the dimension as used in topology. (Such as Lebesgue covering dimension, small and large inductive dimension and similar notions. See also: Dimension theory on Wikipedia.)
  • The tag includes dimension theory as used in commutative algebra. (Such as Krull dimension and some other notions of dimension used for rings and modules. See also: Dimension theory (algebra) on Wikipedia.)
  • If needed, some other types of dimension should be mentioned explicitly. (For example, at the moment the tag-excerpt and the tag-wiki for (dimension-theory) explicitly mention Hausdorff dimension).
  • This probably goes without saying, but the tag-info should also clearly state that this tag is not for dimension of vector spaces (as used in linear algebra).

One of the reasons for this proposal is that the discussion so far already suggests that it is unclear how many separate tags should arise from the existing . (The question suggested two, but it's possible that some users might prefer to split various types of dimensions related to fractals from topological notions of dimension. If we end up with three tags, the boundaries become fuzzy.)

For searching, one can still refine search with addition of some other tags, for example dimension-theory+general-topology, dimension-theory+fractals. dimension-theory+algebraic-geometry or dimension-theory+commutative-algebra, etc. (Although I have to admit that keeping this under single tag makes life a bit more difficult for people that are following that tag and are only interested in one of the meanings of dimension theory.)

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