- Should we divide dimension-theory into 2, one for topology, and one for algebra?
- Are there other tags having the same situation?
Thanks for your kindly discussion.
Thanks for your kindly discussion.
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 dimension-theory 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 linear-algebra (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:
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.
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 dimension-theory 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 dimension-theory 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 linear-algebra (and nothing else, probably). I think that they end up getting tagged with dimension-theory because the askers start typing "dimen..." and see a tag which they think is appropriate, without actually reading the tag description.
I would be more than happy to divide the dimension-theory tag. I might suggest the creation of two new tags:
This split makes the roles of the tags more clear, and may help to prevent novice users from using inappropriate tags.
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 dimension-theory tag. However, we should clearly indicate in the tag-info that:
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 dimension-theory. (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.)