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Let $\mathcal{C}$ be a class of sets/spaces/structures among which we have a dimension. Namely a map $d:\mathcal{C}\rightarrow \mathbb{N}$ defined in a certain manner that motivated the appellative dimension.

What properties would you expect $d$ to have?

E.g. if $Y, X\in\mathcal{C}$ and $Y\subseteq X$ then $\dim Y\leq \dim X$.

Another way of phrasing this question is, what properties are shared by every standard notion of dimension in mathematics?

rschwieb
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Anguepa
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    Why is the codomain of your dimension function $\mathbb{N}$? There are notions of real valued and even complex valued dimensions. That being said, I suspect that the answer to your question is going to be, essentially, that there is nothing that all notions of dimension have in common. They all seek to generalize some intuition about the usual Euclidean spaces, but from very different points of view (e.g. algebraic notions that generalize "degrees of freedom" in a way related to ideals; measure theoretic notions relating volumes and linear measures; etc). – Xander Henderson Nov 15 '18 at 17:29
  • @XanderHenderson I'd welcome an answer where the codomain is not necessarily $\mathbb{N}$. I'd be surprised if indeed there we no properties in common, e.g. at the very least being preserved by the appropriate notion of isomorphism. – Anguepa Nov 15 '18 at 17:40
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    As I said, there is essentially nothing that all notions of dimension have in common. As you suggest, the best that we are likely to be able to say is that any reasonable notion of dimension should be an invariant of some appropriately described category (e.g. the Hausdorff dimension is an invariant in the category of metric spaces where the morphisms are isometries (or, perhaps, bi-Lipschitz maps)). – Xander Henderson Nov 15 '18 at 17:48
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    @XanderHenderson I'm not certain that's entirely true. For example, we probably want dimension to be defined in a setting where basic categorial operations work properly (e.g. we have finite products); we also probably want the dimension of a product to be the sum of the dimensions of the factors and the dimension of the union of two "sufficiently separated" sets to be the maximum of the dimensions. But I do agree overall. – Noah Schweber Nov 15 '18 at 17:53
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    @NoahSchweber I agree that we might want these things from a notion of dimension, but there are notions that likely fail to have those properties. For example, you suggest that we want $$\dim(A\times B) = \dim(A)+\dim(B). $$ I agree that this is a desirable property, but none of the Hausdorff, Minkowski, packing, nor Assouad dimensions satisfy this identity in general. We can get inequalities, but they point in different directions depending on what notion of dimension you choose. – Xander Henderson Nov 15 '18 at 17:57
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    Related: When is a function a dimension? *AND* Dimension of an object? *AND* Ordinal vs cardinal dimension *AND* Vector spaces with fractional dimension *AND* Has the notion of having a complex amount of dimensions ... *AND* Variable Dimensionality Manifolds. – Dave L. Renfro Nov 15 '18 at 18:28

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