Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
gave Cartesian space a TOC and added some statements and references.
Need for a small fix: Entry Cartesian space assumes $\mathbf{R}^n$ is equipped only with topology, while the entry Jacobian assumes that the Cartesian space is equipped with the canonical smooth structure as well. Maybe we should fix this more consistently, but as I prefer not to mention the term Cartesian space in context of Jacobian (I would just say $\mathbf{R}^n$ with smooth structure) I will rather leave to those prefering Cartesian space terminology to make their favorable choice there (in a way not to mess entry smooth space or wherever the notion of Cartesian site is used, I do not know how to make it overall consistent).
There is a subsection “Properties – Smooth structure”. You mean one should move the content there to the Definition-section? Maybe. I won’t object.
The Cartesian site has morphisms respecting the smooth structure by the definition (am I right?). So this is OK with the convention neded in Jacobian. But the section Smooth structures you now quote allows for exotic smooth structures while the notion of Cartesian site needed for Cartesian spaces needs the standard choice. I think that my choice would be to define Cartesian spaces and their morphisms by fixing the standard smooth structure on a real $n$-dimensional space, while the exotic structures should be discussed in an entry dedicated to the real $n$-dimensional space which would not fix that structure. Your treatment of Cartesian site (like in nactwist) fixes the standard structure (and the latter is needed at Jacobian, what is less important) while when people talk about $\mathbf{R}^n$ this is a concrete set which may be viewed with many levels of structure, unlike the object of the category of Cartesian spaces. Maybe there is a more elegant way to resolve this.
The emphasis on topological structure (which didn’t used to be in cartesian space) seems misplaced. A cartesian space is a cartesian power of the real line; it has lots of structure, just as the real line has. One uses whatever structure one needs (as CartSp makes clear).
Edited the definition of cartesian space in accord to Toby’s comment. The definition now reads: A Cartesian space is a finite Cartesian product of the real line with itself. Followed by a cautionary remark: “This definition leaves implicit the category which contains the real line as an object”
I would really like to use the terminology “cartesian n-space” to refer to R^n, following Todd’s comment at item 2 of complex manifold. This is analogous to the Lab referring to the n-dimensional ball as “n-ball”. Ravi Vakil, in Chapter 11 of his notes, describes “dimension” as a slippery concept. A case in point is if we take the implicit category containing R to be Set, but then there is no traditional definition of dimension of a set. It seems to me that for objects like R^n and B^n and the n-simplex, the natural number “n” is more of a “rank” than a dimension. To be pedantic, it is the sheaves which these objects represent that have a dimension.
Another example of this usage is “n-sphere”.
Re #6: the cautionary remark is a good idea. I slightly changed the wording.
I would really like to use the terminology “cartesian n-space” to refer to R^n
I’m fine with that limitation of use. But I also think the phrase “cartesian n-space” ought to be used somewhat sparingly. In almost all cases I think one could simply say “$\mathbb{R}^n$” and let the reader supply the tacit context of how we are viewing $\mathbb{R}$. I don’t think linking to cartesian space really adds anything useful to that, in most cases.
Todd’s comment at item 2 of complex manifold
This linked to Zoran’s comment in the present thread. Colin might have meant this.
A case in point is if we take the implicit category containing R to be Set, but then there is no traditional definition of dimension of a set. It seems to me that for objects like R^n and B^n and the n-simplex, the natural number “n” is more of a “rank” than a dimension. To be pedantic, it is the sheaves which these objects represent that have a dimension.
I find the wording of this remark somewhat odd. The dimension of $\mathbb{R}^n, B^n$, and $\Delta_n$ here simply refers to the number of independent real parameters. It’s pretty clear that the category this “independence” might refer to is not the category of bare sets. There is a notion of dimension of a vector space which is the size of the largest linearly independent set, where “linear independence” refers to the ground field in which the coefficients are considered as living in, and there is a similar notion for affine spaces. The notion of independence is useful more generally in model theory; for the three concrete examples given, I think it’s useful to consider them as definable sets in real semi-algebraic geometry, and there is a definite meaning of dimension there. I don’t know exactly why “dimensions of sheaves” is being brought up.
1 to 9 of 9