Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2010

    gave Cartesian space a TOC and added some statements and references.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 30th 2011

    Need for a small fix: Entry Cartesian space assumes R n\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 R n\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).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2011
    • (edited Nov 30th 2011)

    There is a subsection “Properties – Smooth structure”. You mean one should move the content there to the Definition-section? Maybe. I won’t object.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeNov 30th 2011

    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 nn-dimensional space, while the exotic structures should be discussed in an entry dedicated to the real nn-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 R n\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.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeDec 4th 2011

    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).

    • CommentRowNumber6.
    • CommentAuthorColin Tan
    • CommentTimeJul 19th 2014

    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.

    • CommentRowNumber7.
    • CommentAuthorColin Tan
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    Another example of this usage is “n-sphere”.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    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 “ n\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 n,B n\mathbb{R}^n, B^n, and Δ n\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.

    • CommentRowNumber9.
    • CommentAuthorColin Tan
    • CommentTimeJul 20th 2014
    Thanks for the rewording, Todd.