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nLab > Latest Changes: simplex

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2012

    At simplex I have accompanied the definition of the cellular simplex with that of the topological simplex.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2012

    Added also singular simplex.

    • CommentRowNumber3.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 31st 2012
    Never heard before of a cellular simplex - did not invoke an appropriate image
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2012

    Therfore I put “simplicial” in parenthesis: “Cellular (simplicial) simplex”.

    But I find for a defintion of the term it is unfair to the reader to speak of “simplicial simplex”. While correct, it feels a bit circular.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeSep 1st 2012

    I think “simplicial simplex”, whatever its flaws, is much better than “cellular (simplicial) simplex”. In algebraic topology “cellular” almost exclusively refers to CW complexes and cell complexes.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 1st 2012

    Hm (or h’m, as Toby would write)… abstract or combinatorial simplex? In some older books, there is a distinction made between “abstract simplicial complex” and “geometric simplicial complex”, where the latter is a polyhedron or geometric realization of the former, more combinatorial notion.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2012
    • (edited Sep 6th 2012)

    I have expanded at simplex the section Topological simplex a good bit (just the standard stuff, but written out in some lengthy detail – well, it could be lenghtier still)

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeSep 6th 2012

    Following discussions on another thread, should cellular simplex perhaps rather mean ’topological simplex considered as a cellular / CW-complex’?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2012
    • (edited Sep 6th 2012)

    Maybe the majority feeling here is inclined in that direction.

    But I find the distinction unhelpful. Even if you instist to think of topological simplices with cell decomposition, it’s still the same concept then up to equivalence (if you take homomorphisms to preserves cellular structure, which you should do): the simplicial simplex is the essence of the idea of the cellular topological simplex. That’s why people call elements of a simpliciat set nn-cells, after all.

    But I won’t invest time in fighting for this perspective of mine. Somebody who feels terminologically authorized should please feel invited to edit the entry accordingly.

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeSep 6th 2012

    That’s why people call elements of a simpliciat set n-cells, after all.

    I find they usually call them n-simplices more often than not.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeSep 6th 2012

    Okay, I changed “cellular simplex” to “standard simplicial simplex”.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 6th 2012

    Urs, I added an observation under the cartesian-coordinate description of topological simplex which might be useful – if you’d rather format it differently, please be my guest.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2012

    Thanks, Todd.

    if you’d rather format it differently, please be my guest.

    Yes, I gave it a formal remark home.

    • CommentRowNumber14.
    • CommentAuthorColin Tan
    • CommentTimeOct 27th 2015

    In barycentric coordinates, the (standard) topological nn-simplex is the subset of consisting of those points (x 0,…,x n)\{\mathbb{R}}(x_0,\ldots, x_n) such that x 0+⋯+x n=1x_0 + \cdots + x_n = 1 and x 0,…,x n≥0x_0,\ldots, x_n \ge 0. Hence the affine span of the topological nn-simplex is a codimension one affine subspace of ℝ n+1{\mathbb{R}}^{n+1}.

    Is there a generalization of this concept of simplex where ℝ n+1{\mathbb{R}}^{n+1} is replaced by an affine space of infinite dimension?

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 27th 2015

    It sounds like what you may be after is “free convex set on infinitely many elements”, construing convex sets as models of an algebraic theory. The same way that the nn-simplex would be the convex hull of n+1n+1 in “general position”, or formal convex combinations of n+1n+1 elements. This can be done, yes.

  16. The page currently displays the link for the first use of “simplicial set” as the HTML code for the link. I believe this should fix that.

    Anonymous

    diff, v32, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2022

    added pointer to:

    diff, v37, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeFeb 5th 2024

    added illustration (here) of the first couple of barycentric topological simplices

    diff, v40, current

1 to 18 of 18

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