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At simplex I have accompanied the definition of the cellular simplex with that of the topological simplex.
Added also singular simplex.
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.
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.
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.
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)
Following discussions on another thread, should cellular simplex perhaps rather mean ’topological simplex considered as a cellular / CW-complex’?
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 $n$-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.
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.
Okay, I changed “cellular simplex” to “standard simplicial simplex”.
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.
Thanks, Todd.
if you’d rather format it differently, please be my guest.
Yes, I gave it a formal remark home.
In barycentric coordinates, the (standard) topological $n$-simplex is the subset of consisting of those points $\{\mathbb{R}}(x_0,\ldots, x_n)$ such that $x_0 + \cdots + x_n = 1$ and $x_0,\ldots, x_n \ge 0$. Hence the affine span of the topological $n$-simplex is a codimension one affine subspace of ${\mathbb{R}}^{n+1}$.
Is there a generalization of this concept of simplex where ${\mathbb{R}}^{n+1}$ is replaced by an affine space of infinite dimension?
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 $n$-simplex would be the convex hull of $n+1$ in “general position”, or formal convex combinations of $n+1$ elements. This can be done, yes.
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