Talk:Abstract simplicial complex

?

what is it???

Tosha 20:22, 14 Jun 2004 (UTC)

Algebraic topology section

Moved back to simplicial complex because, as written, it is not about abstract simplcial complexes. A new section here on the algebraic topology of abstract simplicial complexes, without CW complexes etc., would be welcome. Zaslav 11:28, 20 March 2007 (UTC)

error in the definition?

The definition must be wrong, I don't know the correct one, but here is X under a quantor and nowhere else, what makes the statement nonsense, although formally well-formed. —Preceding unsigned comment added by 87.68.82.47 (talk) 22:04, 23 May 2009 (UTC)

Yes, one of the Δ's in the definition should have been an X. I fixed it, but I also rewrote that section to be less notation-heavy: my opinion on this sort of stylistic issue is that with care one can write as precisely in English as one can in notation, but that the result is much more readable. —David Eppstein (talk) 23:46, 23 May 2009 (UTC)

Does the empty set belong to every ASC in the enumeration?

I tried to understand how the Enumeration came about; looking at the example in the linked OEIS entry:

  a(2)=6 from the antichains {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}

this appears to contradict the claim earlier that "the empty set belongs to every abstract simplicial complex".

I'm not at all sure I've understood it correctly, but should the count actually be A000372(n)-1? Or does the "empty set belongs" claim need modifying? Hv (talk) 10:09, 6 August 2010 (UTC)

I've changed the second claim to "the empty set belongs to every non-empty abstract simplicial complex"; I believe this is the intended interpretation. However, I still would like to see an explanation of how {} and {{}} should be interpreted as distinct ASCs. Hv (talk) 09:18, 9 August 2010 (UTC)

Face of a face

Phrasing ".. every face of a face of a complex Δ is itself a face of Δ" one had to define "face of a face" before - right? 77.12.71.207 (talk) 23:27, 9 May 2013 (UTC)

Well, sort of. The point is that a face of a complex is a complex. Indeed, any subset of a complex is a complex. Another way to say what the article is trying to say is to claim that "Any subset of a face is a face". Rswarbrick (talk) 01:12, 10 May 2013 (UTC)

Thanks for quick response. My ask was a pure formal one, and not well founded, as I did add 8 min later in my vanished "Ooops"-PS: .. as you have done: "a face Y is said to belong to another face X if Y ⊂ X" - no dice, sorry. 77.12.71.207 (talk) 23:35, 9 May 2013 (UTC).77.12.98.34 (talk) 22:38, 19 May 2013 (UTC)

Geometric realization

Now think of [0,1]^S as the direct limit of [0,1]^A where A ranges over finite subsets of S

It rather looks like an inverse limit. — Preceding unsigned comment added by 134.157.12.11 (talk) 15:46, 24 April 2014 (UTC)

I'm also reading this for the first time ever, but direct limit seems more appropriate, since the subsets are subsets of the same set, the (direct) limit; whereas, the inverse limit suggests that the set is the product of its subsets, which seems awkward. Hard to tell if it matters much; the article is written in a sketchy survey kind of way, without spelling out much detail. 84.15.191.139 (talk) 16:14, 10 November 2015 (UTC)

WTF is up with the picture.

The picture shows a "geometric representation" of an ASC that isn't a SC. However it is known (and easy to prove, and done in one of the references) that every FINITE ASC has a geometric realisation. So my question is this:

• Do you have a picture that uses the words "geometric representation" (which it doesn't define anywhere) just to fuck with people (relying on "we said representation not realisation so GTFO noob" as a get-out clause) or is there actually a point to the example picture? 2.219.22.164 (talk) 00:53, 9 May 2016 (UTC)

I came here with essentially the same question. An abstract simplicial complex has no structure which could encode the intersections in the picture. While the space depicted in the picture could be used to construct an ASC (as indicated by the vertices, edges, etc identified in the picture), the ASC wouldn't retain the the topological information demonstrated by the picture.

If there was a section devoted to this interplay (or anything relating to the difference indicated by the picture between simplicial complexes and ASCs) that would be one thing, but as it stands, it's unhelpful and only serves to confuse, especially as the only picture present in the article.

I will remove the image and replace it with the more sensible one currently used for simplicial complexes. Donko XI (talk) 20:46, 28 January 2022 (UTC)

Link of empty set?

Link is defined on faces Y of Δ, but the next line talks about link of the empty set, but the empty set is not a face of Δ.

Davyker (talk) 19:41, 5 December 2017 (UTC)

Definitions?

The part about faces looks not so reliable to me. Citations needed much... — Preceding unsigned comment added by Mdcdnaeifkd (talk • contribs) 01:43, 16 November 2019 (UTC)