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I made some much-needed corrections at simplicial complex, directed mostly at errors which had been introduced by yours truly. I also created quasi-topological space (the notion due to Spanier).
I haven’t thought this through, but regarding the process of turning a simplicial complex into a simplicial set, the usual sequence of words seems to involve putting a non-canonical ordering on the set of vertices and then getting ordered simplices from that. But is there anything “wrong” with taking the composite
$SimpComp \hookrightarrow Set^{Fin_{+}^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta^{op}}$where the inclusion is the realization of simplicial complexes as concrete presheaves on nonempty finite sets, and the second arrow is pulling back along the forgetful functor $i$ from nonempty totally ordered finite sets to nonempty finite sets? This looks much more canonical.
Todd, this may be complete nonsense, but something looks a little strange at the second condition of the “Canonical construction” section. Shouldn’t an $\alpha$ be thought of as convex combination of the vertices of one simplex? In that case, shouldn’t the second condition be: for each $\alpha$ in $|K|$, $\sum_v \alpha(v)=1$? (I’d probably feel pretty silly when I hear the answer…)
Good catch, Yaron – those terms $\alpha, v$ got mixed up. I’m having trouble getting into the Lab right now, but once I do I’ll fix it up.
Jim, the term ’polyhedron’ as used on that page was taken from Spanier’s book, where it reads (page 107): a polyhedron is a topological space which admits a triangulation by a simplicial complex. I can add a little disambiguation though.
The notion of abstract simplicial complex was defined in the Idea and Definition section, although the word ’abstract’ was left out. Later in the article, it is effectively redefined to mean a concrete presheaf on nonempty finite sets, which seems for categorical purposes to be a very helpful formulation.
I think the middle category in your canonical composite is called the category of symmetric simplicial sets. It also seems to me that that composite is going to give you the simplicial set that has $(n+1)!$ copies of each $n$-simplex?
I think the middle category in your canonical composite is called the category of symmetric simplicial sets.
Yes, that is so.
It also seems to me that that composite is going to give you the simplicial set that has $(n+1)!$ copies of each $n$-simplex?
Ah, that makes sense now. Thanks!
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