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Volume Mathematics Institute, University of Warwick. But the original simplicial complex definitions have various advantages. Abstractly simplicial complexes are just hereditary collection of sets. Namely collection of sets closed under taking subsets. These are very basic combinatorial objects that appear all over the place. Simplicial complexes thought of with the second definition arise all the time when you have a group acting on some set, and you'd like the group to act on a complex. Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question.
But now they are the key tool in constructing discrete models for topological spaces. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible plus some technical conditions the nerve has the same homotopy type as the space. In a similar spirit to Jeffrey Giansiracusa's answer, let me mention another place in which triangulations in the strict sense of definition 2 are used. Any real algebraic variety admits a triangulation. There are many refinements to this result, including equivariant versions by Soren Illman, which give triangulations of quotients of algebraic varieties by compact groups.
This has been useful to me several times in studying representation varieties.
I think in the second defition, it's easily to generalized to an abstract sense: simplicial object which just require the bundary maps satisfy some axioms. May's book on simplical objects or Weibel's book on homology algebra. In the first case, it's usually used in category sense.
Anywhere, they are the same thing. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 9 years, 10 months ago. Active 6 years, 4 months ago. Viewed 10k times.
My questions are: Are people still using the second definition? If so, in which contexts, and why? What are the advantages of the second definition? Kevin H. Lin Kevin H. Lin Allen Hatcher Allen Hatcher They are important because when you have a simplicial map, the simplices that collapse ought to be sent to something.
The degenerate simplices also provide the automatic subdivision of the Cartesian product of two simplices. Finally in a simplicial group or similar, the non-degenerate simplices are poorly behaved: They are subsets that are not subgroups. The nondegenerate 2-simplices seem to appear by magic.
Jeffrey Giansiracusa Jeffrey Giansiracusa 4, 1 1 gold badge 24 24 silver badges 43 43 bronze badges.
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Charles Rezk Charles Rezk Greg Kuperberg Greg Kuperberg 48k 9 9 gold badges silver badges bronze badges. Gil Kalai Gil Kalai A natural condition which is satisfied even by regular CW complexes is that the combinatorics of the cells determones the topology of the space described by the complex. But some combinatorics can be done even if this condition is violated. Simplicial complexes in the second definition has also the advantage that they can be embedded linearly not just PW linearly to Euclidean spaces. Also cubical complexes are interesting in combinatorics and topology.
Autumn Kent Autumn Kent 9, 3 3 gold badges 45 45 silver badges 74 74 bronze badges. Andrew Ranicki Andrew Ranicki 3, 1 1 gold badge 29 29 silver badges 24 24 bronze badges. Viewed times. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Zuriel Zuriel 2, 12 12 silver badges 28 28 bronze badges. Thank you!! Historically it is possible that the "normal" notations are later than Milnor's, if this is true, people swapped his notation for some reason and it may not be Milnor's fault at all. Dan Rust Dan Rust Just to clarify one thing: In the definition of semi-simplicial complex, apart form the face map, there should also be degeneracy maps, right?
It's hard to know what definition is being used in the extract, especially as some people now equate semi-simplicial complexes with delta sets in terminology, but I don't think this has always been the case. I'll correct my example later although I'm sure you could now do it yourself!