Definition of "simplicial complex"

Question

When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".

However, there is another definition of "simplicial complex", e.g. the one on wikipedia: it's a collection $K$ of simplices such that any face of any simplex in $K$ is also in $K$, and the intersection of two simplices of $K$ is a face of both of the two simplices. There is also the notion of "abstract simplicial complex", which is a collection of subsets of $\{ 1, \dots, n \}$ which is closed under the operation of taking subsets. These kinds of simplicial complexes also have corresponding geometric realizations as topological spaces. I'll refer to both of these definitions as "the second definition".

The second definition looks reasonable at first sight, but then you quickly run into some horrible things, like the fact that triangulating even something simple like a torus requires some ridiculous number of simplices (more than 20?). On the other hand, you can triangulate the torus much more reasonably using the first definition (or alternatively using the definition of "Delta complex" from Hatcher's algebraic topology book, but this is not too far from the first definition anyway).

I believe you can move back and forth between the two definitions without much trouble. (I think you can go from the first to the second by doing some barycentric subdivisions, and going from the second to the first is trivial.)

Due to the fact that the second definition is the one that's listed on wikipedia, I get the impression that people still use this definition. My questions are:

1. Are people still using the second definition? If so, in which contexts, and why?

2. What are the advantages of the second definition?

Answer 1

Simplicial sets and simplicial complexes lie at two ends of a spectrum, with Delta complexes, which were invented by Eilenberg and Zilber under the name "semi-simplicial complexes", lying somewhere in between. Simplicial sets are much more general than simplicial complexes and have the great advantage of allowing quotients and products to be formed without the necessity of subdivision, as is required for simplicial complexes. In this way simplicial sets are like CW complexes, only more combinatorial or categorical. The price to pay for this is that simplicial sets are perhaps less geometric, or at least not as nicely geometric as simplicial complexes. So the choice of which to use may depend in part on how geometric the context is. In some areas simplicial sets are far more natural and useful than simplicial complexes, in others the reverse is true. If one drew a Venn diagram of the people using one or the other structure, the intersection might be very small.

Delta complexes, being something of a compromise, have some of the advantages and disadvantages of each of the other two types of structure. When I wrote my algebraic topology book I had the feeling that Delta complexes had been largely forgotten over the years, so I wanted to re-publicize them, both as a pedagogical tool in introductory algebraic topology courses and as a sort of structure that arises very naturally in many contexts. For example the classifying space of a category is a Delta complex.

Incidentally, I've added 5 pages at the end of the Appendix in the online version of my book going into a little more detail about these various types of simplicial structures. (I owe a debt of thanks to Greg Kuperberg for explaining some of this stuff to me a couple years ago.)

Answer 2

Many thanks to Allen in turn for his generous citation to me. What I had to say is fairly standard, yet I don't think that I'd ever be able to write a book like Hatcher's Algebraic Topology. Here is a link to the appendix that Allen wrote.

Here is the situation in a nutshell. A simplicial set has a small realization. It is a CW complex made of non-degenerate simplices of the simplicial set. The simplices can be glued to themselves or multiply glued to each other. However, the simplicial set structure also implies that the corners of each simplex are consistently locally ordered, and this is not possible with an arbitrary gluing. The consistent local ordering is useful for a variety of purposes. The first time that you'd really care is that it gives you a canonical cup product at the level of simplicial chains.

Geometric topologists, especially 3-manifold topologists, widely use of exactly this structure, except without the consistent local ordering. This is called a generalized triangulation, and you can express it in much the same way as a simplicial set. Instead of using the simplex category, whose morphisms are order-preserving maps between ordered finite sets, you use the symmetric simplex category, whose morphisms are all maps between all finite sets. The small realization of a symmetric simplicial set is exactly a generalized triangulation, if the symmetric simplicial set satisfies a certain freeness condition. The symmetric group $S_n$ acts on the set of formal $n$-simplices, and the condition is that the action on the non-degenerate one should be free. Symmetric simplicial sets appear in a bare handful of papers in the literature.

Every generalized triangulation with consistently locally ordered vertices is represented by a unique simplicial set. This is harmonious view of simplicial sets to make both algebraic and geometric topologists happy. The only problem is that it does not generalize well to other simplicial objects, because the non-degenerate simplices aren't any good in, for instance, a simplicial group.

Simplicial sets are very useful to algebraic topologists. Generalized triangulations are very useful to geometric topologists. Simplicial complexes are useful to combinatorialists: they are hypergraphs with a closure property. Simplicial complexes are not as useful as natural generalizations to either algebraic or geometric topologists: simplicial sets if you order the order the vertices; or generalized triangulations if you don't order the vertices. It is true that certain constructions of simplicial set or generalized triangulations are automatically simplicial complexes. For instance the simplicial set of a poset is automatically that (as Charles Rezk says), or the second barycentric subdivision of any type of CW complex that has a barycentric subdivision. (Because the first barycentric subdivision is automatically a simplicial set with colored vertices.)

Answer 3

Most of the above answers seem to deal with the differences between simplicial sets and simplicial complexes, and the advantages or disadvantages of each. Part of the original question was: are people still using simplicial complexes, and if so, then why? I'd just like to toss in a short answer here.

People certainly are still using abstract simplicial complexes (the "second definition" in the question). Low dimensional topology is full of useful simplicial complexes. For instance, one of the most important objects in the study of mapping class groups and moduli spaces of curves is Harvey's curve complex. Fix a surface. The set of vertices is the set of isotopy classes of (unoriented) simple closed curves in the surface that do not bound discs. A collection of vertices $v_0, \ldots, v_n$ of n+1 distinct vertices spans an n-simplex if they admit representatives that are pairwise disjoint. Or course, if you are only interested in the homotopy type of this complex, then you could just as well work with the nerve of the poset of collections of curves. But for many arguments, it is easier/clearer to work with this simplicial complex and study the mapping class group action on it. Passing to the nerve of the poset of simplices is just a subdivision, and if you already have a perfectly good object to study, why replace it with a something that has many more simplices for you to worry about.

There are plenty of other objects in low dimensional topology and combinatorics that are just naturally given to us as simplicial complexes rather than simplicial sets. Moduli spaces of graphs and things like Culler-Vogtmann Outer Space are (subsets) of the realisations of simplicial complexes, while the spine of outer space is the realisation of a simplicial set. It really is helpful to work with both. For instance, if you want to know about the VCD of outer automorphism groups of free groups then you work with the simplicial set that gives the spine. But the proof that $Out(F_n)$ is a virtual duality group fundamentally uses outer space rather than its spine.

I suppose the moral is that, from a purely homotopical point of view, simplicial complexes and simplicial sets are roughly equivalent, although as Allen points out in his answer, many constructions are easier with simplicial sets. However, not every problem is just about homotopy theory. Sometimes geometry matters, and in these situations simplicial complexes often have the advantage of being much smaller and closer to the geometry that defined the complex in the first place.

Answer 4

You need to be careful here. The geometric realization of a simplicial set may not be the geometric realization of simplicial complex in the obvious way: for instance, the geometric realization of $\Delta[2]/\partial\Delta[2]$ is a union of a point and an open $2$-simplex. This is not a simplicial complex in the sense of your first definition: the closure of the open $2$-simplex isn't a closed $2$-simplex. You can of course produce a subdivision which is a simplicial complex -- but I'm not sure that barycentric subdivision works here.

Simplicial complexes were invented long before simplicial sets (I think they were introduced by Poincare). Simplical complexes are awkward if you are interested in homotopy theory, including for some of the reasons you mention. However, people do think about them:

* It's a classical question as to whether you can triangulate a manifold, i.e., show it is homeomorphic to a (nice) simplicial complex. This question has a subtle answer (see Kirby-Siebenmann invariant), and leads to the notion of "piecewise-linear manifold", which is a tractable class of manifolds between continuous manifolds and smooth manifolds.

* Simplicial complexes show up all over the place in combinatorics, for instance, in the order complex of a poset.

Answer 5

I use the second definition. I do not really understand the first definition; Can you write it in concrete terms? (Hatcher's object are nice and allow more "economical" triangulations.) It is correct that you need sometimes many simplices for triangulating (but only 7 for the torus). 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.

Answer 6

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.

For example, if you have a closed surface $S$ of genus $g \geq 2$, you build the curve complex $\mathcal{C}(S)$ by letting the $0$-skeleton be the set of isotopy classes of essential simple closed curves. Then a $k$-simplex is a collection of $k+1$ curves that may be realized disjointly.

This complex is very useful in studying the mapping class group of $S$, as you might suspect once you realize that it's the nerve of boundary strata of the universal cover of the moduli space.

Answer 7

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. The Wikipedia article http://en.wikipedia.org/wiki/Rips_complex is a good introduction.

Answer 8

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.

Answer 9

I used simplicial complexes of the Wikipedia kind in my 1992 CUP book Algebraic L-theory and Topological Manifolds to construct the algebraic L-theory assembly map. The construction was extended to $\Delta$-sets (in the sense of Rourke and Sanderson - not to be confused with Allen Hatcher's $\Delta$-complexes) in my 2012 joint paper with Michael Weiss The Algebraic L-theory of $\Delta$-sets.

Answer 10

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.), see J.P. 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.