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:


Are people still using the second definition? If so, in which contexts, and why?
What are the advantages of the second definition?


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.)
