An Introduction to Simplicial Sets

Simplicial sets, and more generally simplicial objects in a given category, are central to modern mathematics. While I am not a mathematical historian, I thought I would describe in conceptual outline how naturally simplicial sets arise from the classical study of simplicial complexes. I suspect that something like this recapitulates the historical development. We have described simplicial complexes in several different forms: abstract simplicial complexes, ordered simplicial complexes, geometric simplicial complexes, ordered geometric simplicial complexes and realizations of geometric simplicial complexes. It is possible to go directly from abstract simplicial complexes to realizations without passing through geometric simplicial complexes, although the construction is perhaps less intuitive. I may describe it later. (I did briefly in class.) An abstract simplicial complex is equivalent to a geometric simplicial complex, and neither of these notions involves anything about ordering the vertices. If one has a simplicial complex of either type, one can choose a partial ordering of the vertices that restricts to a linear ordering of the vertices of each simplex, and this gives the notion of an ordered simplicial complex. This can be done most simply, but not most generally, just by choosing a total ordering of the set of all vertices and restricting that ordering to simplices. We have seen in studying products of simplicial complexes that geometric realization behaves especially nicely only in the ordered setting. Both the category S C of simplicial complexes and the category OS C of ordered simplicial complexes have categorical products. Geometric realization preserves products when defined on OS C , but it does not preserve products when defined on S C . The functor K is best viewed as a functor from the category P of partially ordered sets to the category OS C rather than just to the category S C . The functor X , on the other hand, starts in S C and lands in P, which can be identified with the category of Alexandroff T0-spaces. The composite K X is the barycentric subdivision functor Sd : S C −→ OS C , and since the geometric realization functor gives a space |SdK| that can be identified with |K| there is no loss of topological generality working in OS C instead of S C . The most important motivation for working with ordered rather than unordered simplicial complexes is that the ordering leads to the definition of an associated chain complex and thus to a quick definition of homology. I’ll explain that in the talks and add it to the notes if I have time. In the early literature of algebraic topology, a topological space X is called a polyhedron if it is homeomorphic to |K| for a (given) simplicial complex K. Such a homeomorphism |K| −→ X is called a triangulation of X, and X is said to be triangulable if it admits a triangulation. Then we can define the homology of X to be the homology of K. This is a quick definition, and useful where it applies, but