Intuition degeneracy maps I'm trying to get an intuitive understanding for the notion of a simplicial set. Roughly, a simplicial set consists of a set $S_n$ of $n$-simplices for each non-negative $n$, and families of face maps $\{d_i\colon S_n\to S_{n-1}\}_{0\leq i\leq n}$ and degeneracy maps $\{s_i\colon S_n\to S_{n+1}\}_{0\leq i\leq n}$.
I'm interpreting this intuitively as follows: the sets $S_n$ tell us what to glue. Elements of $S_0$ are interpreted as points, elements of $S_1$ as lines, elements of $S_2$ as triangles, elements of $S_3$ as tetrahedrons, and so on. The face maps tell us how to glue. For instance, if we want to glue two triangles $\Delta_1, \Delta_2\in S_2$ both on the "first" line, we let $d_0 \Delta_1 = d_0\Delta_2$.
But here is a specific question I'm wondering about: what's the role of the degeneracy maps? Why do I need to know them in the process of glueing all the elements of the $S_n$ together? It seems the face maps suffice, because they tell us which simplices have which vertices.
 A: Degeneracy maps are indeed totally unnecessary to describe the simplicial set you're interested in, and in fact it's possible to do simplicial homotopy theory with semisimplicial sets, which are precisely simplicial sets without the degeneracy maps. However, the connection with topological spaces is weaker. For instance, the singular simplicial set of the one-point topological space has geometric realization homeomorphic to that space. But the geometric realization of the singular semi-simplicial set of any nonempty topological space is infinite-dimensional, as a CW complex! There's no way to collapse away the simplices that really aren't carrying any information. If you want to work with nice finite semisimplicial sets, like say the one with one $0$-simplex and no higher-dimensional simplices, then you realize that, unfortunately, most semi-simplicial sets cannot map into it! So you have to do some "fattening up" to get such a semisimplicial set to actually behave like a topological point. All of these issues are entirely fixable, but the communicty has preferred the degenerate solution.
