Nerve of category

Question

in my first topology lecture I learned about abstract simplicial complexes which were roughly defined as can be found on wikipedia. Now I am reading up on classifying spaces of (topological) categories which are defined as the geometric realization of the nerve , that is a simplicial set or as it is apparently sometimes called a semi-simplicial complex. Now I am wondering if it's all basically the same, that the nerve of a small category can be identified as a simplicial complex in some way. Another thing is, that the author of the paper I am currently reading stresses he is talking about topological categories, but the definition of the nerve doesn't consider any topology as far as I can see.

Answer 1

In a simplicial complex an $n$-simplex must have $n+1$ distinct vertices and is completely determined by them, that no two simplices can have exactly the same set of vertices. A simplicial set has neither of these restrictions. The usual definition of the nerve a category has as $n$-simplices strings of $n$ composable morphisms; the vertices of such a simplex are the objects these morphisms go between, so as you can see the vertices need not be distinct, and there can be more than one simplex with precisely the same vertices. Thus the nerve is a simplicial set and not a simplicial complex.

As for topological categories, the nerve is naturally a simplicial topological space, not a simplicial set. Say a topological category has space of objects $C_0$ and space of morphisms $C_1$, then you say the space of $n$-simplices is the subspace of $C_0^{n+1} \times C_1^n$ given by $$\{(x_0, f_0, x_1, \ldots, f_n, x_n) : x_i \in C_0, f_i \in C_1, f_i : x_i \to x_{i+1}\}.$$ Simplicial spaces have geometric realizations too, defined in pretty much the same way as the geometric realization of a simplicial set, and the geometric realization of the simplicial space which is the nerve of a topological category is still called its classifying space.