Examples of nerve and classifying space of a category. I saw that from a small category $C$ one can generate a classifying space $\mathcal{B}C$ based on the nerve $NC$ of the category. The definition of both sounds very nice, but applying it on a category, makes me confuse because of the morphisms. Can anyone give me an example (or two) of how you do the nerves and the classifying space step by step? I will appreciate it a lot.
By step by step I mean not saying "the nerve of this category is this one and from this, its classifying space is this one". I mean helpful examples if possible.
 A: Take the category $[3]=\{0\to 1\to 2\}$. It has three objects, so you start with three points in the realization. Then you have three non-trivial arrows, so you glue thee line segments to the the three points according to the (co)domains. You get the circumference of a triangle. Then you look at non-trivian sequences of two arrows. There is only one such thing so you glue a full triangle to what you already have along the edges of the triangle. This happens to precisely patch up the circumference so you get a full triangle. All other sequences of arrows are degenerate, i.e., contain an identity, so you are done. Similarly, show that $\left | [n]\right |$ is the $n$-simplex.
More interesringly, take $I$ to be the free-living iso. You start with two objects, so two points in the realization. Then there are to arrows, so you glue two line segments and get a circle. There are then two sequences of length two, so you glue two $2$-cells, and get a sphere. There are then two sequences of length $3$ so you glue two $3$-cells, and get a sphere one dimension higher. This goes on for ever, so the end-result is infinite dimensional sphere.
