In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”:
Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category $C$. Then the group $π_1(BC)$ has the following presentation: it is generated by symbols $[f]$, one for every morphism in $C$, modulo the relations that
- $[t] = 1$ for every $t ∈ T$, and $[id_c] = 1$ for each object $c$ of $C$
- $[f] · [g] = [f ◦ g]$ for every pair $(f, g)$ of composable morphisms in $C$.
$BC$ denotes the classifying space of $C$, i.e. the geometric realization of its nerve. The notion of maximal tree is that from graph theory: we are considering subgraphs of the obvious unordered graph underlying the category $C$.
As I am not too well acquainted with the theory of simplicial objects, I would be grateful if someone could give me a sketch of this "straightforward” proof, or tell me how much of the theory I have to know to understand it.
If you have not seen the proof that the fundamental group of a graph is free, then this notation can be misleading. When he says the elements are represented by $[f]$ for all morphisms in $C$, first these correspond to 1-cells in the nerve, but these are not necessarily loops hence not elements of the fundamental group. However the maximal tree has the property that given any two points on the graph there is a unique path in the tree that connects them. So if we fix a point $x_0$, then $[f]$ corresponds to the path $a\star f\star b$ where $a$ and $b$ are paths that connect $x_0$ to the endpoints of $f$. This should explain the first point.
For the second point note that the 1-cells corresponding to $f$, $g$ and $f\circ g$ form the sides of a $2$-cell (whose realisation is a triangle) which should give you a homotopy from $[f]\star[g]$ to $[f\circ g]$ alnog the face of the triangle.