I am trying to solve an exercise that has as a goal the description of the group completion of a monoid.
In order to do that, we start with the definition of the nerve of a small category $\mathcal{C}$ as the simplicial set $\text{N}(\mathcal{C}): \Delta^{\text{op}} \to \text{Set}, [n] \mapsto \text{Hom}_{\text{Cat}}([n], \mathcal{C})$.
I understood that we can identify the $n$-simplex $\text{N}(\mathcal{C})_n$ with the diagram $C_0 \xrightarrow{f_1} C_1 \xrightarrow{f_2} C_2 \to \cdots \xrightarrow{f_n} C_n$. So, the vertices of the nerve can be identified with the objects of $\mathcal{C}$, and the edges with the morphisms.
Moreover, the $i$-th face is given by $C_0 \xrightarrow{f_1} C_1 \to \cdots \to C_{i-1} \xrightarrow{f_{i+1} \circ f_i} C_{i+1} \to \cdots \xrightarrow{f_n}C_n$ (with the clear special cases for the $0$-th and $n$-th faces), while the $i$-th degeneracy map is given by $C_0 \xrightarrow{f_1} \cdots C_{i-1} \xrightarrow{f_i} C_i \xrightarrow{\text{id}_{C_i}} C_i \xrightarrow{f_{i+1}} C_{i+1}\to \cdots \xrightarrow{f_n} C_n$.
In order to understand the exercise, I have to construct a functor $\varphi_{\mathcal{C}}: \mathcal{C} \to \Pi_1(|\text{N}(\mathcal{C})|)$ which sends objects and morphisms to the corresponding vertices and edges in $|\text{N}(\mathcal{C})|$: this shouldn't be difficult, but apparently I didn't understand how the geometric realization of the nerve works. Can you please help me in this part of the exercise?
After that, I have to show that, for a monoid $M$, $\varphi_{\text{B}M}$ exhibits $\pi_1(| \text{N}(\text{B}M)|, \ast)$ as the group completion of $M$: references on the topic will be highly appreciated.
Thank you for your help.
Best Answer
In general if $X$ is a simplicial set, any $n$-simplex $\sigma\in X_n$ corresponds to a map of simplicial sets $\Delta^n\to X$. This follows from the Yoneda lemma. After realization this leads to a map $\vert\sigma\vert\colon\Delta^n_\mathrm{top} = \vert\Delta^n\vert\to \vert X\vert$. This map can alternatively described as follows. Recall that $\vert X\vert$ is the quotient of $\coprod_{n=0}^\infty X_n\times \Delta^n_\mathrm{top}$. Then $\vert\sigma\vert$ is the composite $$\Delta^n_\mathrm{top}\cong \{\sigma\}\times \Delta^n_\mathrm{top}\subset X_n\times \Delta^n_\mathrm{top}\to \vert X\vert.$$ We identify $\Delta^1_\mathrm{top}$ with the unit interval via the homeomorphism $$[0,1]\xrightarrow{\cong}\Delta^1_\mathrm{top},t\mapsto (1-t,t),$$ and we see in particular that any $1$-simplex $\sigma$ of $X$ gives rise to the path $\vert\sigma\vert$ with starting point $\vert d_1(\sigma)\vert$ and endpoint $\vert d_0(\sigma)\vert$. Also, $\vert\sigma\vert$ is constant if $\sigma$ is a degenerate $1$-simplex. Finally, note that if $\sigma$ is a $2$-simplex, then the composition of the paths $\vert d_2(\sigma)\vert$ and $\vert d_0(\sigma)\vert$ is path homotopic to $\vert d_1(\sigma)\vert$.
In the special case of $N(\mathcal{C})$ the $0$-simplices are the objects and the $1$-simplices are the morphisms. Note that if $f\colon x\to y$ is a morphism, then $d_1(f) = x$ and $d_0(f) = y$. Also, if $\sigma$ is the $2$-simplex $x\xrightarrow{f} y\xrightarrow{g} z$, then $d_0(\sigma) = g, d_1(\sigma) = g\circ f$ and $d_2(\sigma) = f$. From the previous discussion we immediately see that we obtain a functor $\mathcal{C}\to \Pi_1(\vert N(\mathcal{C})\vert$.
Finally, if $M$ and $BM$ is the category with one object $\ast$ and a morphism for any element of $M$, then $\vert N(BM)\vert$ is a CW-complex with one vertex and an edge for every $m\in M\setminus\{1\}$. A $2$-simplex $\ast\xrightarrow{m}\ast\xrightarrow{n}$ is non-degenerate iff $m\neq 1\neq n$. We see that $\pi_1(\vert N(BM)\vert)$ is the free group with a generator $e_m$ for every $m\in M\setminus\{1\}$ modulo the relation $e_m\cdot e_n = e_{m\cdot n}$, which is exactly the group completion of $M$.