Example 2.5.6.6. Let $M_{\ast }$ be a chain complex, and let $x,y \in M_0$ be a pair of $0$-cycles, which we identify with vertices of the simplicial set $\mathrm{K}(M_{\ast })$. The following conditions are equivalent:
- $(a)$
The vertices $x$ and $y$ belong to the same connected component of the simplicial set $\mathrm{K}(M_{\ast } )$ (Definition 1.2.1.8).
- $(b)$
There exists an edge $e$ of the simplicial set $\mathrm{K}(M_{\ast })$ connecting $x$ to $y$ (so that $d^{1}_1(e) = x$ and $d^{1}_0(e) = y$).
- $(c)$
The cycles $x$ and $y$ are homologous: that is, there exists an element $u \in M_{1}$ satisfying $\partial (u) = x - y$.
The equivalence of $(a) \Leftrightarrow (b)$ follows from the fact that $\mathrm{K}( M_{\ast } )$ is a Kan complex (see Remark 1.4.6.13), while the equivalence $(b) \Leftrightarrow (c)$ follows immediately from the construction of the simplicial set $\mathrm{K}(M_{\ast } )$. It follows that the set of connected components $\pi _0( \mathrm{K}(M_{\ast } ))$ can be identified with the $0$th homology group $\mathrm{H}_{0}( M )$.