The normalized chain complex of a simplicial set

Question

The paper The Geometry of Rewriting Systems, by K. Brown, mentions the notion of a normalized chain complex associated to a simplicial set. How is this complex defined? (and perharps the non-normalized chain complex as well)

Answer 1

Goerss and Jardine define this in their book Simplicial Homotopy Theory, chapter III, section 2. It is also defined at the nLab. Both define it for simplicial groups, Goerss and Jardine just for simplicial abelian groups. These two sources use slightly different conventions.

Let $A$ be a simplicial abelian group. Goerss and Jardine define the normalized chain complex $NA$ by $$ NA_n = \bigcap_{i=0}^{n-1} \ker(d_i), \quad \partial = (-1)^n d_n $$ while on nLab they use $$ NA_n = \bigcap_{i=1}^{n} \ker(d_i), \quad \partial = d_0 $$ A standard result (Theorem 2.1 in Goerss and Jardine) is that $NA$ is isomorphic to the chain complex obtained by modding out by the subgroup generated by the degeneracies.