Steenrod coalgebras III. The fundamental group

In this note, we extend earlier work by showing that if $X$ and $Y$ are delta-complexes (i.e. simplicial sets without degeneracy operators), a morphism $g:N(X)\to N(Y)$ of Steenrod coalgebras (normalized chain-complexes equipped with extra structure) induces one of 2-skeleta $\widehat{g}:X_2\to Y_2$, inducing a homomorphism ${\pi }_1(\widehat{g}):{\pi }_1(X)\to {\pi }_1(Y)$ that is an isomorphism if $g$ is an isomorphism. This implies a corresponding conclusion for a morphism $g:C(X)\to C(Y)$ of Steenrod coalgebras on unnormalized chain-complexes of simplicial sets.