A so-called singular $p$-symplex in an $n$-manifold is defined to be a differentiable map
$$ \sigma_p: \Delta_p \rightarrow M^n $$
where $\Delta_p = (P_0,P_1, \ldots,P_p)$ is a standard $p$-simplex given by its $p+1$ vertices.
It is said that there are no restrictions on the rank of the map $\sigma_p$, so if I understand it correctly it does not necessarily have to be singular.
So what exactly is singular about a singular $p$-simplex then?