  A simplicial set is said to be non-singular if the representing map of each
non-degenerate simplex is degreewise injective. The inclusion into the category
of simplicial sets, of the full subcategory whose objects are the non-singular
simplicial sets, admits a left adjoint functor called desingularization. In
this paper, we provide an iterative description of desingularization that is
useful for theoretical purposes as well as for doing calculations.
