$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper *Weak Equivalences and Simplicial Presheaves* (http://www.math.uiuc.edu/K-theory/0564/wesp.pdf) Daniel Dugger and Daniel Isaksen give the following criterion (due to Reedy):

A map $f:X\to Y$ between fibrant simplicial sets is a weak equivalence if and only if it has the relative homotopy lifting property (RHLP, see below) with respect to all generating cofibrations $\partial\Delta^n\hookrightarrow \Delta^n.$

They also mention that a similar result would hold for non-fibrant objects $X,Y$ if one allows to subdivide $\partial \Delta^n$ and $\Delta^n.$ What exactly would this lifting criterion for general maps $f:X\to Y$ be?

Here is the definition of the RHLP: A square $$\begin{CD} A @>>> X\\@VVV@VVV\\B@>>>Y\end{CD}$$ has a relative homotopy lifting if there exist a lift $B\to X$ making the upper triangle commute on the nose and a simplicial homotopy relative to $K$ from $B\to X\to Y$ to the given map $B\to Y.$ The map $X\to Y$ has the RHLP with respect to $A\to B$ if every such square has a relative homotopy lifting.