$\require{AMScd}$ 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.



recognising weak equivalences of simplicial sets
$\newcommand{\Ex}{\mathrm{Ex}}\newcommand{\Sd}{\mathrm{Sd}}$Consider the square


$$\begin{CD}
X @>{\sim}>> \Ex^\infty X \\
@V{f}VV @VV{\Ex^\infty f}V \\
Y @>>{\sim}> \Ex^\infty Y \textrm{.} \\
\end{CD}$$


The map $f$ is a weak equivalence if and only if $\Ex^\infty f$ is if and only if there is a relative lift in every diagram of the form


$$\begin{CD}
\partial\Delta[n] @>>> \Ex^\infty X \\
@VVV @VV{\Ex^\infty f}V \\
\Delta[n] @>>> \Ex^\infty Y \textrm{.} \\
\end{CD}$$


Since such a lifting problem and its solution involve only finitely many simplices of $\Ex^\infty X$ and $\Ex^\infty Y$, there is a $j$ such that the lifting problem actually lives in


$$\begin{CD}
\partial\Delta[n] @>>> \Ex^j X \\
@VVV @VV{\Ex^j f}V \\
\Delta[n] @>>> \Ex^j Y \\
\end{CD}$$


and a $k$ such that its solution lives in


$$\begin{CD}
\partial\Delta[n] @>>> \Ex^j X @>>> \Ex^{j+k} X\\
@VVV @VV{\Ex^j f}V @VV{\Ex^{j+k} f}V \\
\Delta[n] @>>> \Ex^j Y @>>> \Ex^{j+k} Y \textrm{.} \\
\end{CD}$$


By adjointness, this means that $f$ is a weak equivalence if and only if for every $j$ and every lifting problem


$$\begin{CD}
\Sd^j\partial\Delta[n] @>>> X \\
@VVV @VV{f}V \\
\Sd^j\Delta[n] @>>> Y \\
\end{CD}$$


there is a $k$ and a "relative lift" in


$$\begin{CD}
\Sd^{j+k}\partial\Delta[n] @>>> \Sd^j\partial\Delta[n] @>>> X \\
@VVV @VVV @VV{f}V \\
\Sd^{j+k}\Delta[n] @>>> \Sd^j\Delta[n] @>>> Y \textrm{.} \\
\end{CD}$$


However, you have to be careful here since a homotopy $\Delta[n] \times \Delta[1] \to \Ex^{j+k} Y$ translates to a map $\Sd^{j+k}(\Delta[n] \times \Delta[1]) \to Y$ which is different than $\Sd^{j+k}\Delta[n] \times \Delta[1] \to Y$ or $\Sd^{j+k}\Delta[n] \times \Sd^{j+k}\Delta[1] \to Y$ so "relative lift" has a slightly different meaning now.


