  We present three novel classifications of the weak sequential (and strong)
limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called
the QM condition which is a kind of separation property for pre-images of
closed connected sets and show that $u$ satisfies this property exactly when it
is the limit of Sobolev homeomorphisms. Further, we prove that $u\in
W^{1,p}_{\operatorname{id}}((-1,1)^2,\mathbb{R}^2)$ is the limit of a sequence
of homeomorphisms exactly when there are classically monotone mappings
$g_{\delta}:[-1,1]^2\to \mathbb{R}^2$ and very small open sets $U_{\delta}$
such that $g_{\delta} = u$ on $[-1,1]^2 \setminus U_{\delta}$. Also, we
introduce the so-called three curve condition, which is in some sense
reminiscent of the NCL condition of \cite{CPR} but for $u^{-1}$ instead of for
$u$, and prove that a map is the $W^{1,p}$ limit of planar Sobolev
homeomorphisms exactly when it satisfies this property. This improves on
results in \cite{DPP} answering the question from \cite{IO2}.
