I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.

For example, several iterated orthogonals of $ \emptyset\longrightarrow \{*\}$ in $Top$ correspond to meaningful topological properties. I want to see more examples of "meaningful" iterated orthogonals.

Here are details and definitions.

Recall two morphisms $f:A\rightarrow B$ and $g:C\rightarrow D$ are *orthogonal*, write $f \bot g$, iff for
for any morphisms $t:A\rightarrow C$ and $b:B\rightarrow D$ such that $f\circ b=t\circ g$ there is $d:B\rightarrow D$
such that $t=f\circ d$ and $b=d\circ g$. For a class $C$ of morphisms in a category,
define its {\em left} and {\em right orthogonals}
$$ C^l := \{ f :\text{ for each }g \in C\ f \bot g \} $$
$$ C^r := \{ g :\text{ for each }f \in C\ f \bot g \} $$
$$ C^{lr}:=(C^l)^r,\, C^{ll}:=(C^l)^l, ... $$

Take $C=\{ \emptyset\longrightarrow \{*\} \}$ in $Top$.

$C^r$ is the class of surjections, $C^{rr}$ is the class of subsets, $C^{rl}$ is the class of maps of form $A\longrightarrow A\sqcup D$, $D$ is discrete; $\{\bullet\}\longrightarrow A$ is in $C^{rll}$ iff $A$ is connected; $Y$ is totally disconnected iff $\{\bullet\}\xrightarrow y Y$ is in $C^{rllr}$ for each map $\{\bullet\}\xrightarrow y Y$ (or, in other words, each point $y\in Y$). $C^l$ consists of maps $f:A\longrightarrow B$ such that either $A=B=\emptyset$ or $A\neq \emptyset$, $B$ arbitrary. $C^{l}$ is the class of maps $A\longrightarrow B$ such that either $A\neq\emptyset$ or $A=B=\emptyset$. $C^{ll}$ is the class of isomorphisms. $C^{lr}$ is the class of maps $\emptyset\longrightarrow B$, $B$ is arbitrary. $C^{lrl}$ is the class of maps which admit a section. $C^{lll}=C^{llr}=..$ is the class of all maps.

This question is analogous to natural examples of sequences of adjoint functors, iterated adjoint functors and infinite chains of adjoint functors