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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

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Many examples are provided in a paper by Misha Gavrilovich, called "Point-set topology as diagram chasing computations". This paper proves that the following properties are all definable via lifting properties just like the examples in your question. The notation $\{0<1\}$ refers to a two-point space where one of the points is open. The notation $\{a > x < b\}$ refers to a three-point space where two of the points are open. The notation $I_2$ (which the author denotes $\{0 \stackrel{>}{<} 1\}$) refers to a two point space where neither point is open.

  • surjectivity and injectivity
  • connectedness (X is connected if $X\to \ast$ lifts against $S^0\to \ast$)
  • separation axiom $T_0$ ($X$ is $T_0$ iff $X\to \ast$ has the right lifting property with respect to $I_2 \to \ast$)
  • separation axiom $T_1$ ($X$ is $T_1$ iff $X\to \ast$ has the right lifting property with respect to $\{0 < 1\} \to \{0\}$)
  • separation axiom $T_2$ in topology ($X$ is Hausdorff iff $S^0 \to X$ lifts against $\{a > x < b\} \to \{a\}$)
  • having dense image (if $X\to Y$ has the left lifting property with respect to $\{a\} \to \{a\to b\}$)
  • when $X$ has the subspace topology in $Y$ (if $X\to Y$ has the left lifting property with respect to $\{a < b\} \to \{a\}$; the author calls this the "induced (pullback) topology")
  • real-valued functions being bounded on a connected domain
  • abelian groups, perfect groups, and finite groups of order prime to p

The paper also discusses lifting problems and Banach spaces, compactness, completeness of metric spaces, closed maps, projective and injective modules, the Sylow theorem, and the Feit-Thomson theorem.

Also, section 3 shows that "sequential compactness can be viewed as a lifting property followed by a rule to erase errors"

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  • $\begingroup$ Thanks, there are more examples by the same author in later papers. But can't they be traced back earlier ? $\endgroup$ – user126830 Oct 28 '18 at 7:14

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