# sequences of iterated orthogonals (lifting property) in a category

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

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"

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