examples of lifting properties

Question

A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property (a category-theoretic construction I define below) "starting" to a single (counter)example or a simple class of morphisms, for example a finite group being nilpotent, solvable, p-group, a topological space being compact, discrete, T4 (normal).

I would like to see more examplees, to help me understand if there is a bigger picture behind.

Let me give the definitions.

For a property $C$ of arrows (morphisms) in a category, define its {\em left and right orthogonals} as

$$ C^\perp := \{ f :\text{ for each }g \in C\ f \,\rightthreetimes\, g \} $$ $$ {}^\perp C := \{ g :\text{ for each }f \in C\ f \,\rightthreetimes\, g \} $$

here $f \,\rightthreetimes\, g$ reads " $f$ has the left lifting property wrt $g$ ", " $f$ is (left) orthogonal to $g$ ", i.e. for $f:A\longrightarrow B$, $g:X\longrightarrow Y$, $f \,\rightthreetimes\, g$ iff for each $i:A\longrightarrow X$, $j:B\longrightarrow Y$ such that $ig=fj$ ("the square commutes"), there is $j':B\longrightarrow X$ such that $fj'=i$ and $j'g=j$ ("there is a diagonal making the diagram commute").

Examples:

In the category Sets of sets the right orthogonal ${}^\perp \{\emptyset \longrightarrow \{*\}\}$ of the simplest non-surjection $\emptyset \longrightarrow \{*\}$ is the class of surjections. The triple left orthogonal $ ((\{\emptyset \longrightarrow \{*\}\}^\perp)^\perp)^\perp$ is the class of functions which split.

The left and right orthogonals of $ \{x_{1},x_{2}\}\longrightarrow \{*\} $, the simplest non-injection, are both precisely the class of injections.

A finite group $H$ is nilpotent iff $H\longrightarrow H\times H$ is in ${}^\perp(\{ 0\longrightarrow G : G\text{ arbitrary} \}^\perp)$

A Hausdorff space $K$ is compact iff $K\longrightarrow \{*\}$ is in ${}^\perp({}^\perp(\{a\}\longrightarrow \{a{<}b\})_{<5})^{\perp})$; here $^\perp(\{a\}\longrightarrow \{a{<}b\})_{<5}$ denotes maps in $^\perp(\{a\}\longrightarrow \{a{<}b\})$ between spaces of size less than 5.

I give more examples in the answers to my own question I posted, as they require some notation.

Answer 1

A word on the bigger picture. Examples are absolutely ubiquitous in category theory. First note that if $\mathcal{C} \perp \mathcal{D}$, then $\mathcal{C}$ and $\mathcal{D}$ determine each other uniquely up to the closure operations ${}^\perp(()^\perp)$ and $(^\perp())^\perp$, so to understand what's going on largely boils down to understanding classes of morphisms of the form ${}^\perp(\mathcal{C}^\perp)$. Now consider:

Theorem Let $\mathcal{K}$ be a locally presentable category and $\mathcal{K}^{[1]}$ its category of morphisms. Then the following are equivalent:

1. Accessible, accessibly-embedded, weakly-reflective full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$.

2. Accessible, accessibly-embedded full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$ which are closed under coproduct, pushout along arbitrary morphisms of $\mathcal{L}$, composition, transfinite composition, and retracts.

3. Full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$ of the form $\mathcal{L} = {}^\perp(\mathcal{C}^\perp)$ for some small set $\mathcal{C} \subseteq \mathcal{K}^{[1]}$

The proof is via the small object argument. Note that the "closure" conditions of (2) are always satisfied by a subcategory closed under ${}^\perp(()^\perp)$ in an arbitrary category.

And consider this: if $f: A \to B$ is a morphism and you can form the pushout $B\cup_A B$ of $f$ along itself, then the lifting property with respect to the map $B \cup_A B \to B$ is equivalent to lifts with respect to $f$ being unique, which means that the orthogonality relation we're talking about (usually called weak orthogonality) can be used to express strong orthogonality (where unique lifts exist). And we get analogously:

Theorem Let $\mathcal{K}$ be a locally presentable category and $\mathcal{K}^{[1]}$ its category of morphisms. Then the following are equivalent:

1. Accessible, accessibly-embedded, reflective full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$.

2. Accessible, accessibly-embedded full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$ which are closed under colimits and pushouts and retracts along arbitrary morphisms of $\mathcal{L}$.

3. Full subcategories $\mathcal{L} \subseteq \mathcal{K}^{[1]}$ of the form $\mathcal{L} = {}^\perp(\mathcal{C}^\perp)$ for some small set $\mathcal{C} \subseteq \mathcal{K}^{[1]}$ (where for the moment I'm using the more standard convention that ${}^\perp$ denotes strong orthogonality.

These theorems have implications for full subcategories of $\mathcal{K}$ rather than $\mathcal{K}^{[1]}$ by identifying an object $X$ with either $X \to 1$ or $\emptyset \to X$, as appropriate.

These two example theorems are meant to be illustrative. The provisos "accessible, accessibly-embedded" are technical, and can be omitted in the presence of the set-theoretical Vopenka's Principle.

In another direction, in a non-locally-presentable category like $\mathsf{Top}$ many of the same principles still apply, allowing one to concluded from "closure" properties of a class of morphisms as in (2) above that it is generated under ${}^\perp(()^\perp)$ by a small class of morphisms. $\mathsf{Top}$ itself also satisfies a weakening of the notion of local presentability called local boundedness; see the references on the nlab page for analogs of the above theorems in the locally bounded case. There is also a way to apply the above theorems directly to $\mathsf{Top}$, which also illustrates an example of the phenomenon you're looking for:

Example $\mathsf{Top}$ is the union of a chain of full, coreflective subcategories $\mathsf{Top} = \cup_\kappa \mathsf{Top}_\kappa$ which are locally presentable. Here $\kappa$ is a regular cardinal and $\mathsf{Top}_\kappa$ is the category of spaces of $<\kappa$-tightness, (I may be off by taking a successor cardinal here) and in fact

$\mathsf{Top}_\kappa = {}^\perp(\mathsf{Disc}_\kappa^\perp)$

where here $\perp$ denotes strong orthogonality and $\mathsf{Disc}_\kappa$ is the category of discrete spaces of cardinality $<\kappa$, considered as a full subcategory of $\mathsf{Top}$.

The moral is: any nice (~ "sufficiently cocomplete") subcategory $\mathcal{L}$ of a sufficiently-good cocomplete category $\mathcal{K}$ is definable in terms of lifting properties, unless some weird set-theoretical phenomenon is occurring.

Answer 2

To improve readability of iterated orthogonals, I write $C^l$ instead of $C^\perp$ and $C^r$ instead of $C^r$.

(i) $(\emptyset\longrightarrow \{*\})^r$, $(0\longrightarrow R)^r$, and $\{0\longrightarrow \Bbb Z\}^r$ are the classes of surjections in Sets, R-modules, and Groups, resp, (where $\{*\}$ is the one-element set, and in the category of groups, $0$ denotes the trivial group)

(ii) $(\{a,b\}\longrightarrow \{*\})^l=(\{a,b\}\longrightarrow \{*\})^r$, $(R\longrightarrow 0)^r$, $\{\Bbb Z\longrightarrow 0\}^r$ are the classes of injections in Sets, R-modules, and Groups, resp

(iii) in R-mod, a module $P$ is projective iff $0\longrightarrow P$ is in $(0\longrightarrow R)^{rl}$

a module $I$ is injective iff $I\longrightarrow 0$ is in $(R\longrightarrow 0)^{rr}$

(iv) in the category of groups, a finite group $H$ is nilpotent iff $H\longrightarrow H\times H$ is in $\{\, 0\longrightarrow G : G\text{ arbitrary} \}^{lr}$

a finite group $H$ is solvable iff $0\longrightarrow H$ is in $\{\, 0\longrightarrow A : A\text{ abelian }\}^{lr}= \{\, [G,G]\longrightarrow {G} : G\text{ arbitrary }\}^{lr}$

a finite group ${H}$ is of order prime to $p$ iff $H\longrightarrow 0$ is in $\{\Bbb Z/p\Bbb Z\longrightarrow 0\}^{rr}$

a finite group $H$ is a p-group iff $H\longrightarrow 0$ is in $\{\Bbb Z/p\Bbb Z\longrightarrow 0\}^{rr}$

a group $H$ is torsion-free iff $0\longrightarrow H$ is in $\{ n\Bbb Z\longrightarrow \Bbb Z: n>0 \}^r$

(v) in the category of metric spaces and uniformly continuous maps, a metric space $X$ is complete iff $\{1/n\}_n\longrightarrow \{1/n\}_n\cup \{0\} \,\rightthreetimes\, X\longrightarrow \{0\}$ where the metric on $\{1/n\}_n$ and $\{1/n\}_n\cup \{0\}$ is induced from the real line

a subset $A \subset X$ is closed iff $\{1/n\}_n\longrightarrow \{1/n\}_n\cup \{0\} \,\rightthreetimes\, A\longrightarrow X$

for a connected topological space X, each function on $X$ is bounded iff $ \emptyset\longrightarrow X \,\rightthreetimes\, \cup_n (-n,n) \longrightarrow \Bbb R$ (disjoint union)

Answer 3

Properties of topological spaces.

Here I need to use some notation for finite topological spaces. I use the fact that a finite topological space may be thought of as a category such that $card Hom(x,y)\leq 1$ for any objects $x,y$. By convention, $ \{a{\rightarrow}b\}$ denote the space where $a$ is open and $b$ is closed; more generally, a subset is closed iff there are no arrows going outside the subset. in maps, each point goes to "itself". The arrow $(\{a{\rightarrow}b\}\longrightarrow \{a=b\}$ denotes the map to a singleton gluing together points $a$ and $b$.

$(\{a{\rightarrow}b\})^r_{<5}$ denotes maps in the class $(\{a{\rightarrow}b\})^r$ between spaces of size less than $5$.

a Hausdorff space $K$ is compact iff $K\longrightarrow \{*\}$ is in $((\{a\}\longrightarrow \{a{\rightarrow}b\})^r_{<5})^{lr}$

a Hausdorff space $K$ is compact iff $K\longrightarrow \{*\}$ is in $$ \{\, \{a\leftrightarrow b\}\longrightarrow \{a=b\},\, \{a{\rightarrow}b\}\longrightarrow \{a=b\},\, \{b\}\longrightarrow \{a{\rightarrow}b\},\,\{a{\leftarrow}o{\rightarrow}b\}\longrightarrow \{a=o=b\}\,\,\}^{lr}$$

a space $D$ is discrete iff $ \emptyset \longrightarrow D$ is in $ (\emptyset\longrightarrow \{*\})^{rl} $

a space $D$ is antidiscrete iff $ {D} \longrightarrow \{*\} $ is in $(\{a,b\}\longrightarrow \{a=b\})^{rr}= (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{lr} $

a space $K$ is connected or empty iff $K\longrightarrow \{*\}$ is in $(\{a,b\}\longrightarrow \{a=b\})^l $

a space $K$ is connected and non-empty iff for some arrow $\{*\}\longrightarrow K$ it holds that $ \{*\}\longrightarrow K$ is in
$ (\emptyset\longrightarrow \{*\})^{rll} = (\{a\}\longrightarrow \{a,b\})^l$

a space $K$ is non-empty iff $K\longrightarrow \{*\}$ is in $ (\emptyset\longrightarrow \{*\})^l$

a space $K$ is empty iff $K \longrightarrow \{*\}$ is in $ (\emptyset\longrightarrow \{*\})^{ll}$

a space $K$ is $T_0$ iff $K \longrightarrow \{*\}$ is in $ (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^r$

a space $K$ is $T_1$ iff $K \longrightarrow \{*\}$ is in $ (\{a{\rightarrow}b\}\longrightarrow \{a=b\})^r$

a space $X$ is Hausdorff iff $\{x,y\} \longrightarrow {X} \,\rightthreetimes\, \{ {x} {\rightarrow} {o} {\leftarrow} {y} \} \longrightarrow \{ x=o=y \}$

a non-empty space $X$ is regular (T3) iff for each arrow $ \{x\} \longrightarrow X$ it holds $ \{x\} \longrightarrow {X} \,\rightthreetimes\, \{x{\rightarrow}X{\leftarrow}U{\rightarrow}F\} \longrightarrow \{x=X=U{\rightarrow}F\}$

a space $X$ is normal (T4) iff $\emptyset \longrightarrow {X} \,\rightthreetimes\, \{a{\leftarrow}U{\rightarrow}x{\leftarrow}V{\rightarrow}b\}\longrightarrow \{a{\leftarrow}U=x=V{\rightarrow}b\}$

a space $X$ is completely normal iff $\emptyset\longrightarrow {X} \,\rightthreetimes\, [0,1]\longrightarrow \{0{\leftarrow}x{\rightarrow}1\}$ where the map $[0,1]\longrightarrow \{0{\leftarrow}x{\rightarrow}1\}$ sends $0$ to $0$, $1$ to $1$, and the rest $(0,1)$ to $x$ a space $X$ is path-connected iff $\{0,1\} \longrightarrow [0,1] \,\rightthreetimes\, {X} \longrightarrow \{*\}$

a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow K$ it holds $\{x,y\} \hookrightarrow {K} \,\rightthreetimes\, {X} \longrightarrow \{*\}$

$(\emptyset\longrightarrow \{*\})^r$ is the class of surjections

$(\emptyset\longrightarrow \{*\})^{rr}$ is the class of subsets, i.e. injective maps $A\hookrightarrow B$ where the topology on $A$ is induced from $B$

$(\{b\}\longrightarrow \{a{\rightarrow}b\})^l$ is the class of maps with dense image

$(\{b\}\longrightarrow \{a{\rightarrow}b\})^{lr}$ is the class of closed subsets $A \subset X$, $A$ a closed subset of $X$

$((\{a\}\longrightarrow \{a{\rightarrow}b\})^r_{<5})^{lr}$ is roughly the class of proper maps, i.e. a map between T4 spaces is in the class iff it is proper