Lifting property

In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

Formal definition

A morphism $i$ in a category has the left lifting property with respect to a morphism $p$ , and $p$ also has the right lifting property with respect to $i$ , sometimes denoted $i\perp p$ or $i\downarrow p$ , iff the following implication holds for each morphism $f$ and $g$ in the category:

if the outer square of the following diagram commutes, then there exists $h$ completing the diagram, i.e. for each $f:A\to X$ and $g:B\to Y$ such that $p\circ f=g\circ i$ there exists $h:B\to X$ such that $h\circ i=f$ and $p\circ h=g$ .

A commutative diagram in the shape of a square with an anti-diagonal line, which graphically representing the relations stated in the preceding text. There are four letters representing vertices, here listed from left to right, then from top to bottom order, which are "A" (the top-left corner of the square), "X" (the top-right corner of the square), "B" (the bottom-left corner of the square), and "Y" (the bottom-right corner of the square). Additionally, there are five arrows which connect these letters, listed here using the same order as before: a solid-stroke, left to right arrow labeled "f" from A to X (the top-side line of the square); a solid-stroke, top to bottom arrow labeled "i" from A to B (the left-side line of the square); a dotted-stroke, bottom-left to top-right arrow labeled "h" from B to X (the anti-diagonal line of the square); a solid-stroke, top to bottom arrow labeled "p" from X to Y (the right-side line of the square); and a solid-stroke, left to right arrow labeled "g" from B to Y (the bottom-side line of the square).

This is sometimes also known as the morphism $i$ being orthogonal to the morphism $p$ ; however, this can also refer to the stronger property that whenever $f$ and $g$ are as above, the diagonal morphism $h$ exists and is also required to be unique.

For a class $C$ of morphisms in a category, its left orthogonal $C^{\perp \ell }$ or $C^{\perp }$ with respect to the lifting property, respectively its right orthogonal $C^{\perp r}$ or ${}^{\perp }C$ , is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class $C$ . In notation,

{\begin{aligned}C^{\perp \ell }&:=\{i\mid \forall p\in C,i\perp p\}\\C^{\perp r}&:=\{p\mid \forall i\in C,i\perp p\}\end{aligned}}

Taking the orthogonal of a class $C$ is a simple way to define a class of morphisms excluding non-isomorphisms from $C$ , in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right orthogonal $\{\emptyset \to \{*\}\}^{\perp r}$ of the simplest non-surjection $\emptyset \to \{*\},$ is the class of surjections. The left and right orthogonals of $\{x_{1},x_{2}\}\to \{*\},$ the simplest non-injection, are both precisely the class of injections,

\{\{x_{1},x_{2}\}\to \{*\}\}^{\perp \ell }=\{\{x_{1},x_{2}\}\to \{*\}\}^{\perp r}=\{f\mid f{\text{ is an injection }}\}.

It is clear that $C^{\perp \ell r}\supset C$ and $C^{\perp r\ell }\supset C$ . The class $C^{\perp r}$ is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) & composition of morphisms, and contains all isomorphisms (that is, invertible morphisms) of the underlying category. Meanwhile, $C^{\perp \ell }$ is closed under retracts, pushouts, (small) coproducts & transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.

Examples

A number of notions can be defined by passing to the left or right orthogonal several times starting from a list of explicit examples, i.e. as $C^{\perp \ell },C^{\perp r},C^{\perp \ell r},C^{\perp \ell \ell }$ , where $C$ is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class $C$ is a kind of negation of the property of being in $C$ , and that right-lifting is also a kind of negation. Hence the classes obtained from $C$ by taking orthogonals an odd number of times, such as $C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }$ etc., represent various kinds of negation of $C$ , so $C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }$ each consists of morphisms which are far from having property $C$ .

Examples of lifting properties in algebraic topology

A map $f:U\to B$ has the path lifting property iff $\{0\}\to [0,1]\perp f$ where $\{0\}\to [0,1]$ is the inclusion of one end point of the closed interval into the interval $[0,1]$ .

A map $f:U\to B$ has the homotopy lifting property iff $X\to X\times [0,1]\perp f$ where $X\to X\times [0,1]$ is the map $x\mapsto (x,0)$ .

Examples of lifting properties coming from model categories

Fibrations and cofibrations.

Let Top be the category of topological spaces, and let $C_{0}$ be the class of maps $S^{n}\to D^{n+1}$ , embeddings of the boundary $S^{n}=\partial D^{n+1}$ of a ball into the ball $D^{n+1}$ . Let $WC_{0}$ be the class of maps embedding the upper semi-sphere into the disk. $WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[1]

Let sSet be the category of simplicial sets. Let $C_{0}$ be the class of boundary inclusions $\partial \Delta [n]\to \Delta [n]$ , and let $WC_{0}$ be the class of horn inclusions $\Lambda ^{i}[n]\to \Delta [n]$ . Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, $WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}$ .^[2]

Let $\mathbf {Ch} (R)$ be the category of chain complexes over a commutative ring $R$ . Let $C_{0}$ be the class of maps of form

\cdots \to 0\to R\to 0\to 0\to \cdots \to \cdots \to R{\xrightarrow {\operatorname {id} }}R\to 0\to 0\to \cdots ,

and

WC_{0}

be

\cdots \to 0\to 0\to 0\to 0\to \cdots \to \cdots \to R{\xrightarrow {\operatorname {id} }}R\to 0\to 0\to \cdots .

Then

WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}

are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[3]

Elementary examples in various categories

In Set,

$\{\emptyset \to \{*\}\}^{\perp r}$ is the class of surjections,

$(\{a,b\}\to \{*\})^{\perp r}=(\{a,b\}\to \{*\})^{\perp \ell }$ is the class of injections.

In the category $R{\text{-}}\mathbf {Mod}$ of modules over a commutative ring $R$ ,

$\{0\to R\}^{\perp r},\{R\to 0\}^{\perp r}$ is the class of surjections, resp. injections,

A module $M$ is projective, resp. injective, iff $0\to M$ is in $\{0\to R\}^{\perp r\ell }$ , resp. $M\to 0$ is in $\{R\to 0\}^{\perp rr}$ .

In the category $\mathbf {Grp}$ of groups,

$\{\mathbb {Z} \to 0\}^{\perp r}$ , resp. $\{0\to \mathbb {Z} \}^{\perp r}$ , is the class of injections, resp. surjections (where $\mathbb {Z}$ denotes the infinite cyclic group),

A group $F$ is a free group iff $0\to F$ is in $\{0\to \mathbb {Z} \}^{\perp r\ell },$

A group $A$ is torsion-free iff $0\to A$ is in $\{n\mathbb {Z} \to \mathbb {Z} :n>0\}^{\perp r},$

A subgroup $A$ of $B$ is pure iff $A\to B$ is in $\{n\mathbb {Z} \to \mathbb {Z} :n>0\}^{\perp r}.$

For a finite group $G$ ,

$\{0\to {\mathbb {Z} }/p{\mathbb {Z} }\}\perp 1\to G$ iff the order of $G$ is prime to $p$ iff $\{{\mathbb {Z} }/p{\mathbb {Z} }\to 0\}\perp G\to 1$ ,

$G\to 1\in (0\to {\mathbb {Z} }/p{\mathbb {Z} })^{\perp rr}$ iff $G$ is a $p$ -group,

$H$ is nilpotent iff the diagonal map $H\to H\times H$ is in $(1\to *)^{\perp \ell r}$ where $(1\to *)$ denotes the class of maps $\{1\to G:G{\text{ arbitrary}}\},$

a finite group $H$ is soluble iff $1\to H$ is in $\{0\to A:A{\text{ abelian}}\}^{\perp \ell r}=\{[G,G]\to G:G{\text{ arbitrary }}\}^{\perp \ell r}.$

In the category $\mathbf {Top}$ of topological spaces, let $\{0,1\}$ , resp. $\{0\leftrightarrow 1\}$ denote the discrete, resp. antidiscrete space with two points 0 and 1. Let $\{0\to 1\}$ denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let $\{0\}\to \{0\to 1\},\{1\}\to \{0\to 1\}$ etc. denote the obvious embeddings.

a space $X$ satisfies the separation axiom T₀ iff $X\to \{*\}$ is in $(\{0\leftrightarrow 1\}\to \{*\})^{\perp r},$

a space $X$ satisfies the separation axiom T₁ iff $\emptyset \to X$ is in $(\{0\to 1\}\to \{*\})^{\perp r},$

$(\{1\}\to \{0\to 1\})^{\perp \ell }$ is the class of maps with dense image,

$(\{0\to 1\}\to \{*\})^{\perp \ell }$ is the class of maps $f:X\to Y$ such that the topology on $A$ is the pullback of topology on $B$ , i.e. the topology on $A$ is the topology with least number of open sets such that the map is continuous,

$(\emptyset \to \{*\})^{\perp r}$ is the class of surjective maps,

$(\emptyset \to \{*\})^{\perp r\ell }$ is the class of maps of form $A\to A\cup D$ where $D$ is discrete,

$(\emptyset \to \{*\})^{\perp r\ell \ell }=(\{a\}\to \{a,b\})^{\perp \ell }$ is the class of maps $A\to B$ such that each connected component of $B$ intersects $\operatorname {Im} A$ ,

$(\{0,1\}\to \{*\})^{\perp r}$ is the class of injective maps,

$(\{0,1\}\to \{*\})^{\perp \ell }$ is the class of maps $f:X\to Y$ such that the preimage of a connected closed open subset of $Y$ is a connected closed open subset of $X$ , e.g. $X$ is connected iff $X\to \{*\}$ is in $(\{0,1\}\to \{*\})^{\perp \ell }$ ,

for a connected space $X$ , each continuous function on $X$ is bounded iff $\emptyset \to X\perp \cup _{n}(-n,n)\to \mathbb {R}$ where $\cup _{n}(-n,n)\to \mathbb {R}$ is the map from the disjoint union of open intervals $(-n,n)$ into the real line $\mathbb {R} ,$

a space $X$ is Hausdorff iff for any injective map $\{a,b\}\hookrightarrow X$ , it holds $\{a,b\}\hookrightarrow X\perp \{a\to x\leftarrow b\}\to \{*\}$ where $\{a\leftarrow x\to b\}$ denotes the three-point space with two open points $a$ and $b$ , and a closed point $x$ ,

a space $X$ is perfectly normal iff $\emptyset \to X\perp [0,1]\to \{0\leftarrow x\to 1\}$ where the open interval $(0,1)$ goes to $x$ , and $0$ maps to the point $0$ , and $1$ maps to the point $1$ , and $\{0\leftarrow x\to 1\}$ denotes the three-point space with two closed points $0,1$ and one open point $x$ .

In the category of metric spaces with uniformly continuous maps.

A space $X$ is complete iff $\{1/n\}_{n\in \mathbb {N} }\to \{0\}\cup \{1/n\}_{n\in \mathbb {N} }\perp X\to \{0\}$ where $\{1/n\}_{n\in \mathbb {N} }\to \{0\}\cup \{1/n\}_{n\in \mathbb {N} }$ is the obvious inclusion between the two subspaces of the real line with induced metric, and $\{0\}$ is the metric space consisting of a single point,