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 $p\circ h=g$.

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.

• 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, $WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}$.

• 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.

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,

• A subspace $i:A\to X$ is closed iff $\{1/n\}_{n\in \mathbb {N} }\to \{0\}\cup \{1/n\}_{n\in \mathbb {N} }\perp A\to X.$

References

• Hovey, Mark (1999). Model Categories.
• J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories