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 $$p\circ h = g$$.


 * Model_category_lifting.png

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{align}

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{align}$$

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,


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


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


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


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

For a finite group $$G$$,


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


 * $$G\to 1 \in (0\to {\Z}/p{\Z})^{\perp rr}$$ 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 $$\emptyset\to X \perp \cup_n (-n,n) \to \R$$ where $$\cup_n (-n,n) \to \R$$ 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 $$\{1/n\}_{n \in \N} \to \{0\}\cup \{1/n\}_{n \in \N} \perp X\to \{0\}$$ where $$\{1/n\}_{n \in \N} \to \{0\}\cup \{1/n\}_{n \in \N}$$ 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 $$\{1/n\}_{n \in \N} \to \{0\}\cup \{1/n\}_{n \in \N} \perp A\to X.$$