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 [math]\displaystyle{ i }[/math] in a category has the left lifting property with respect to a morphism [math]\displaystyle{ p }[/math], and [math]\displaystyle{ p }[/math] also has the right lifting property with respect to [math]\displaystyle{ i }[/math], sometimes denoted [math]\displaystyle{ i\perp p }[/math] or [math]\displaystyle{ i\downarrow p }[/math], iff the following implication holds for each morphism [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] in the category:

• if the outer square of the following diagram commutes, then there exists [math]\displaystyle{ h }[/math] completing the diagram, i.e. for each [math]\displaystyle{ f:A\to X }[/math] and [math]\displaystyle{ g:B\to Y }[/math] such that [math]\displaystyle{ p\circ f = g \circ i }[/math] there exists [math]\displaystyle{ h:B\to X }[/math] such that [math]\displaystyle{ h\circ i = f }[/math] and [math]\displaystyle{ p\circ h = g }[/math].

This is sometimes also known as the morphism [math]\displaystyle{ i }[/math] being orthogonal to the morphism [math]\displaystyle{ p }[/math]; however, this can also refer to the stronger property that whenever [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are as above, the diagonal morphism [math]\displaystyle{ h }[/math] exists and is also required to be unique.

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

[math]\displaystyle{ \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} }[/math]

Taking the orthogonal of a class [math]\displaystyle{ C }[/math] is a simple way to define a class of morphisms excluding non-isomorphisms from [math]\displaystyle{ C }[/math], in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right orthogonal [math]\displaystyle{ \{\emptyset \to \{*\}\}^{\perp r} }[/math] of the simplest non-surjection [math]\displaystyle{ \emptyset\to \{*\}, }[/math] is the class of surjections. The left and right orthogonals of [math]\displaystyle{ \{x_1,x_2\}\to \{*\}, }[/math] the simplest non-injection, are both precisely the class of injections,

[math]\displaystyle{ \{\{x_1,x_2\}\to \{*\}\}^{\perp\ell} = \{\{x_1,x_2\}\to \{*\}\}^{\perp r} = \{ f \mid f \text{ is an injection } \}. }[/math]

It is clear that [math]\displaystyle{ C^{\perp\ell r} \supset C }[/math] and [math]\displaystyle{ C^{\perp r\ell} \supset C }[/math]. The class [math]\displaystyle{ C^{\perp r} }[/math] 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, [math]\displaystyle{ C^{\perp \ell} }[/math] 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 [math]\displaystyle{ C^{\perp\ell}, C^{\perp r}, C^{\perp\ell r}, C^{\perp\ell\ell} }[/math], where [math]\displaystyle{ C }[/math] is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class [math]\displaystyle{ C }[/math] is a kind of negation of the property of being in [math]\displaystyle{ C }[/math], and that right-lifting is also a kind of negation. Hence the classes obtained from [math]\displaystyle{ C }[/math] by taking orthogonals an odd number of times, such as [math]\displaystyle{ C^{\perp\ell}, C^{\perp r}, C^{\perp\ell r\ell}, C^{\perp\ell\ell\ell} }[/math] etc., represent various kinds of negation of [math]\displaystyle{ C }[/math], so [math]\displaystyle{ C^{\perp\ell}, C^{\perp r}, C^{\perp\ell r\ell}, C^{\perp\ell\ell\ell} }[/math] each consists of morphisms which are far from having property [math]\displaystyle{ C }[/math].

Examples of lifting properties in algebraic topology

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

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

Examples of lifting properties coming from model categories

Fibrations and cofibrations.

• Let Top be the category of topological spaces, and let [math]\displaystyle{ C_0 }[/math] be the class of maps [math]\displaystyle{ S^n\to D^{n+1} }[/math], embeddings of the boundary [math]\displaystyle{ S^n=\partial D^{n+1} }[/math] of a ball into the ball [math]\displaystyle{ D^{n+1} }[/math]. Let [math]\displaystyle{ WC_0 }[/math] be the class of maps embedding the upper semi-sphere into the disk. [math]\displaystyle{ WC_0^{\perp\ell}, WC_0^{\perp\ell r}, C_0^{\perp\ell}, C_0^{\perp\ell r} }[/math] are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.

• Let sSet be the category of simplicial sets. Let [math]\displaystyle{ C_0 }[/math] be the class of boundary inclusions [math]\displaystyle{ \partial \Delta[n] \to \Delta[n] }[/math], and let [math]\displaystyle{ WC_0 }[/math] be the class of horn inclusions [math]\displaystyle{ \Lambda^i[n] \to \Delta[n] }[/math]. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, [math]\displaystyle{ WC_0^{\perp\ell}, WC_0^{\perp\ell r}, C_0^{\perp\ell}, C_0^{\perp\ell r} }[/math].

• Let [math]\displaystyle{ \mathbf{Ch}(R) }[/math] be the category of chain complexes over a commutative ring [math]\displaystyle{ R }[/math]. Let [math]\displaystyle{ C_0 }[/math] be the class of maps of form

[math]\displaystyle{ \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, }[/math]

and [math]\displaystyle{ WC_0 }[/math] be

[math]\displaystyle{ \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. }[/math]

Then [math]\displaystyle{ WC_0^{\perp\ell}, WC_0^{\perp\ell r}, C_0^{\perp\ell}, C_0^{\perp\ell r} }[/math] are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.

Elementary examples in various categories

In Set,

• [math]\displaystyle{ \{\emptyset\to \{*\}\}^{\perp r} }[/math] is the class of surjections,

• [math]\displaystyle{ (\{a,b\}\to \{*\})^{\perp r}=(\{a,b\}\to \{*\})^{\perp\ell} }[/math] is the class of injections.

In the category [math]\displaystyle{ R\text{-}\mathbf{Mod} }[/math] of modules over a commutative ring [math]\displaystyle{ R }[/math],

• [math]\displaystyle{ \{0\to R\}^{\perp r}, \{R\to 0\}^{\perp r} }[/math] is the class of surjections, resp. injections,

• A module [math]\displaystyle{ M }[/math] is projective, resp. injective, iff [math]\displaystyle{ 0\to M }[/math] is in [math]\displaystyle{ \{0\to R\}^{\perp r\ell} }[/math], resp. [math]\displaystyle{ M\to 0 }[/math] is in [math]\displaystyle{ \{R\to 0\}^{\perp rr} }[/math].

In the category [math]\displaystyle{ \mathbf{Grp} }[/math] of groups,

• [math]\displaystyle{ \{\Z \to 0\}^{\perp r} }[/math], resp. [math]\displaystyle{ \{0\to \Z\}^{\perp r} }[/math], is the class of injections, resp. surjections (where [math]\displaystyle{ \Z }[/math] denotes the infinite cyclic group),

• A group [math]\displaystyle{ F }[/math] is a free group iff [math]\displaystyle{ 0\to F }[/math] is in [math]\displaystyle{ \{0\to \Z \}^{\perp r\ell}, }[/math]

• A group [math]\displaystyle{ A }[/math] is torsion-free iff [math]\displaystyle{ 0\to A }[/math] is in [math]\displaystyle{ \{ n \Z\to \Z : n\gt 0 \}^{\perp r}, }[/math]

• A subgroup [math]\displaystyle{ A }[/math] of [math]\displaystyle{ B }[/math] is pure iff [math]\displaystyle{ A \to B }[/math] is in [math]\displaystyle{ \{ n\Z\to \Z : n\gt 0 \}^{\perp r}. }[/math]

For a finite group [math]\displaystyle{ G }[/math],

• [math]\displaystyle{ \{0\to {\Z}/p{\Z}\} \perp G\to 1 }[/math] iff the order of [math]\displaystyle{ G }[/math] is prime to [math]\displaystyle{ p }[/math],

• [math]\displaystyle{ G\to 1 \in (0\to {\Z}/p{\Z})^{\perp rr} }[/math] iff [math]\displaystyle{ G }[/math] is a [math]\displaystyle{ p }[/math]-group,

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

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

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

• a space [math]\displaystyle{ X }[/math] satisfies the separation axiom T₀ iff [math]\displaystyle{ X\to \{*\} }[/math] is in [math]\displaystyle{ (\{0\leftrightarrow 1\} \to \{*\})^{\perp r}, }[/math]

• a space [math]\displaystyle{ X }[/math] satisfies the separation axiom T₁ iff [math]\displaystyle{ \emptyset\to X }[/math] is in [math]\displaystyle{ ( \{0\to 1\}\to \{*\})^{\perp r}, }[/math]

• [math]\displaystyle{ (\{1\}\to \{0\to 1\})^{\perp\ell} }[/math] is the class of maps with dense image,

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

• [math]\displaystyle{ (\emptyset\to \{*\})^{\perp r} }[/math] is the class of surjective maps,

• [math]\displaystyle{ (\emptyset\to \{*\})^{\perp r\ell} }[/math] is the class of maps of form [math]\displaystyle{ A\to A\cup D }[/math] where [math]\displaystyle{ D }[/math] is discrete,

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

• [math]\displaystyle{ (\{0,1\}\to \{*\})^{\perp r} }[/math] is the class of injective maps,

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

• for a connected space [math]\displaystyle{ X }[/math], each continuous function on [math]\displaystyle{ X }[/math] is bounded iff [math]\displaystyle{ \emptyset\to X \perp \cup_n (-n,n) \to \R }[/math] where [math]\displaystyle{ \cup_n (-n,n) \to \R }[/math] is the map from the disjoint union of open intervals [math]\displaystyle{ (-n,n) }[/math] into the real line [math]\displaystyle{ \mathbb{R}, }[/math]

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

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

In the category of metric spaces with uniformly continuous maps.

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

• A subspace [math]\displaystyle{ i:A\to X }[/math] is closed iff [math]\displaystyle{ \{1/n\}_{n \in \N} \to \{0\}\cup \{1/n\}_{n \in \N} \perp A\to X. }[/math]

References

• Hovey, Mark (1999). Model Categories. https://archive.org/details/arxiv-math9803002.