nLab
lift

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Idea

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. A number of elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

Often it is useful to think of lifting properties as a expressing a kind of qualitative negation (“Quillen negation”): The morphisms with the left/right lifting property against those in a class PP tend to be characterized by properties opposite of those in PP. For example, a morphism in Sets is surjective iff it has the right lifting property against the archetypical non-surjective map {*}\varnothing \to \{*\}, and injective iff it has either left or right lifting property against the archetypical non-injective map {x 1,x 2}{*}\{x_1,x_2\}\to \{*\}. (For more such examples see at separation axioms in terms of lifting properties.)

Definition

Definition

A morphism ii in a category has the ‘’left lifting property’‘ with respect to a morphism pp, and pp also has the ‘’right lifting property’‘ with respect to ii, sometimes denoted ipi\,\,⧄\,\, p or ipi\downarrow p, iff the following implication holds for each morphism ff and gg in the category:

  • if the outer square of the following diagram commutes, then there exists hh completing the diagram, i.e. for each f:AXf:A\to X and g:BYg:B\to Y such that pf=gip\circ f = g \circ i there exists h:BXh:B\to X such that hi=fh\circ i = f and ph=gp\circ h = g.

This is sometimes also known as the morphism ii being ‘’weakly orthogonal to’‘ the morphism pp; however, ‘’orthogonal to’‘ will refer to the stronger property that whenever ff and gg are as above, the diagonal morphism hh exists and is also required to be unique.

For a class CC of morphisms in a category, its ‘’left weak orthogonal’‘ or its ‘’left Quillen negation’‘ C C^{⧄ \ell} with respect to the lifting property, respectively its ‘’right weak orthogonal’‘ and its ‘’right Quillen negation’‘ C rC^{⧄r} is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class CC. In notation,

C :={ipC,ip}C r:={piC,ip},C lr:=(C l) r C^{⧄ \ell} := \{ i \mid \forall p\in C, i\,\,⧄\,\, p\} \,\,\, C^{⧄ r} := \{ p \mid \forall i\in C, i\,\,⧄\,\, p\}, \,\,\, C^{⧄ lr} := (C^{⧄ l})^{⧄ r}

Taking the Quillen negation of a class CC is a simple way to define a class of morphisms excluding isomorphisms from CC, in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right Quillen negation {{*}} r\{\emptyset \to \{*\}\}^{⧄ r} of the simplest non-surjection {*},\emptyset\to \{*\}, is the class of surjections. The left and right Quillen negation of {x 1,x 2}{*},\{x_1,x_2\}\to \{*\}, the simplest non-injection, are both precisely the class of injections, {{x 1,x 2}{*}} ={{x 1,x 2}{*}} r={ff is an injection }.\{\{x_1,x_2\}\to \{*\}\}^{⧄ \ell} = \{\{x_1,x_2\}\to \{*\}\}^{⧄ r} = \{ f \mid f \text{ is an injection } \}.

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

Examples of lifting properties

Decyphering notation in most of the examples below leads to standard definitions or reformulations. The intution behind most examples below is that the class of morphisms consists of simple or archetypal examples related to the property defined.

Elementary examples

Sets

In Set,

Modules

In the category RMod of modules over a commutative ring RR,

Groups

In the category Grp of groups,

For a finite group GG, in the category of finite groups,

Moreover,

In algebraic topology and in model categories

Lifting properties are paramount in homotopy theory and algebraic topology. In “abstract homotopy theory” lifting properties are encoded in the structures of model categories, whose defintion revolves all around compatible classes of weak factorization systems. In particular:

Serre fibrations of topological spaces

The classical model structure on topological spaces Top QuTop_{Qu} is controlled by the following lifting properties:

consider let C 0C_0 be the class of maps S nD n+1S^n\to D^{n+1}, embeddings of the boundary S n=D n+1S^n=\partial D^{n+1} of a ball into the ball D n+1D^{n+1}. Let WC 0WC_0 be the class of maps embedding the upper semi-sphere into the disk. WC 0 ,WC 0 r,C 0 ,C 0 rWC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r} are the classes of Serre fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. Hovey, Model Categories, Def. 2.4.3, Th.2.4.9

Hurewicz fibrations of topological spaces

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

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

Kan fibrations of simplicial sets

The classical model structure on simplicial sets sSet QusSet_{Qu} is controlled by the following lifting properties:

Let C 0C_0 be the class of boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n], and let WC 0WC_0 be the class of horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n]. Then the classes of Kan fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, WC 0 ,WC 0 r,C 0 ,C 0 rWC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}. (Model Categories, Def. 3.2.1, Th.3.6.5)

Degreewise surjections of chain complexes

A model structure on chain complexes is controlled by the following lifting properties:

Topology

Many elementary properties in general topology, such as compactness, being dense or open, can be expressed as iterated Quillen negation of morphisms of finite topological spaces in the category Top of topological spaces. This leads to a concise, if useless, notation for a number of properties. Items below use notation for morphisms of finite topological spaces defined in the page on separation axioms in terms of lifting properties, and some examples are explained there in detail.

Uniform spaces

In the category of uniform spaces or metric spaces with uniformly continuous maps.

In topological spaces

The following lifting properties are calculated in the category of (all) topological spaces.

Iterated lifting properties

Separation axioms

Here follows a list of examples of well-known properties defined by iterated Quillen negation starting from maps between finite topological spaces, often with less than 5 elements. See at separation axioms in terms of lifting properties for more on the following.

Model theory

In model theory, a number of the Shelah’s divining lines, namely NOP,NSOP,NSOP i,NTP,NTP iNOP, NSOP, NSOP_i, NTP, NTP_i, and NATPNATP are expressed as Quillen lifting properties of form

A B M A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top

where \top is the terminal object, and MM is a situs associated with a model and a formula, and AA and BB are objects of combinatorial nature, in the category of simplicial objects in the category of filters.