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The lifting property is a property of a pair of morphism morphisms s in acategory. 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 tend to be characterized by properties opposite of those in . For example, a morphism in Sets is surjective iff it has the right lifting property against the archetypical non-surjective map , and injective iff it has either left or right lifting property against the archetypical non-injective map . (For more such examples see at separation axioms in terms of lifting properties.)
\begin{definition}\label{LiftingPropertiesOfMorphisms} (lifting properties of morphisms) \linebreak A morphism in a category has the left lifting property with respect to a morphism , and also has the right lifting property with respect to , sometimes denoted i\,\,⧄\,\, p
or , iff the following implication holds for each morphism and in the category:
\begin{tikzcd} A \ar[dd, i] \ar[rr, f] && X\ar[dd,p] \ \ B \ar[rr,g] \ar[uurr, dashed, { \exists }{description}] &&Y \end{tikzcd} \end{definition}
This is sometimes also known as the morphism being ‘’weakly orthogonal to’‘ the morphism ; however, ‘’orthogonal to’‘ will refer to the stronger property that whenever and are as above, the diagonal morphism exists and is also required to be unique.
\begin{remark} \label{LiftingForObjects} (lifting properties of objects) \linebreak One also speaks of objects having left or right lifting properties (for instance in the definition of projective objects and injective objects, respectively, or in the characterization of cofibrant objects and fibrant objects, respectively), by which one then means, respectively: \end{remark}
\begin{tikzcd} \color{lightgray} \varnothing \ar[dd, \exists !, lightgray] \ar[rr, \exists !, lightgray] && X \ar[dd,p] \ \ B \ar[rr,g] \ar[uurr, dashed, { \exists }{description}] && Y \mathrlap{\,,} \end{tikzcd}
\begin{tikzcd} A \ar[dd, i] \ar[rr, f] && X\ar[dd,\exists !, lightgray] \ \ B \ar[rr, \exists !, lightgray] \ar[uurr, dashed, { \exists }{description}] && \color{lightgray} \ast \mathrlap{\,,} \end{tikzcd}
\begin{definition} \label{QuillenNegation} (orthogonal class/Quillen negation) \linebreak Given a class of morphisms in a category , its
\multiscripts{^⧄}{M}{}
or
\multiscripts{}{M}{^⧄}
is the class of all morphisms which have the left, respectively right, lifting property (in the sense of Def. \ref{LiftingPropertiesOfMorphisms}) with respect to each morphism in the class M
:
M^{⧄}
\;\coloneqq\;
\Big\{
p \,\in\, Mor(\mathcal{C})
\;\big\vert\;
\underset{i \in M}{\forall}
\; i \,⧄\, p
\Big\},
\;\;\;\;\;
{}^{⧄}M
\;\coloneqq\;
\Big\{
i \,\in\, Mor(\mathcal{C})
\;\big\vert\;
\underset{p \in M}{\forall}
\; i \,⧄\, p
\Big\}
\,.
\end{definition}
Decyphering notation in most of the examples below leads to standard definitions or reformulations. The intuition behind most examples below is that the class of morphisms consists of simple or archetypal examples related to the property defined.
We use the notation of Def. \ref{QuillenNegation}.
In Set,
\big\{
\varnothing \to \{*\}\big
\}^{⧄}
\;\;\;
=
\;\;\;
Srjctv
is the class of surjective functions,
\big( \{a,b\}\to \{*\} \big)^{⧄}
\;\;\;
=
\;\;\;
Injctv
is the class of injective functions.
In the category RMod of modules over a commutative ring (recalling that we use thenotation of Def. \ref{QuillenNegation}):
The surjective homomorphisms are those with the right lifting property against the initial homomorphism from the zero module into the ground ring:
\{ 0 \to R \}^{⧄}
\;\;\;=\;\;\;
Srjctv
\,.
The injective homomorphisms are those with the right lifting property against the terminal homomorphism from the ground ring into the zero module:
\{ R \to 0 \}^{⧄}
\;\;\;=\;\;\;
Injctv
\,.
An -module is projective iff (by direct unwinding of the definitions of projective objects and lifts) the initial morphism (out of the zero module into the ground ring) has the left lifting property against all surjective homomorphisms.
With the notation of Def. \ref{QuillenNegation} this reads as follows:
M\;\text{projective}
\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;
\{ 0 \to M \}
\;⧄\;
Srjctv
\;\;\;\;\;
\overset{
\text{(eq:SurjectiveModuleMapsAsRightOrthClass)}
}{
\Leftrightarrow
}
\;\;\;\;\;
\{ 0 \to M \}
\;⧄\;
\Big(
\{ 0 \to R \}^{⧄}
\Big)
\;\;\;\;\;
\Leftrightarrow
\;\;\;\;\;
\{ 0 \to M \}
\;\in\;
\multiscripts{^{⧄}}
{
\Big(
\{ 0 \to R \}^{^⧄}
\Big)
}{}
In the category Grp of groups,
\{\mathbb{Z} \to 0\}^{⧄ r}
, resp. \{0\to \mathbb{Z}\}^{⧄ r}
, is the class of injections, resp. surjections (where denotes the infinite cyclic group),
A group is a free group iff is in \{0\to \mathbb{Z} \}^{⧄ r\ell},
A group is torsion-free iff is in \{ n \mathbb{Z} \to \mathbb{Z} : n\ge0 \}^{⧄ r},
A subgroup of is pure? iff is in \{ n\mathbb{Z}\to \mathbb{Z} : n\ge0 \}^{⧄ r}.
(*\to 1)^{⧄ l}
is the class of retracts
(1\to *)^{⧄ r}
is the class of split homomorphisms
(0\longrightarrow \mathbb{Z})^{⧄ r}
is the class of surjections
(\mathbb{Z}\to 1)^{⧄ r}
is the class of injections
a group is free iff is in (0\longrightarrow \mathbb{Z})^{⧄rl}
a group is Abelian iff is in ( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄ r}
group can be obtained from by adding commutation relations, i.e.~the kernel of is generated by commutators , , iff is in ( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄rl}
subgroup of is the normal span of substitutions in words of the free group iff is in ( \mathbb{F}_n \to \mathbb{F}_n/\le\!w_1,...,w_i\!\ge)^{⧄rl}
\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell l}
is the class of homomorphisms whose kernel is perfect
For a finite group , in the category of finite groups,
\{0\to {\mathbb{Z}}/p{\mathbb{Z}}\} \,\,⧄\,\, G\to 1
iff the order of is prime to ,
G\to 1 \in (0\to {\mathbb{Z}}/p{\mathbb{Z}})^{⧄ rr}
iff is a $p$-group,
is nilpotent iff the diagonal map is in (1\to *)^{⧄ \ell r}
where denotes the class of maps
a finite group is soluble? iff is in \{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}.
Moreover,
\{0\to G : G\,\,\text{ arbitrary}\}^{⧄ \ell r}
is the class of subnormal subgroups
\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}
, and is the class of subgroups such that there is a chain of subnormal subgroups such that is Abelian, for .
\{1 \to S\}^{⧄ \ell r}
is the class of subgroups such that there is a chain of subnormal subgroups such that embeds into , for .
(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄r}
is the class of homomorphisms whose kernel has no elements of order
(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄rr}
is the class of surjective homomorphisms whose kernel is a -group
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:
the cofibrations in a model category are precisely the class with the left lifting property against the acyclic fibrations,
the fibrations in a model category are precisely the class with the right lifting property against the acyclic cofibrations,
The classical model structure on topological spaces is controlled by the following lifting properties:
consider let be the class of maps , embeddings of the boundary of a ball into the ball . Let be the class of maps embedding the upper semi-sphere into the disk. WC_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
A map has the ‘’path lifting property’‘ iff \{0\}\to [0,1] \,\,⧄\,\, f
where is the inclusion of one end point of the closed interval into the interval .
A map has the homotopy lifting property iff X \to X\times [0,1] \,\,⧄\,\, f
where is the map .
The classical model structure on simplicial sets is controlled by the following lifting properties:
Let be the class of boundary inclusions , and let be the class of horn inclusions . Then the classes of Kan fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}
. (Model Categories, Def. 3.2.1, Th.3.6.5)
A model structure on chain complexes is controlled by the following lifting properties:
WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}
are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. (Model Categories, Def. 2.3.3, Th.2.3.11)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.
In the category of uniform spaces or metric spaces with uniformly continuous maps.
A space is complete iff \{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, X\to \{0\}
where is the obvious inclusion between the two subspaces of the real line with induced metric, and is the metric space consisting of a single point,
A subspace is closed iff \{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, A\to X.
The following lifting properties are calculated in the category of (all) topological spaces. Below we use notation defined in the page on lifting properties
(\emptyset\longrightarrow \{o\})^{⧄r}
is the class of surjections
(\emptyset\longrightarrow \{o\})^{⧄r}
is the class of maps where or
(\emptyset\longrightarrow \{o\})^{⧄rr}=\{\{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}=\{\{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}
is the class of subsets, i.e. injective maps where the topology on is induced from
(\emptyset\longrightarrow \{o\})^{⧄lr}
is the class of maps , arbitrary
(\emptyset\longrightarrow \{o\})^{⧄lrr}
is the class of maps which admit a section
(\emptyset\longrightarrow \{o\})^{⧄l}
consists of maps such that either or
(\emptyset\longrightarrow \{o\})^{⧄rl}
is the class of maps of form where is discrete
(\emptyset\longrightarrow \{o\})^{⧄rll}
is the class of maps such that each connected subset of intersects the image of ; for “nice” spaces it means that the map is surjective, where “nice” means that connected componets are both open and closed.
(\emptyset\longrightarrow \{o\})^{⧄rllr}
is the class of maps of form where denotes the disjoint union of and .
\{ \{z\leftrightarrow x\leftrightarrow y\rightarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄l} = \{\{c\}\longrightarrow \{o\rightarrow c\}\}^{⧄lr}
is the class of closed inclusions where is closed
\{ \{z\leftrightarrow x\leftrightarrow y\leftarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}
is the class of open inclusions where is open
\{ \{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄l}
is the class of closed maps where the topology on is pulled back from
\{ \{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄l}
is the class of open maps where the topology on is pulled back from
(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄l}
is the class of maps with dense image
(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}=\{ \{z\leftrightarrow x \leftrightarrow y\rightarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}
is the class of closed subsets , a closed subset of
\{ \{z\leftrightarrow x \leftrightarrow y\leftarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}
is the class of open subsets , a open subset of
(\{a\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}
is the class of subsets such that is the intersection of open subsets containing
((\{a\}\longrightarrow \{a \searrow b\})^{⧄r}_{\le 4})^{⧄lr}
is roughly the class of proper maps
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.
a space is non-empty iff is in (\emptyset\longrightarrow \{o\})^{⧄l}
a space is empty iff is in (\emptyset\longrightarrow \{o\})^{⧄ll}
a space is iff is in (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄r}
a space is iff is in (\{a{ \searrow}b\}\longrightarrow \{a=b\})^{⧄r}
a space is Hausdorff iff for each injective map it holds \{x,y\} \hookrightarrow {X} \,⧄\, \{ {x} { \searrow} {o} { \swarrow} {y} \} \longrightarrow \{ x=o=y \}
a non-empty space is regular (T3) iff for each arrow it holds \{x\} \longrightarrow {X} \,⧄\, \{x{ \searrow}X{ \swarrow}U{ \searrow}F\} \longrightarrow \{x=X=U{ \searrow}F\}
a space is normal (T4) iff \emptyset \longrightarrow {X} \,⧄\, \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}
a space is completely normal iff \emptyset\longrightarrow {X} \,⧄\, [0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\}
where the map sends to , to , and the rest to
a space is hereditary normal iff \emptyset \to X ⧄
\{ x \leftarrow au \leftrightarrow u' \leftarrow u \leftarrow uv \rightarrow v \rightarrow v'\leftrightarrow bv \rightarrow x \}
\longrightarrow
\{ x \leftarrow au \leftrightarrow u' = u \leftarrow uv \rightarrow v = v'\leftrightarrow bv \rightarrow x \}
a space is path-connected iff \{0,1\} \longrightarrow [0,1] \,⧄\, {X} \longrightarrow \{o\}
a space is path-connected iff for each Hausdorff compact space and each injective map it holds \{x,y\} \hookrightarrow {K} \,⧄\, {X} \longrightarrow \{o\}
A map is a quotient iff X\to Y \,\,⧄\,\, \{o \rightarrow c\}\longrightarrow \{o\leftrightarrow c\}
For every pair of disjoint closed subsets of , the closures of their images of do not intersect, if X\to Y \,\,⧄\,\, \{x\leftarrow o\rightarrow y\}\longrightarrow \{x=o=y\}
A topological space is extremally disconnected iff \emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{u\rightarrow a=b\leftarrow v\}
A topological space is zero-dimensional iff \emptyset\to X \,\,⧄\,\, \{a\leftarrow u,v\rightarrow b\}\longrightarrow \{a\leftarrow u=v\rightarrow b\}
A topological space is ultranormal iff \emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{a\leftarrow u=v\rightarrow v\}
is in (\emptyset\longrightarrow \{o\})^{⧄rll}
iff is connected
is totally disconnected iff is in (\emptyset\longrightarrow \{o\})^{⧄rllr}
for each map (or, in other words, each point ).
a Hausdorff space is compact iff is in ((\{o\}\longrightarrow \{o{ \searrow}c\})^{⧄r}_{\le5})^{⧄lr}
a Hausdorff space is compact iff is in $
\{\, \{a\leftrightarrow b\}\longrightarrow \{a=b\},\, \{o{ \searrow}c\}\longrightarrow \{o=c\},\,
\{c\}\longrightarrow \{o{ \searrow}c\},\,\{a{ \swarrow}o{ \searrow}b\}\longrightarrow \{a=o=b\}\,\,\}^{⧄lr}
$
a topological space is compactly generated iff is in \big(\{\{0 \leftrightarrow 1\}\to\{0=1\}\}\cup\{\varnothing \to K \,\,:\,\, K\,\, \text{ compact}\}\big)^{⧄rl}
a space is discrete iff is in (\emptyset\longrightarrow \{o\})^{⧄rl}
a space is codiscrete iff is in(\{a,b\}\longrightarrow \{a=b\})^{⧄rr}= (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄lr}
a space is connected or empty iff is in (\{a,b\}\longrightarrow \{a=b\})^{⧄l}
a space is totally disconnected and non-empty iff is in (\{a,b\}\longrightarrow \{a=b\})^{⧄lr}
a space is connected and non-empty iff for some arrow it holds that is in (\emptyset\longrightarrow \{o\})^{⧄rll} = (\{a\}\longrightarrow \{a,b\})^{⧄l}
A topological space has Lebesgue dimension at most iff for each finite set \emptyset\to X \,\,⧄\,\,
\{ (F,J): 1\leq |F|\leq n+1, F\subset J\subset I\}\longrightarrow
\{ J: 1\leq |J|, J\subset I\}
where the order on the domain is given by iff and .
A topological space has Lebesgue dimension at most iff for each closed subset of
A\to X \,\,⧄\,\, \mathbb{S}^n\to \{o\}
where denotes the -sphere.
A finite CW complex is contractible iff X \longrightarrow {\{\bullet\}} \in \{ \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}\}^{⧄rl}
The map defining Separation Axiom above is a trivial Serre fibration, hence their {}^{⧄rl}
-orthogonals are classes of trivial fibrations.
\begin{conjecture} If is a “nice” map, then is a trivial fibration iff
f\in\{ \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}
\}^{⧄rl}
\end{conjecture}
One can make the same conjecture for the map defining Separation Axiom (hereditary normal) since it is also a trivial Serre fibration.
In model theory, a number of the Shelah’s divining lines, namely , and are expressed as Quillen lifting properties of form
where is the terminal object, and is a situs associated with a model and a formula, and and are objects of combinatorial nature, in the category of simplicial objects in the category of filters.
Last revised on February 4, 2024 at 19:32:13. See the history of this page for a list of all contributions to it.