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 $p\circ h = g$ .

::File: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.{{cite book | last = Hovey | first = Mark |title = Model Categories | url = https://archive.org/details/arxiv-math9803002 }} Def. 2.4.3, Th.2.4.9

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, $WC_0^{\perp\ell}, WC_0^{\perp\ell r}, C_0^{\perp\ell}, C_0^{\perp\ell r}$ .{{cite book | last = Hovey | first = Mark |title = Model Categories | url = https://archive.org/details/arxiv-math9803002 }} Def. 3.2.1, Th.3.6.5

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.{{cite book | last = Hovey | first = Mark |title = Model Categories | url = https://archive.org/details/arxiv-math9803002 }} Def. 2.3.3, Th.2.3.11

=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 $\{1/n\}_{n \in \N} \to \{0\}\cup \{1/n\}_{n \in \N} \perp A\to X.$

Notes

References

{{cite book | last = Hovey | first = Mark |title = Model Categories | url = https://archive.org/details/arxiv-math9803002 |date=1999}}
J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories

Category:Category theory