Lifting property

Short description: Concept category theory (mathematics)

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 $p\circ h=g$.

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,

$$\begin{array}{rl}{C}^{\perp \ell }& :=\{i\mid \mathrm{\forall }p\in C,i\perp p\}\\ C^{\perp r}& :=\{p\mid \mathrm{\forall }i\in C,i\perp p\}\end{array}$$

Properties

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.

In the category Set of sets, the right orthogonal $\{\mathrm{\varnothing }\to \{*\}{\}}^{\perp r}$ of the simplest non-surjection $\mathrm{\varnothing }\to \{*\}$ 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 (that is, if $X$ and $Y$ are objects, $C\perp Y$, and $X$ is a retract of $Y$, then $C\perp X$), 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.

Let $i$, $j$, and $k$ be morphisms such that $i\circ j$ exists. Then:

• If $i\circ j\perp k$ and $j$ is an epimorphism, then $i\perp k$.

• If $k\perp i\circ j$ and $i$ is a monomorphism, then $k\perp j$.

These two properties are useful when the category is equipped with a weak factorisation system consisting of epimorphisms and monomorphisms.

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 $C^{\perp \ell },C^{\perp r},C^{\perp \ell r},C^{\perp \ell \ell }$, etc., where $C$ is a given class of morphisms. A useful intuition is to think that the left and right lifting properties against a class $C$ are a way of expressing a negation of some property of the morphisms in $C$. In this vein, performing a "double negation" can be seen as a kind of "closure" or "completion" procedure.

Elementary examples in various categories

In Set

Let $1$ denote any fixed singleton set, such as $\{0\}$, and let $2$ denote any fixed set with two elements, such as $\{0,1\}$.

• If $i:1\to 2$ denotes either of the two functions from $1$ to $2$, then $\{i{\}}^{\perp l}=\{i{\}}^{\perp r}$ is the class of surjections.

• If $j:2\to 1$ is the unique function from $2$ to $1$, then $\{j{\}}^{\perp r}=\{j{\}}^{\perp l}$ is the class of injections.

In the category of modules over a commutative ring R

Let $0$ denote the zero module and for each $R$-module $M$, let $0\to M$ and $M\to 0$ denote the two unique morphisms between $0$ and $M$.

• $\{0\to R{\}}^{\perp r}$ is the class of surjective module homomorphisms.

• $\{R\to 0{\}}^{\perp r}$ is the class of injective module homomorphisms.

• A module $M$ is projective if and only if $0\to M$ is in $\{0\to R{\}}^{\perp rl}$.

• A module $M$ is injective if and only if $M\to 0$ is in $\{R\to 0{\}}^{\perp rr}$.

In the category of groups

Let $\mathbb{\mathbb{Z}}$ denote the infinite cyclic group of integers under addition.

• $\{0\to \mathbb{\mathbb{Z}}{\}}^{\perp r}$ is the class of surjective group homomorphisms.

• $\{\mathbb{\mathbb{Z}}\to 0{\}}^{\perp r}$ is the class of injective group homomorphisms.

• A group $F$ is a free group if and only if $0\to F$ is in $\{0\to \mathbb{\mathbb{Z}}{\}}^{\perp rl}$.

• A group $A$ is torsion-free if and only if $0\to A$ is in $\{n\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}:n>0{\}}^{\perp r}$.

• A subgroup $A$ of a group $B$ is pure if and only if $A\to B$ is in $\{n\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}:n>0{\}}^{\perp r}$.

For a finite group $G$,

• $\{0\to \mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}\}\perp 1\to G$ iff the order of $G$ is prime to $p$ iff $\{\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}\to 0\}\perp G\to 1$.

• $G\to 1\in (0\to \mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}{)}^{\perp rr}$ 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 of topological spaces

Let $\{0,1\}$ and $\{0\leftrightarrow 1\}$ denote a two-element set with the discrete topology and the indiscrete topology, respectively. Let $\{0\to 1\}$ denote the Sierpinski space of two points, in which the set $\{0\}$ is open (and not closed) and the set $\{1\}$ is closed (and not open), and let $\{0\}\to \{0\to 1\},\{1\}\to \{0\to 1\}$, etc. denote the obvious embeddings.

• A space $X$ is a T₀ space if and only if $X\to \{*\}$ is in $(\{0\leftrightarrow 1\}\to \{*\}{)}^{\perp r}$.

• A space $X$ is a T₁ space if and only if $\mathrm{\varnothing }\to X$ is in $(\{0\to 1\}\to \{*\}{)}^{\perp r}$.

• $(\{1\}\to \{0\to 1\}{)}^{\perp l}$ 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,

• $(\mathrm{\varnothing }\to \{*\}{)}^{\perp r}$ is the class of surjective maps,

• $(\mathrm{\varnothing }\to \{*\}{)}^{\perp r\ell }$ is the class of maps of form $A\to A\cup D$ where $D$ is discrete,

• $(\mathrm{\varnothing }\to \{*\}{)}^{\perp r\ell \ell }=(\{a\}\to \{a,b\}{)}^{\perp \ell }$ is the class of maps $A\to B$ such that each connected component of $B$ intersects $\mathrm{Im}A$,

• $(\{0,1\}\to \{*\}{)}^{\perp r}$ 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 $\mathrm{\varnothing }\to X\perp {\cup }_n(-n,n)\to \mathbb{ℝ}$ where ${\cup }_n(-n,n)\to \mathbb{ℝ}$ 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 $\mathrm{\varnothing }\to X\perp [0,1]\to \{0\leftarrow x\to 1\}$ 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 $\{1/n{\}}_{n\in \mathbb{\mathbb{N}}}\to \{0\}\cup \{1/n{\}}_{n\in \mathbb{\mathbb{N}}}\perp X\to \{0\}$ where $\{1/n{\}}_{n\in \mathbb{\mathbb{N}}}\to \{0\}\cup \{1/n{\}}_{n\in \mathbb{\mathbb{N}}}$ 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 $\{1/n{\}}_{n\in \mathbb{\mathbb{N}}}\to \{0\}\cup \{1/n{\}}_{n\in \mathbb{\mathbb{N}}}\perp A\to X.$

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 $W{C}_0$ be the class of maps embedding the upper semi-sphere into the disk. $W{C}_0^{\perp \ell },W{C}_0^{\perp \ell r},C_0^{\perp \ell },C_0^{\perp \ell r}$ 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 $\partial \mathrm{\Delta }[n]\to \mathrm{\Delta }[n]$, and let $W{C}_0$ be the class of horn inclusions ${\mathrm{\Lambda }}^i[n]\to \mathrm{\Delta }[n]$. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, $W{C}_0^{\perp \ell },W{C}_0^{\perp \ell r},C_0^{\perp \ell },C_0^{\perp \ell r}$.

• Let $\mathbf{𝐂}\mathbf{𝐡}(R)$ be the category of chain complexes over a commutative ring $R$. Let $C_0$ be the class of maps of form

$$\cdots \to 0\to R\to 0\to 0\to \cdots \to \cdots \to R\stackrel{\mathrm{id}}{\to }R\to 0\to 0\to \cdots ,$$

and $W{C}_0$ be

$$\cdots \to 0\to 0\to 0\to 0\to \cdots \to \cdots \to R\stackrel{\mathrm{id}}{\to }R\to 0\to 0\to \cdots .$$

Then $W{C}_0^{\perp \ell },W{C}_0^{\perp \ell r},C_0^{\perp \ell },C_0^{\perp \ell r}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.

References

• Hovey, Mark (1999). Model Categories. https://archive.org/details/arxiv-math9803002. 
• J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories