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.

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

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

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

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

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

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

In the category Set of sets, the right orthogonal ${\displaystyle \{\emptyset \to \{*\}\}^{\perp r}}$ of the simplest non-surjection ${\displaystyle \emptyset \to \{*\}}$ is the class of surjections. The left and right orthogonals of ${\displaystyle \{x_{1},x_{2}\}\to \{*\},}$ the simplest non-injection, are both precisely the class of injections,

$${\displaystyle \{\{x_{1},x_{2}\}\to \{*\}\}^{\perp \ell }=\{\{x_{1},x_{2}\}\to \{*\}\}^{\perp r}=\{f\mid f{\text{ is an injection }}\}.}$$

It is clear that ${\displaystyle C^{\perp \ell r}\supset C}$ and ${\displaystyle C^{\perp r\ell }\supset C}$. The class ${\displaystyle C^{\perp r}}$ is always closed under retracts (that is, if ${\displaystyle X}$ and ${\displaystyle Y}$ are objects, ${\displaystyle C\perp Y}$, and ${\displaystyle X}$ is a retract of ${\displaystyle Y}$, then ${\displaystyle C\perp X}$), 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, ${\displaystyle C^{\perp \ell }}$ 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 ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$ be morphisms such that ${\displaystyle i\circ j}$ exists. Then:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

For a finite group ${\displaystyle G}$,

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

• ${\displaystyle G\to 1\in (0\to {\mathbb {Z} }/p{\mathbb {Z} })^{\perp rr}}$ iff ${\displaystyle G}$ is a ${\displaystyle p}$-group.

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

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

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

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

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

• ${\displaystyle (\{1\}\to \{0\to 1\})^{\perp l}}$ is the class of maps with dense image.

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

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

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

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

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

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

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

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

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

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

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

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

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

Fibrations and cofibrations.

• Let Top be the category of topological spaces, and let ${\displaystyle C_{0}}$ be the class of maps ${\displaystyle S^{n}\to D^{n+1}}$, embeddings of the boundary ${\displaystyle S^{n}=\partial D^{n+1}}$ of a ball into the ball ${\displaystyle D^{n+1}}$. Let ${\displaystyle WC_{0}}$ be the class of maps embedding the upper semi-sphere into the disk. ${\displaystyle WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[1]

• Let sSet be the category of simplicial sets. Let ${\displaystyle C_{0}}$ be the class of boundary inclusions ${\displaystyle \partial \Delta [n]\to \Delta [n]}$, and let ${\displaystyle WC_{0}}$ be the class of horn inclusions ${\displaystyle \Lambda ^{i}[n]\to \Delta [n]}$. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, ${\displaystyle WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}}$.^[2]

• Let ${\displaystyle \mathbf {Ch} (R)}$ be the category of chain complexes over a commutative ring ${\displaystyle R}$. Let ${\displaystyle C_{0}}$ be the class of maps of form

$${\displaystyle \cdots \to 0\to R\to 0\to 0\to \cdots \to \cdots \to R{\xrightarrow {\operatorname {id} }}R\to 0\to 0\to \cdots ,}$$

and ${\displaystyle WC_{0}}$ be

$${\displaystyle \cdots \to 0\to 0\to 0\to 0\to \cdots \to \cdots \to R{\xrightarrow {\operatorname {id} }}R\to 0\to 0\to \cdots .}$$

Then ${\displaystyle WC_{0}^{\perp \ell },WC_{0}^{\perp \ell r},C_{0}^{\perp \ell },C_{0}^{\perp \ell r}}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[3]

• Hovey, Mark (1999). Model Categories.
• J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories