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.

Thus, 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, 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.

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 ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r},C^{\perp \ell \ell }}$ , where ${\displaystyle C}$  is a class consisting of several explicitly given morphisms. A useful intuition is to think that the property of left-lifting against a class ${\displaystyle C}$  is a kind of negation of the property of being in ${\displaystyle C}$ , and that right-lifting is also a kind of negation. Hence the classes obtained from ${\displaystyle C}$  by taking orthogonals an odd number of times, such as ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }}$  etc., represent various kinds of negation of ${\displaystyle C}$ , so ${\displaystyle C^{\perp \ell },C^{\perp r},C^{\perp \ell r\ell },C^{\perp \ell \ell \ell }}$  each consists of morphisms which are far from having property ${\displaystyle C}$ .

Examples of lifting properties in algebraic topology edit

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 ${\displaystyle x\mapsto (x,0)}$ .

Examples of lifting properties coming from model categories edit

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]

Elementary examples in various categories edit

In Set,

• ${\displaystyle \{\emptyset \to \{*\}\}^{\perp r}}$  is the class of surjections,

• ${\displaystyle (\{a,b\}\to \{*\})^{\perp r}=(\{a,b\}\to \{*\})^{\perp \ell }}$  is the class of injections.

In the category ${\displaystyle R{\text{-}}\mathbf {Mod} }$  of modules over a commutative ring ${\displaystyle R}$ ,

• ${\displaystyle \{0\to R\}^{\perp r},\{R\to 0\}^{\perp r}}$  is the class of surjections, resp. injections,

• A module ${\displaystyle M}$  is projective, resp. injective, iff ${\displaystyle 0\to M}$  is in ${\displaystyle \{0\to R\}^{\perp r\ell }}$ , resp. ${\displaystyle M\to 0}$  is in ${\displaystyle \{R\to 0\}^{\perp rr}}$ .

In the category ${\displaystyle \mathbf {Grp} }$  of groups,

• ${\displaystyle \{\mathbb {Z} \to 0\}^{\perp r}}$ , resp. ${\displaystyle \{0\to \mathbb {Z} \}^{\perp r}}$ , is the class of injections, resp. surjections (where ${\displaystyle \mathbb {Z} }$  denotes the infinite cyclic group),

• A group ${\displaystyle F}$  is a free group iff ${\displaystyle 0\to F}$  is in ${\displaystyle \{0\to \mathbb {Z} \}^{\perp r\ell },}$ 

• A group ${\displaystyle A}$  is torsion-free iff ${\displaystyle 0\to A}$  is in ${\displaystyle \{n\mathbb {Z} \to \mathbb {Z} :n>0\}^{\perp r},}$ 

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

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

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

• ${\displaystyle G\to 1\in (0\to {\mathbb {Z} }/p{\mathbb {Z} })^{\perp rr}}$  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
  $${\displaystyle \{0\to A:A{\text{ abelian}}\}^{\perp \ell r}=\{[G,G]\to G:G{\text{ arbitrary }}\}^{\perp \ell r}.}$$

   

In the category ${\displaystyle \mathbf {Top} }$  of topological spaces, let ${\displaystyle \{0,1\}}$ , resp. ${\displaystyle \{0\leftrightarrow 1\}}$  denote the discrete, resp. antidiscrete space with two points 0 and 1. Let ${\displaystyle \{0\to 1\}}$  denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let ${\displaystyle \{0\}\to \{0\to 1\},\{1\}\to \{0\to 1\}}$  etc. denote the obvious embeddings.

• a space ${\displaystyle X}$  satisfies the separation axiom T₀ iff ${\displaystyle X\to \{*\}}$  is in ${\displaystyle (\{0\leftrightarrow 1\}\to \{*\})^{\perp r},}$ 

• a space ${\displaystyle X}$  satisfies the separation axiom T₁ iff ${\displaystyle \emptyset \to X}$  is in ${\displaystyle (\{0\to 1\}\to \{*\})^{\perp r},}$ 

• ${\displaystyle (\{1\}\to \{0\to 1\})^{\perp \ell }}$  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 ${\displaystyle x}$ .

In the category of metric spaces with uniformly continuous maps.

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

• Hovey, Mark (1999). Model Categories.