Lifting property explained

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 tothe 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} or C ^{\perp} with respect to the lifting property, respectively its right orthogonal C ^{\perp} 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,

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} 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,

$$\backslash ^\; =\; \backslash ^\; =\; \backslash .$$

It is clear that

C ^{\perp\ell} \supset C and C ^{\perp} \supset C . The class C ^{\perp} 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} 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} , C ^{\perp\ell} , 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 negationof 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} , C ^{\perp\ell} , C ^{\perp\ell\ell\ell} etc., represent various kinds of negation of C , so C ^{\perp\ell} , C ^{\perp} , C ^{\perp\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 x [ 0 , 1 ] \perp f where X\to X x [ 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 ^{\perp\ell} _{0} , WC ^{\perp\ell r} _{0} , C ^{\perp\ell} _{0} , C ^{\perp\ell r} _{0} are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[1]

• 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 Λ ^{i[n]} \to \Delta[n] . Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, WC ^{\perp\ell} _{0} , WC ^{\perp\ell r} _{0} , C ^{\perp\ell} _{0} , C ^{\perp\ell r} _{0} .^[2]

• Let

Ch (R) be the category of chain complexes over a commutative ring R . Let C _{0} be the class of maps of form

$$\backslash cdots\backslash to\; 0\backslash to\; R\; \backslash to\; 0\; \backslash to\; 0\; \backslash to\; \backslash cdots\; \backslash to\; \backslash cdots\; \backslash to\; R\; \backslash xrightarrow\; R\; \backslash to\; 0\; \backslash to\; 0\; \backslash to\; \backslash cdots,$$

and

WC _{0} be

$$\backslash cdots\; \backslash to\; 0\backslash to\; 0\; \backslash to\; 0\; \backslash to\; 0\; \backslash to\; \backslash cdots\; \backslash to\; \backslash cdots\; \backslash to\; R\; \backslash xrightarrow\; R\; \backslash to\; 0\; \backslash to\; 0\; \backslash to\; \backslash cdots.$$

Then

WC ^{\perp\ell} _{0} , WC ^{\perp\ell r} _{0} , C ^{\perp\ell} _{0} , C ^{\perp\ell r} _{0} are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.^[3]

Elementary examples in various categories

In Set,

\{\emptyset\to \{*\}\} ^{\perp} is the class of surjections,(\{a,b\}\to \{*\}) ^{\perp} = (\{a,b\}\to \{*\}) ^{\perp\ell} is the class of injections.

In the category

R - Mod of modules over a commutative ring R ,\{ 0 \to R\} ^{\perp} , \{R\to 0 \} ^{\perp} is the class of surjections, resp. injections,

• A module

M is projective, resp. injective, iff 0 \to M is in \{ 0 \to R\} ^{\perp} , resp. M\to 0 is in \{R\to 0 \} ^{\perp} .

In the category

Grp of groups, \{\Z \to 0 \} ^{\perp} , resp. \{ 0 \to \Z\} ^{\perp} , 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} ,

• A group

A is torsion-free iff 0 \to A is in \{ n \Z\to \Z : n> 0 \} ^{\perp} , A of B is pure iff A \to B is in \{ n\Z\to \Z : n> 0 \} ^{\perp} . 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} iff G is a p -group,H is nilpotent iff the diagonal map H\to H x H is in ( 1 \to *) ^{\perp\ell} where ( 1 \to *) denotes the class of maps \{ 1 \to G : G arbitrary \},

• a finite group

H is soluble iff 1 \to H is in $$\backslash ^=\backslash ^.$$

In the category

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} ,

• a space

X satisfies the separation axiom T₁ iff \emptyset\to X is in ( \{ 0 \to 1 \}\to \{*\}) ^{\perp} , (\{ 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} is the class of surjective maps, (\emptyset\to \{*\}) ^{\perp} is the class of maps of form A\to A\cup D where D is discrete,(\emptyset\to \{*\}) ^{\perp} = (\{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} 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 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} \to \{ 0 \}\cup \{ 1 /n\} _{n} \perp X\to \{ 0 \} where \{ 1 /n\} _{n} \to \{ 0 \}\cup \{ 1 /n\} _{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} \to \{ 0 \}\cup \{ 1 /n\} _{n} \perp A\to X.

References

• Book: Hovey, Mark . Model Categories . 1999.

Notes and References

1. Book: Hovey, Mark . Model Categories . Def. 2.4.3, Th.2.4.9
2. Book: Hovey, Mark . Model Categories . Def. 3.2.1, Th.3.6.5
3. Book: Hovey, Mark . Model Categories . Def. 2.3.3, Th.2.3.11