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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
or
, 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
and
such that
there exists
such that
and
.
![Model category lifting.png](//upload.wikimedia.org/wikipedia/commons/8/88/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
or
with respect to the lifting property, respectively its right orthogonal
or
, 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,
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a65bf73d0e2170962d1ec996ee04fae0bd18165)
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
of the simplest non-surjection
is the class of surjections. The left and right orthogonals of
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 }}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4daad315d5835cb08b1fefb15161570d288ed8ba)
It is clear that
and
. The class
is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) and composition of morphisms, and contains all isomorphisms of C. Meanwhile,
is closed under retracts, pushouts, (small) coproducts and 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
, where
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
etc., represent various kinds of negation of C, so
each consists of morphisms which are far from having property
.
Examples of lifting properties in algebraic topology[]
A map
has the path lifting property iff
where
is the inclusion of one end point of the closed interval into the interval
.
A map
has the homotopy lifting property iff
where
is the map
.
Examples of lifting properties coming from model categories[]
Fibrations and cofibrations.
- Let Top be the category of topological spaces, and let
be the class of maps
, embeddings of the boundary
of a ball into the ball
. Let
be the class of maps embedding the upper semi-sphere into the disk.
are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.[1]
- Let sSet be the category of simplicial sets. Let
be the class of boundary inclusions
, and let
be the class of horn inclusions
. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively,
.[2]
![{\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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/236312d187bb5272a0d1d74f31c73b83b358b628)
- and
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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaea525ae83d23a24f1ca801c69b9aa3ab4d171)
- Then
are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.[3]
Elementary examples in various categories[]
In Set,
is the class of surjections,
is the class of injections.
In the category R-Mod of modules over a commutative ring R,
is the class of surjections, resp. injections,
- A module M is projective, resp. injective, iff
is in
, resp.
is in
.
In the category Grp of groups,
, resp.
, is the class of injections, resp. surjections (where
denotes the infinite cyclic group),
- A group F is a free group iff
is in ![{\displaystyle \{0\to \mathbb {Z} \}^{\perp r\ell },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/731f1810cfcc328fbbd74beb540e2dfc52af2cbe)
- A group A is torsion-free iff
is in ![{\displaystyle \{n\mathbb {Z} \to \mathbb {Z} :n>0\}^{\perp r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc1147ba19ccd5ca3429538f48a29d99f1c2ed5)
- A subgroup A of B is pure iff
is in ![{\displaystyle \{n\mathbb {Z} \to \mathbb {Z} :n>0\}^{\perp r}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b044eb65da776ed17f45b91e6fce504000e4fe5)
For a finite group G,
iff the order of G is prime to p,
iff G is a p-group,
- H is nilpotent iff the diagonal map
is in
where
denotes the class of maps ![{\displaystyle \{1\to G:G{\text{ arbitrary}}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30af185e1ecc3c7e144f35b0a530b7576ba2a391)
- a finite group H is soluble iff
is in ![{\displaystyle \{0\to A:A{\text{ abelian}}\}^{\perp \ell r}=\{[G,G]\to G:G{\text{ arbitrary }}\}^{\perp \ell r}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1189354731e225534fd0eacb3dce9bcd44f1e66e)
In the category Top of topological spaces, let
, resp.
denote the discrete, resp. antidiscrete space with two points 0 and 1. Let
denote the Sierpinski space of two points where the point 0 is open and the point 1 is closed, and let
etc. denote the obvious embeddings.
- a space X satisfies the separation axiom T0 iff
is in ![{\displaystyle (\{0\leftrightarrow 1\}\to \{*\})^{\perp r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/195ea6155c437004a127708236c2d9d0a3eaebc3)
- a space X satisfies the separation axiom T1 iff
is in ![{\displaystyle (\{0\to 1\}\to \{*\})^{\perp r},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f187bb8150af434b18d38c62137c2f6fec8dc702)
is the class of maps with dense image,
is the class of maps
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,
is the class of surjective maps,
is the class of maps of form
where D is discrete,
is the class of maps
such that each connected component of B intersects
,
is the class of injective maps,
is the class of maps
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
is in
,
- for a connected space X, each continuous function on X is bounded iff
where
is the map from the disjoint union of open intervals
into the real line ![{\displaystyle \mathbb {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd)
- a space X is Hausdorff iff for any injective map
, it holds
where
denotes the three-point space with two open points a and b, and a closed point x,
- a space X is perfectly normal iff
where the open interval
goes to x, and
maps to the point
, and
maps to the point
, and
denotes the three-point space with two closed points
and one open point x.
In the category of metric spaces with uniformly continuous maps.
- A space X is complete iff
where
is the obvious inclusion between the two subspaces of the real line with induced metric, and
is the metric space consisting of a single point,
- A subspace
is closed iff ![{\displaystyle \{1/n\}_{n\in \mathbb {N} }\to \{0\}\cup \{1/n\}_{n\in \mathbb {N} }\perp A\to X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71326e1fba735a328141508612ba960286a2fb3)
Notes[]
References[]