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

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A morphism   in a category has the left lifting property with respect to a morphism  , and   also has the right lifting property with respect to  , sometimes denoted   or  , iff the following implication holds for each morphism   and   in the category:

  • if the outer square of the following diagram commutes, then there exists   completing the diagram, i.e. for each   and   such that   there exists   such that   and  .
 

This is sometimes also known as the morphism   being orthogonal to the morphism  ; however, this can also refer to the stronger property that whenever   and   are as above, the diagonal morphism   exists and is also required to be unique.

For a class   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  . In notation,

 

Taking the orthogonal of a class   is a simple way to define a class of morphisms excluding non-isomorphisms from  , 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,

 

It is clear that   and  . The class   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,   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

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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   is a kind of negation of the property of being in  , and that right-lifting is also a kind of negation. Hence the classes obtained from   by taking orthogonals an odd number of times, such as   etc., represent various kinds of negation of  , so   each consists of morphisms which are far from having property  .

Examples of lifting properties in algebraic topology

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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

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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]
  • Let   be the category of chain complexes over a commutative ring  . Let   be the class of maps of form
 
and   be
 
Then   are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.[3]

Elementary examples in various categories

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In Set,

  •   is the class of surjections,
  •   is the class of injections.

In the category   of modules over a commutative ring  ,

  •   is the class of surjections, resp. injections,
  • A module   is projective, resp. injective, iff   is in  , resp.   is in  .

In the category   of groups,

  •  , resp.  , is the class of injections, resp. surjections (where   denotes the infinite cyclic group),
  • A group   is a free group iff   is in  
  • A group   is torsion-free iff   is in  
  • A subgroup   of   is pure iff   is in  

For a finite group  ,

  •   iff the order of   is prime to   iff  ,
  •   iff   is a  -group,
  •   is nilpotent iff the diagonal map   is in   where   denotes the class of maps  
  • a finite group   is soluble iff   is in  

In the category   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   satisfies the separation axiom T0 iff   is in  
  • a space   satisfies the separation axiom T1 iff   is in  
  •   is the class of maps with dense image,
  •   is the class of maps   such that the topology on   is the pullback of topology on  , i.e. the topology on   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   is discrete,
  •   is the class of maps   such that each connected component of   intersects  ,
  •   is the class of injective maps,
  •   is the class of maps   such that the preimage of a connected closed open subset of   is a connected closed open subset of  , e.g.   is connected iff   is in  ,
  • for a connected space  , each continuous function on   is bounded iff   where   is the map from the disjoint union of open intervals   into the real line  
  • a space   is Hausdorff iff for any injective map  , it holds   where   denotes the three-point space with two open points   and  , and a closed point  ,
  • a space   is perfectly normal iff   where the open interval   goes to  , and   maps to the point  , and   maps to the point  , and   denotes the three-point space with two closed points   and one open point  .

In the category of metric spaces with uniformly continuous maps.

  • A space   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  

Notes

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

References

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