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.

Formal definition

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 .
A commutative diagram in the shape of a square with an anti-diagonal line, which graphically representing the relations stated in the preceding text. There are four letters representing vertices, here listed from left to right, then from top to bottom order, which are "A" (the top-left corner of the square), "X" (the top-right corner of the square), "B" (the bottom-left corner of the square), and "Y" (the bottom-right corner of the square). Additionally, there are five arrows which connect these letters, listed here using the same order as before: a solid-stroke, left to right arrow labeled "f" from A to X (the top-side line of the square); a solid-stroke, top to bottom arrow labeled "i" from A to B (the left-side line of the square); a dotted-stroke, bottom-left to top-right arrow labeled "h" from B to X (the anti-diagonal line of the square); a solid-stroke, top to bottom arrow labeled "p" from X to Y (the right-side line of the square); and a solid-stroke, left to right arrow labeled "g" from B to Y (the bottom-side line of the square).

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

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

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]
  • 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

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 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

  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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