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Lifting property

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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.

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Mathematics

Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Category theory

Category theory

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Category (mathematics)

Category (mathematics)

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

Homotopy theory

Homotopy theory

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).

Algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

Daniel Quillen

Daniel Quillen

Daniel Gray "Dan" Quillen was an American mathematician. He is known for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.

Factorization system

Factorization system

In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.

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.

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Set (mathematics)

Set (mathematics)

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Pullback (category theory)

Pullback (category theory)

In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often writtenP = X ×Z Y

Product (category theory)

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Pushout (category theory)

Pushout (category theory)

In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an object P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .

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

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Homotopy lifting property

Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E.

Embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

Chain complex

Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

Commutative ring

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

Module (mathematics)

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.

Injective module

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook.

Group (mathematics)

Group (mathematics)

In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.

Cyclic group

Cyclic group

In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

Free group

Free group

In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms. The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses.

Finite group

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.

Order (group theory)

Order (group theory)

In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a. If no such m exists, the order of a is infinite.

Discrete space

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

Source: "Lifting property", Wikipedia, Wikimedia Foundation, (2023, January 25th), https://en.wikipedia.org/wiki/Lifting_property.

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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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