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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. It is used in
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 topol ...
within
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 classif ...
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 In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
, an axiomatic framework for
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 topol ...
introduced by Daniel Quillen. It is also used in the definition of a
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. Definition A factoriza ...
, 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 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'' C^ or C^\perp with respect to the lifting property, respectively its ''right orthogonal'' C^ 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, :\begin C^ &:= \ \\ C^ &:= \ \end 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 \emptyset\to \, is the class of surjections. The left and right orthogonals of \\to \, the simplest non-injection, are both precisely the class of injections, :\^ = \^ = \. It is clear that C^ \supset C and C^ \supset C. The class C^ 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, C^ 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 C^, C^, C^, C^, 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 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 C^, C^, C^, C^ etc., represent various kinds of negation of ''C'', so C^, C^, C^, C^ 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 \\to ,1\perp f where \ \to ,1/math> is the inclusion of one end point of the closed interval into the interval ,1/math>. A map f:U\to B has the homotopy lifting property iff X \to X\times ,1\perp f where X\to X\times ,1/math> 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 space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, and let C_0 be the class of maps S^n\to D^,
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
s of the boundary S^n=\partial D^ of a ball into the ball D^. Let WC_0 be the class of maps embedding the upper semi-sphere into the disk. WC_0^, WC_0^, C_0^, C_0^ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. * Let sSet be the category of
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s. Let C_0 be the class of boundary inclusions \partial \Delta \to \Delta /math>, and let WC_0 be the class of horn inclusions \Lambda^i \to \Delta /math>. Then the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, WC_0^, WC_0^, C_0^, C_0^. * Let Ch(''R'') be the category of chain complexes over a commutative ring ''R''. Let C_0 be the class of maps of form :: \cdots\to 0\to R \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow R \to 0 \to 0 \to \cdots, : and WC_0 be :: \cdots \to 0\to 0 \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow R \to 0 \to 0 \to \cdots. :Then WC_0^, WC_0^, C_0^, C_0^ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. Def. 2.3.3, Th.2.3.11


Elementary examples in various categories

In Set, * \^ is the class of surjections, * (\\to \)^=(\\to \)^ 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
, iff 0\to M is in \^, resp. M\to 0 is in \^. In the category Grp of groups, * \^, resp. \^, is the class of injections, resp. surjections (where \Z denotes the infinite
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
), * A group ''F'' is a
free group In mathematics, the free group ''F'S'' 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 (e.g. ''st'' = ''suu''− ...
iff 0\to F is in \^, * A group ''A'' is torsion-free iff 0\to A is in \^, * A
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
''A'' of ''B'' is pure iff A \to B is in \^. For a finite group ''G'', * \ \perp G\to 1 iff the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of ''G'' is prime to ''p'', * G\to 1 \in (0\to /p)^ iff ''G'' is a ''p''-group, * ''H'' is nilpotent iff the diagonal map H\to H\times H is in (1\to *)^ where (1\to *) denotes the class of maps \, * a finite group ''H'' is
soluble In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution. The extent of the solub ...
iff 1\to H is in \^=\^. In the category Top of topological spaces, let \, resp. \ denote the
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
, 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 \\to \, \ \to \ etc. denote the obvious embeddings. * a space ''X'' satisfies the separation axiom T0 iff X\to \ is in (\ \to \)^, * a space ''X'' satisfies the separation axiom T1 iff \emptyset\to X is in ( \\to \)^, * (\\to \)^ is the class of maps with dense
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
, * (\\to \)^ is the class of maps f:X\to Y such that the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
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 \)^ is the class of surjective maps, * (\emptyset\to \)^ is the class of maps of form A\to A\cup D where ''D'' is discrete, * (\emptyset\to \)^ = (\\to \)^ is the class of maps A\to B such that each connected component of ''B'' intersects \operatorname A, * (\\to \)^ is the class of injective maps, * (\\to \)^ is the class of maps f:X\to Y such that the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or throug ...
of a connected closed open subset of ''Y'' is a connected closed open
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of ''X'', e.g. ''X'' is connected iff X\to \ is in (\ \to \)^, * 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 In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
of open intervals (-n,n) into the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
\mathbb, * a space ''X'' is Hausdorff iff for any injective map \\hookrightarrow X, it holds \\hookrightarrow X \perp \\to\ 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 \emptyset\to X \perp ,1\to \ where the open interval (0,1) goes to ''x'', and 0 maps to the point 0, and 1 maps to the point 1, and \ denotes the three-point space with two closed points 0, 1 and one open point ''x''. In the category of
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
s with uniformly continuous maps. * A space ''X'' is complete iff \_ \to \\cup \_ \perp X\to \ where \_ \to \\cup \_ 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 i:A\to X is closed iff \_ \to \\cup \_ \perp A\to X.


Notes


References

* {{cite book , last = Hovey , first = Mark , title = Model Categories , url = https://archive.org/details/arxiv-math9803002 , date=1999 Category theory