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universal property
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{{other uses|Universal (disambiguation)}}{{Short description|Characterizing property of mathematical constructions}}(File:Universal morphism definition.svg|thumb|The typical diagram of the definition of a universal morphism.)In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see {{slink||Formal definition}}, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see {{slink||Connection with comma categories}}, below). Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a commutative ring {{mvar|R}}, the field of fractions of the quotient ring of {{mvar|R}} by a prime ideal {{mvar|p}} can be identified with the residue field of the localization of {{mvar|R}} at {{mvar|p}}; that is R_p/pR_pcong operatorname {Frac}(R/p) (all these constructions can be defined by universal properties).Other objects that can be defined by universal properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, DedekindâMacNeille completion, product topologies, StoneâÄech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces.- the content below is remote from Wikipedia
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Motivation
Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.- The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construction is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier to deal with by its universal property.
- Universal properties define objects uniquely up to a unique isomorphism.Jacobson (2009), Proposition 1.6, p. 44. Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.
- Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property of group rings.
- Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.
Formal definition
To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see Examples below). Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples.Let F: mathcal{C} to mathcal{D} be a functor between categories mathcal{C} and mathcal{D}. In what follows, let X be an object of mathcal{D}, A and A’ be objects of mathcal{C}, and h: A to A’ be a morphism in mathcal{C}. Then, the functor F maps A, A’ and h in mathcal{C} to F(A), F(A’) and F(h) in mathcal{D}.A universal morphism from X to F is a unique pair (A, u: X to F(A)) in mathcal{D} which has the following property, commonly referred to as a universal property:For any morphism of the form f: X to F(A’) in mathcal{D}, there exists a unique morphism h: A to A’ in mathcal{C} such that the following diagram commutes:(File:Universal morphism definition.svg|center|The typical diagram of the definition of a universal morphism.){{anchor|Terminal morphism}}We can dualize this categorical concept. A universal morphism from F to X is a unique pair (A, u: F(A) to X) that satisfies the following universal property:For any morphism of the form f: F(A’) to X in mathcal{D}, there exists a unique morphism h: A’ to A in mathcal{C} such that the following diagram commutes:(File:Universal definition dualized.svg|center|The most important arrow here is u: F(A) to X which establishes the universal property.)Note that in each definition, the arrows are reversed. Both definitions are necessary to describe universal constructions which appear in mathematics; but they also arise due to the inherent duality present in category theory.In either case, we say that the pair (A, u) which behaves as above satisfies a universal property.Connection with comma categories
Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right).Let F: mathcal{C} to mathcal{D} be a functor and X an object of mathcal{D}. Then recall that the comma category (X downarrow F) is the category where- Objects are pairs of the form (B, f: X to F(B)), where B is an object in mathcal{C}
- A morphism from (B, f: X to F(B)) to (B’, f’: X to F(B’)) is given by a morphism h: B to B’ in mathcal{C} such that the diagram commutes:
- Objects are pairs of the form (B, f: F(B) to X) where B is an object in mathcal{C}
- A morphism from (B, f:F(B) to X) to (B’, f’:F(B’) to X) is given by a morphism h: B to B’ in mathcal{C} such that the diagram commutes:
Examples
Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.Tensor algebras
Let mathcal{C} be the category of vector spaces K-Vect over a field K and let mathcal{D} be the category of algebras K-Alg over K (assumed to be unital and associative). Let
U : K-Alg → K-Vect
be the forgetful functor which assigns to each algebra its underlying vector space.Given any vector space V over K we can construct the tensor algebra T(V). The tensor algebra is characterized by the fact:
“Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T(V) to A.”
This statement is an initial property of the tensor algebra since it expresses the fact that the pair (T(V),i), where i:V to U(T(V)) is the inclusion map, is a universal morphism from the vector space V to the functor U.Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg. This means that T is left adjoint to the forgetful functor U (see the section below on relation to adjoint functors).Products
A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist.Let X and Y be objects of a category mathcal{C} with finite products. The product of X and Y is an object X × Y together with two morphisms
pi_1 : X times Y to X
pi_2 : X times Y to Y
such that for any other object Z of mathcal{C} and morphisms f: Z to X and g: Z to Y there exists a unique morphism h: Z to X times Y such that f = pi_1 circ h and g = pi_2 circ h.To understand this characterization as a universal property, take the category mathcal{D} to be the product category mathcal{C} times mathcal{C} and define the diagonal functor
Delta: mathcal{C} to mathcal{C} times mathcal{C}
by Delta(X) = (X, X) and Delta(f: X to Y) = (f, f). Then (X times Y, (pi_1, pi_2)) is a universal morphism from Delta to the object (X, Y) of mathcal{C} times mathcal{C}: if (f, g) is any morphism from (Z, Z) to (X, Y), then it must equala morphism Delta(h: Z to X times Y) = (h,h) from Delta(Z) = (Z, Z)to Delta(X times Y) = (X times Y, X times Y) followed by (pi_1, pi_2). As a commutative diagram:(File:Universal-property-products.svg|center|484x484px|Commutative diagram showing how products have a universal property.)For the example of the Cartesian product in Set, the morphism (pi_1, pi_2) comprises the two projections pi_1(x,y) = x and pi_2(x,y) = y. Given any set Z and functions f,g the unique map such that the required diagram commutes is given by h = langle x,yrangle(z) = (f(z), g(z)).ARXIV, Fong, Brendan, Spivak, David I., 2018-10-12, Seven Sketches in Compositionality: An Invitation to Applied Category Theory, math.CT, 1803.05316, Limits and colimits
Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.Let mathcal{J} and mathcal{C} be categories with mathcal{J} a small index category and let mathcal{C}^mathcal{J} be the corresponding functor category. The diagonal functor
Delta: mathcal{C} to mathcal{C}^mathcal{J}
is the functor that maps each object N in mathcal{C} to the constant functor Delta(N): mathcal{J} to mathcal{C} (i.e. Delta(N)(X) = N for each X in mathcal{J} and Delta(N)(f) = 1_N for each f: X to Y in mathcal{J}) and each morphism f : N to M in mathcal{C} to the natural transformation Delta(f):Delta(N)toDelta(M) in mathcal{C}^{mathcal{J}} defined as, for every object X of mathcal{J}, the component Delta(f)(X):Delta(N)(X)toDelta(M)(X) = f:Nto Mat X. In other words, the natural transformation is the one defined by having constant component f:Nto M for every object of mathcal{J}.Given a functor F: mathcal{J} to mathcal{C} (thought of as an object in mathcal{C}^mathcal{J}), the limit of F, if it exists, is nothing but a universal morphism from Delta to F. Dually, the colimit of F is a universal morphism from F to Delta.Properties
Existence and uniqueness
Defining a quantity does not guarantee its existence. Given a functor F: mathcal{C} to mathcal{D} and an object X of mathcal{D}, there may or may not exist a universal morphism from X to F. If, however, a universal morphism (A, u) does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism: if (A’, u’) is another pair, then there exists a unique isomorphism k: A to A’ such that u’ = F(k) circ u.This is easily seen by substituting (A, u’) in the definition of a universal morphism.It is the pair (A, u) which is essentially unique in this fashion. The object A itself is only unique up to isomorphism. Indeed, if (A, u) is a universal morphism and k: A to A’ is any isomorphism then the pair (A’, u’), where u’ = F(k) circ u is also a universal morphism.Equivalent formulations
The definition of a universal morphism can be rephrased in a variety of ways. Let F: mathcal{C} to mathcal{D} be a functor and let X be an object of mathcal{D}. Then the following statements are equivalent:- (A, u) is a universal morphism from X to F
- (A, u) is an initial object of the comma category (X downarrow F)
- (A, F(bullet)circ u) is a representation of text{Hom}_mathcal{D}(X, F(-)), where its components (F(bullet)circ u)_B:text{Hom}_{mathcal{C}}(A, B) to text{Hom}_{mathcal{D}}(X, F(B)) are defined by
- (A, u) is a universal morphism from F to X
- (A, u) is a terminal object of the comma category (F downarrow X)
- (A, ucirc F(bullet)) is a representation of text{Hom}_mathcal{D}(F(-), X), where its components (ucirc F(bullet))_B:text{Hom}_{mathcal{C}}(B, A)to text{Hom}_{mathcal{D}}(F(B), X) are defined by
(ucirc F(bullet))_B(f:Bto A):F(B)to X = ucirc F(f):F(B)to X
for each object B in mathcal{C}.Relation to adjoint functors
Suppose (A_1, u_1) is a universal morphism from X_1 to F and (A_2, u_2) is a universal morphism from X_2 to F. By the universal property of universal morphisms, given any morphism h: X_1 to X_2 there exists a unique morphism g: A_1 to A_2 such that the following diagram commutes:(File:Connection between universal elements inducing a functor.svg|center|Universal morphisms can behave like a natural transformation between functors under suitable conditions.)If every object X_i of mathcal{D} admits a universal morphism to F, then the assignment X_i mapsto A_i and h mapsto g defines a functor G: mathcal{D} to mathcal{C}. The maps u_i then define a natural transformation from 1_mathcal{D} (the identity functor on mathcal{D}) to Fcirc G. The functors (F, G) are then a pair of adjoint functors, with G left-adjoint to F and F right-adjoint to G.Similar statements apply to the dual situation of terminal morphisms from F. If such morphisms exist for every X in mathcal{C} one obtains a functor G: mathcal{C} to mathcal{D} which is right-adjoint to F (so F is left-adjoint to G).Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let F and G be a pair of adjoint functors with unit eta and co-unit epsilon (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in mathcal{C} and mathcal{D}:- For each object X in mathcal{C}, (F(X), eta_X) is a universal morphism from X to G. That is, for all f: X to G(Y) there exists a unique g: F(X) to Y for which the following diagrams commute.
- For each object Y in mathcal{D}, (G(Y), epsilon_Y) is a universal morphism from F to Y. That is, for all g: F(X) to Y there exists a unique f: X to G(Y) for which the following diagrams commute.
History
Universal properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.See also
- Free object
- Natural transformation
- Adjoint functor
- Monad (category theory)
- Variety of algebras
- Cartesian closed category
Notes
References
- Paul Cohn, Universal Algebra (1981), D.Reidel Publishing, Holland. {{isbn|90-277-1213-1}}.
- BOOK, Mac Lane, Saunders, Saunders Mac Lane, Categories for the Working Mathematician, 2nd, 1998, Graduate Texts in Mathematics 5, Springer, 0-387-98403-8,
- Borceux, F. Handbook of Categorical Algebra: vol 1 Basic category theory (1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications) {{isbn|0-521-44178-1}}
- N. Bourbaki, Livre II : Algèbre (1970), Hermann, {{isbn|0-201-00639-1}}.
- Milies, César Polcino; Sehgal, Sudarshan K.. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. {{isbn|978-1-4020-0238-0}}
- Jacobson. Basic Algebra II. Dover. 2009. {{isbn|0-486-47187-X}}
External links
- nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view
- André Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics
- BOOK, Chris, Hillman, A Categorical Primer, 2001, 10.1.1.24.3264, :, formal introduction to category theory.
- J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats
- Stanford Encyclopedia of Philosophy: “Category Theory“âby Jean-Pierre Marquis. Extensive bibliography.
- List of academic conferences on category theory
- Baez, John, 1996,“The Tale of n-categories.” An informal introduction to higher order categories.
- WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties.
- The catsters, a YouTube channel about category theory.
- Video archive of recorded talks relevant to categories, logic and the foundations of physics.
- www.j-paine.org/cgi-bin/webcats/webcats.php" title="web.archive.org/web/20080916162345www.j-paine.org/cgi-bin/webcats/webcats.php">Interactive Web page which generates examples of categorical constructions in the category of finite sets.
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