Universal property

Summary

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.

The typical diagram of the definition of a universal morphism.

Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § 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 R, the field of fractions of the quotient ring of R by a prime ideal p can be identified with the residue field of the localization of R at p; that is (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, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces.

Motivation

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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.[1] 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.[2]
  • 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

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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   be a functor between categories   and  . In what follows, let   be an object of  ,   and   be objects of  , and   be a morphism in  .

Then, the functor   maps  ,   and   in   to  ,   and   in  .

A universal morphism from   to   is a unique pair   in   which has the following property, commonly referred to as a universal property:

For any morphism of the form   in  , there exists a unique morphism   in   such that the following diagram commutes:

 
The typical diagram of the definition of a universal morphism.

We can dualize this categorical concept. A universal morphism from   to   is a unique pair   that satisfies the following universal property:

For any morphism of the form   in  , there exists a unique morphism   in   such that the following diagram commutes:

 
The most important arrow here is   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   which behaves as above satisfies a universal property.

Connection with comma categories

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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   be a functor and   an object of  . Then recall that the comma category   is the category where

  • Objects are pairs of the form  , where   is an object in  
  • A morphism from   to   is given by a morphism   in   such that the diagram commutes:
 
A morphism in the comma category is given by the morphism   which also makes the diagram commute.

Now suppose that the object   in   is initial. Then for every object  , there exists a unique morphism   such that the following diagram commutes.

 
This demonstrates the connection between a universal diagram being an initial object in a comma category.

Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a universal morphism from   to  . Therefore, we see that a universal morphism from   to   is equivalent to an initial object in the comma category  .

Conversely, recall that the comma category   is the category where

  • Objects are pairs of the form   where   is an object in  
  • A morphism from   to   is given by a morphism   in   such that the diagram commutes:
 
This simply demonstrates the definition of a morphism in a comma category.

Suppose   is a terminal object in  . Then for every object  , there exists a unique morphism   such that the following diagrams commute.

 
This shows that a terminal object in a specific comma category corresponds to a universal morphism.

The diagram on the right side of the equality is the same diagram pictured when defining a universal morphism from   to  . Hence, a universal morphism from   to   corresponds with a terminal object in the comma category  .

Examples

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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

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Let   be the category of vector spaces  -Vect over a field   and let   be the category of algebras  -Alg over   (assumed to be unital and associative). Let

  :  -Alg -Vect

be the forgetful functor which assigns to each algebra its underlying vector space.

Given any vector space   over   we can construct the tensor algebra  . The tensor algebra is characterized by the fact:

“Any linear map from   to an algebra   can be uniquely extended to an algebra homomorphism from   to  .”

This statement is an initial property of the tensor algebra since it expresses the fact that the pair  , where   is the inclusion map, is a universal morphism from the vector space   to the functor  .

Since this construction works for any vector space  , we conclude that   is a functor from  -Vect to  -Alg. This means that   is left adjoint to the forgetful functor   (see the section below on relation to adjoint functors).

Products

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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   and   be objects of a category   with finite products. The product of   and   is an object   ×   together with two morphisms

  :  
  :  

such that for any other object   of   and morphisms   and   there exists a unique morphism   such that   and  .

To understand this characterization as a universal property, take the category   to be the product category   and define the diagonal functor

 

by   and  . Then   is a universal morphism from   to the object   of  : if   is any morphism from   to  , then it must equal a morphism   from   to   followed by  . As a commutative diagram:

 
Commutative diagram showing how products have a universal property.

For the example of the Cartesian product in Set, the morphism   comprises the two projections   and  . Given any set   and functions   the unique map such that the required diagram commutes is given by  .[3]

Limits and colimits

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Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.

Let   and   be categories with   a small index category and let   be the corresponding functor category. The diagonal functor

 

is the functor that maps each object   in   to the constant functor   (i.e.   for each   in   and   for each   in  ) and each morphism   in   to the natural transformation   in   defined as, for every object   of  , the component

 
at  . In other words, the natural transformation is the one defined by having constant component   for every object of  .

Given a functor   (thought of as an object in  ), the limit of  , if it exists, is nothing but a universal morphism from   to  . Dually, the colimit of   is a universal morphism from   to  .

Properties

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Existence and uniqueness

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Defining a quantity does not guarantee its existence. Given a functor   and an object   of  , there may or may not exist a universal morphism from   to  . If, however, a universal morphism   does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism: if   is another pair, then there exists a unique isomorphism   such that  . This is easily seen by substituting   in the definition of a universal morphism.

It is the pair   which is essentially unique in this fashion. The object   itself is only unique up to isomorphism. Indeed, if   is a universal morphism and   is any isomorphism then the pair  , where   is also a universal morphism.

Equivalent formulations

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The definition of a universal morphism can be rephrased in a variety of ways. Let   be a functor and let   be an object of  . Then the following statements are equivalent:

  •   is a universal morphism from   to  
  •   is an initial object of the comma category  
  •   is a representation of  , where its components   are defined by

 

for each object   in  

The dual statements are also equivalent:

  •   is a universal morphism from   to  
  •   is a terminal object of the comma category  
  •   is a representation of  , where its components   are defined by

 

for each object   in  

Relation to adjoint functors

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Suppose   is a universal morphism from   to   and   is a universal morphism from   to  . By the universal property of universal morphisms, given any morphism   there exists a unique morphism   such that the following diagram commutes:

 
Universal morphisms can behave like a natural transformation between functors under suitable conditions.

If every object   of   admits a universal morphism to  , then the assignment   and   defines a functor  . The maps   then define a natural transformation from   (the identity functor on  ) to  . The functors   are then a pair of adjoint functors, with   left-adjoint to   and   right-adjoint to  .

Similar statements apply to the dual situation of terminal morphisms from  . If such morphisms exist for every   in   one obtains a functor   which is right-adjoint to   (so   is left-adjoint to  ).

Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let   and   be a pair of adjoint functors with unit   and co-unit   (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in   and  :

  • For each object   in  ,   is a universal morphism from   to  . That is, for all   there exists a unique   for which the following diagrams commute.
  • For each object   in  ,   is a universal morphism from   to  . That is, for all   there exists a unique   for which the following diagrams commute.
 
The unit and counit of an adjunction, which are natural transformations between functors, are an important example of universal morphisms.

Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of   (equivalently, every object of  ).

History

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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

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Notes

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  1. ^ Jacobson (2009), Proposition 1.6, p. 44.
  2. ^ See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property of group rings.
  3. ^ Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].

References

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  • Paul Cohn, Universal Algebra (1981), D.Reidel Publishing, Holland. ISBN 90-277-1213-1.
  • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer. ISBN 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
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  • 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
  • Hillman, Chris (2001). A Categorical Primer. CiteSeerX 10.1.1.24.3264: formal introduction to category theory.
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  • Stanford Encyclopedia of Philosophy: "Category Theory"—by Jean-Pierre Marquis. Extensive bibliography.
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  • 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.
  • Interactive Web page which generates examples of categorical constructions in the category of finite sets.