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universal mapping property
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(Definition)
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A strong attribute of categories is a uniform description of many seemingly unrelated concepts from diverse interest areas of mathematics. For example, Cartesian products produce new sets from old just as direct product produce new groups from old and Cartesian products of topologies are again topologies. To make
these notions uniform one can provide categorical definitions. Because categories assume as their constituents only abstract objects and morphisms, all the properties of such constructions must be given as properties of morphisms between objects. Those constructions which can be characterized by means of the existence of a unique morphism are generally grouped under the heading of universal mapping properties [1,
p. 57]. A precise definition will follow below.
As is common with category theory, there are generally not many theorems that can be proved at a categorical level and set to apply to all universal mapping properties. The exception is a generic proof of uniqueness of objects with a universal mapping property. It should be emphasized that most universal mapping properties are a result of the underlying structure of the categories in question and not provided for abstract reasons. (For concrete examples consider the construction of free groups or free Lie algebras.)
Remark 1 There are contingents that use the phrase universal property especially in the vernacular. Mac Lane does so in passing but his use is not as a defined term but rather as an intuitive concept to be refined to a strict definition later [3, p. 2]
Because the property applies to the uniqueness of mappings the title universal mapping property remains the choice of most mathematical references in print. Authors interested in shorter titles will sometimes substitute Mac Lane's concept of universal arrows[3, III.1] - a rigorous definition which will be explained below and captures the spirit of the universal mapping property as presently described. The preferred term of morphism for arrow motivates the modern title universal morphism.
Universal mapping properties arise in such constructions as: direct products and direct sums, free groups, free algebras, product topology, Stone-Čech compactification, tensor product, exterior algebra, inverse and direct limit, pullbacks and pushouts.
We follow [3, III.1] for our treatment of universal arrows and universal mapping properties.
Definition 2 Let
be a functor from a category
to a category
, and let be an object of
. A universal morphism from to consists of a pair where is an object of
and
is a morphism in
, such that the following universal mapping property is satisfied:
Whenever is an object of
and
is a morphism in
, then there exists a unique morphism such that the following diagram commutes.
Figure 1: A universal morphism from to
![\begin{figure} % latex2html id marker 201 \begin{displaymath}\begin{xy} *!C\xybo... ...]^{\phi}\ar[dr]_f & & \ & U(Y) & Y. } } \end{xy}\end{displaymath} \end{figure} \begin{figure} % latex2html id marker 201 \begin{displaymath}\begin{xy} *!C\xybo... ...]^{\phi}\ar[dr]_f & & \ & U(Y) & Y. } } \end{xy}\end{displaymath} \end{figure}](http://images.planetmath.org:8080/cache/objects/7566/l2h/img26.png) |
The existence of the morphism g intuitively expresses the fact that is “general enough”, while the uniqueness of the morphism ensures that is “not too general”.
One can also consider the categorical dual of the above definition by reversing all the arrows. Let
be a functor, and let be an object of
. A co-universal morphism from to consists of a pair where is an object of
and
is a morphism in
, such that the following universal mapping property is satisfied:
Whenever is an object of
and
is a morphism in
, then there exists a unique morphism
such that the following diagram commutes.
Figure 2: A co-universal morphism from to
![\begin{figure} % latex2html id marker 219 \centering \begin{displaymath}\begin{x... ... & & X.\ A & F(A)\ar[ur]_{\phi} & } } \end{xy}\end{displaymath} \end{figure} \begin{figure} % latex2html id marker 219 \centering \begin{displaymath}\begin{x... ... & & X.\ A & F(A)\ar[ur]_{\phi} & } } \end{xy}\end{displaymath} \end{figure}](http://images.planetmath.org:8080/cache/objects/7566/l2h/img48.png) |
To avoid ambiguity, some authors may call one of these constructions a universal morphism and the other one a co-universal morphism.
Defining a quantity does not guarantee its existence. Given a functor and an object as above, there may or may not exist a universal morphism from to (or from to ). If, however, a universal morphism does exists, then it is unique up to a unique
isomorphism. That is, if
is another such pair then there exists a unique isomorphism
such that
. This is easily seen by substituting
for in the definition of the universal mapping property.
The definition of a universal morphism can be rephrased in a variety of ways. Let be a functor from
to
, and let be an object of
. Then the following statements are equivalent
The dual statements are also equivalent
is a universal morphism from to ;
is a terminal object of the comma category
;
is a representable functor|representation of
.
Suppose that
is a universal morphism from to and that
is also a universal morphism from to . By the universal mapping property, given any morphism
there exists a unique morphism
such that the following diagram commutes
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 . Similar statements apply to the dual situation of morphisms from . If
such morphisms exist for every
one obtains a functor
which is right-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
, the pair
is a universal morphism from to . That is, for all
, there exists a unique
for which the diagrams below commute.
- Dually, for each object
, the pair
is a universal morphism from to . That is, for all
there exists a unique
for which the following diagrams commute.
Figure 3: Universal mapping property of a pair of adjoint functors
![\begin{figure} % latex2html id marker 291 \begin{displaymath}\begin{xy} *!C\xybo... ... & G(Y) & FG(Y)\ar[ur]_{\epsilon_Y} } } \end{xy}\end{displaymath} \end{figure} \begin{figure} % latex2html id marker 291 \begin{displaymath}\begin{xy} *!C\xybo... ... & G(Y) & FG(Y)\ar[ur]_{\epsilon_Y} } } \end{xy}\end{displaymath} \end{figure}](http://images.planetmath.org:8080/cache/objects/7566/l2h/img124.png) |
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, for every object of
).
Let
be the category of vector spaces over a field , and let
be the category of -algebras (assumed to be unital and associative). Let be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space over we can construct the tensor algebra of . The universal mapping property of the tensor algebra expresses the fact that the pair
, where
is the natural inclusion map, is a universal morphism from to . Since this construction works for any vector space , we conclude that is a functor from
to
. This functor is left-adjoint to the forgetful functor .
Suppose
is a category with zero morphisms
(such as the category of groups) and let
be a morphism in
. A kernel of is any morphism
such that
- the composition
is the zero morphism from to ;
- given any morphism
such that is the zero morphism, there is a unique morphism
such that .
To understand this in the framework of the general setting above, we let
be the category of morphisms in
. The objects of
are morphisms
in
, and a morphism from
to
is a commutative square whose sides are a pair of morphisms
and
. Diagramatically,
Let
be the functor that maps an object
to the zero morphism
, and that maps a morphism
to the trivial square with sides
. Now, let be an object
(which is the same thing as a morphism
in the category
). A kernel
, if it exists, is the same thing as an object
and a morphism in
that satisfies the co-universal property expressed by the diagram below. I.e., a kernel is the same thing as a universal morphism from to .
Limits and colimits are important special cases of universal constructions. Let
and
be categories with
small (
is to be thought of as an index category) and let
be the corresponding functor category of functors from
to
. The diagonal functor
is the functor that maps each object
to the constant functor
(i.e.,
for each
.) 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 co-universal morphism from to .
For the most part, the objects in a category which are constructed in terms of universal mapping properties are not new to the theorist of these categories. For example, topologist long knew that a new topology results from a Cartesian product of topologies. However the general vocabulary allows for easier detection and understanding of functors between different categories. For example, the homology functors take the coproducts - wedge products in topology - to coproducts of modules - direct sums in module categories. Proofs of these facts are in no way facilitated by category theory but the terminology is at least uniform and thus easier to conceptualize and express.
Once one recognizes a certain construction as given by a universal mapping property, one gains several benefits.
- Universal mapping properties define objects up to a unique isomorphism. One strategy to prove that two objects are isomorphic is therefore to show that they satisfy the same universal mapping property.
- The concrete details of a given construction may be messy, but if the construction satisfies a universal mapping property, one can forget all those details; all there is to know about the construct is already contained in the universal mapping property. Proofs often become short and elegant if the universal mapping property is used rather than the concrete details.
- If the universal construction can be carried out for every
in
, then we know that we obtain a functor from
to . For example, forming kernels is functorial; every commutative square
from the morphism to the morphism induces a morphism from the kernel of to the kernel of .
- Furthermore, if such a functor can be formed, it is a right or left adjoint to
. But right adjoints commute with limits, and left adjoints commute with colimits! So, looking back at the previous example, we can immediately conclude that the kernel of a product is equal to the product of the kernels.
Universal mapping 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.
- 1
- Thomas W. Hungerford Algebra, Springer-Verlag, New York, (1974).
- 2
- Paul M. Cohen, Universal Algebra, D.Reidel Publishing, Holland, (1981). ISBN 90-277-1213-1.
- 3
- Saunders Mac Lane Categories for the Working Mathematician 2nd ed. Graduate Texts in Mathematics 5. Springer, (1998). ISBN 0-387-98403-8.
This entries was adapted, for the most part, from the Wikipedia entry entitled Universal Property. In turn much of Wikipedia's entry appears in [3, III.1] were the interested reader is directed for further details.
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Cross-references: Bourbaki, product, right, induces, contained, isomorphic, strategy, modules, wedge products, homology, constant functor, diagonal functor, functor category, index category, colimits, limits, sides, square, commutative, composition, kernel, zero morphisms, inclusion map, tensor algebra, algebra, forgetful functor, associative, unital, field, vector spaces, solution, adjoint pair, co-unit, unit, similar, adjoint functors, identity functor, natural transformation, maps, terminal object, representable functor, comma category, initial object, equivalent, variety, isomorphism, functor, pushouts, pullbacks, direct limit, inverse, exterior algebra, tensor product, compactification, product topology, free algebras, direct sums, arrow, mappings, strict, term, free Lie algebras, free groups, structure, generic, level, category theory, objects, morphisms, abstract objects, definitions, categorical, topologies, groups, direct product, Cartesian products, areas, categories, attribute
There are 61 references to this entry.
This is version 11 of universal mapping property, born on 2006-01-19, modified 2007-06-30.
Object id is 7566, canonical name is UniversalProperty.
Accessed 6230 times total.
Classification:
| AMS MSC: | 18A40 (Category theory; homological algebra :: General theory of categories and functors :: Adjoint functors ) |
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Pending Errata and Addenda
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