In category theory, a branch of mathematics, a **universal property** is an important property which is satisfied by a **universal morphism** (see Formal Definition). Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category (see Connection with Comma Categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

- Motivation
- Formal definition
- Connection with Comma Categories
- Examples
- Tensor algebras
- Products
- Limits and colimits
- Properties
- Existence and uniqueness
- Equivalent formulations
- Relation to adjoint functors
- History
- See also
- Notes
- References
- External links

Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory.

This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, Dedekind–MacNeille completion, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

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 painful to actually construct, but using its universal property makes it much easier to deal with.
- 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.

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 , while and are objects of .

Thus, 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:

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:

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.

Universal morphisms can be described more concisely as initial and terminal objects in a comma category.

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:

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

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:

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

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 .

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.

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).

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 .

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 to (i.e. for each in ).

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 .

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.

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

The dual statements are also equivalent:

- is a universal morphism from to
- is a terminal object of the comma category
- is a representation of

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:

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.

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 ).

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.

- ↑ Jacobson (2009), Proposition 1.6, p. 44.
- ↑ See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property of group rings.

In mathematics, specifically category theory, a **functor** is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In category theory, a branch of mathematics, the abstract notion of a **limit** captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a **colimit** generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In category theory, a branch of mathematics, an **initial object** of a category C is an object I in C such that for every object X in C, there exists precisely one morphism *I* → *X*.

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.

In category theory, the **coproduct**, or **categorical sum**, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, the idea of a **free object** is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure. It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set *A* can be thought of as being a "generic" algebraic structure over *A*: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

In mathematics, a **comma category** is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category".

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 written

In mathematics, particularly category theory, a **representable functor** is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

In mathematics, the **derived category***D*(*A*) of an abelian category *A* is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on *A*. The construction proceeds on the basis that the objects of *D*(*A*) should be chain complexes in *A*, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, a **triangulated category** is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

In mathematics, specifically in category theory, an **exponential object** or **map object** is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an **exponential law**.

This is a glossary of properties and concepts in category theory in mathematics.

**Kan extensions** are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.

In category theory, a branch of mathematics, the **diagonal functor** is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects *within* the category : a product is a universal arrow from to . The arrow comprises the projection maps.

In category theory, a branch of mathematics, the **cone of a functor** is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

In category theory, a branch of mathematics, a **diagram** is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a *function* from a fixed index *set* to the class of *sets*. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a *functor* from a fixed index *category* to some *category*.

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*Categories for the Working Mathematician*. Graduate Texts in Mathematics 5 (2nd ed.). Springer. ISBN 0-387-98403-8. - Borceux, F.
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*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

- nLab, a wiki project on mathematics, physics and philosophy with emphasis on the
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