Universal property

In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. These properties are called universal properties. Universal properties are studied abstractly using the language of 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, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

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 construct 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 up to a unique isomorphism. 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.
* 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

Let "U": "D" → "C" be a functor from a category "D" to a category "C", and let "X" be an object of "C". A universal morphism from "U" to "X" consists of a pair ("A", φ) where "A" is an object of "D" and φ: "U"("A") → "X" is a morphism in "C", such that the following universal property is satisfied:
*Whenever "Y" is an object of "D" and "f": "U"("Y") → "X" is a morphism in "C", then there exists a "unique" morphism "g": "Y" → "A" such that the following diagram commutes:

The existence of the morphism "g" intuitively expresses the fact that "A" is "general enough", while the uniqueness of the morphism ensures that "A" is "not too general".

One can also consider the categorical dual of the above definition by reversing all the arrows. A universal morphism from "X" to "U" consists of a pair ("A", φ) where "A" is an object of "D" and φ: "X" → "U"("A") is a morphism in "C", such that the following universal property is satisfied:
*Whenever "Y" is an object of "D" and "f": "X" → "U"("Y") is a morphism in "C", then there exists a "unique" morphism "g": "A" → "Y" such that the following diagram commutes:

Note that some authors may call one of these constructions a "universal morphism" and the other one a "co-universal morphism". Which is which depends on the author, although in order to be consistent with the naming of limits and colimits the former construction should be named universal and the latter couniversal. However, the rest of this article uses the latter construction.

Examples

Below are a few worked examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

Tensor algebras

Let "C" be the category of vector spaces "K"-Vect over a field "K" and let "D" be the category of algebras "K"-Alg over "K" (assumed to be unital and associative). Let:"U" : "K"-Alg → "K"-Vectbe 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") of "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 a universal property of the tensor algebra since it expresses the fact that the pair ("T"("V"), "i"), where "i" : "V" → "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 functor is left adjoint to the forgetful functor (see the section below on relation to adjoint functors).

Products

Categorical products can be characterized by a universal property. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top.

Let "X" and "Y" be objects of a category "D". The product of "X" and "Y" is an object "X" × "Y" together with two morphisms:π₁ : "X" × "Y" → "X":π₂ : "X" × "Y" → "Y"such that for any other object "Z" of "D" and morphisms "f" : "Z" → "X" and "g" : "Z" → "Y" there exists a unique morphism "h" : "Z" → "X" × "Y" such that "f" = π₁∘"h" and "g" = π₂∘"h".

To understand this characterization as a universal property we take the category "C" to be the product category "D" × "D" and define the diagonal functor:Δ : "D" → "D" × "D"by Δ("X") = ("X", "X") and Δ("f" : "X" → "Y") = ("f", "f"). Then ("X" × "Y", (π₁, π₂)) is a universal morphism from Δ to the object ("X", "Y") of "D" × "D". This is just a restatement of the above since the pair ("f", "g") represents an (arbitrary) morphism from Δ("Z") to ("X", "Y").

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 "J" and "C" be categories with a "J" small index category and let "C"^"J" be the corresponding functor category. The "diagonal functor":Δ : "C" → "C"^"J"is the functor that maps each object "N" in "C" to the constant functor Δ("N"): "J" → "C" to "N" (i.e. Δ("N")("X") = "N" for each "X" in "J").

Given a functor "F" : "J" → "C" (thought of as an object in "C"^"J"), the "limit" of "F", if it exists, is nothing but a universal morphism from Δ to "F". Dually, the "colimit" of "F" is a universal morphism from "F" to Δ.

Properties

Existence and uniqueness

Defining a quantity does not guarantee its existence. Given a functor "U" and an object "X" as above, there may or may not exist a universal morphism from "X" to "U" (or from "U" to "X"). If, however, a universal morphism ("A", φ) does exists then it is essentially unique. Specifically, it is unique up to a "unique" isomorphism: if ("A"′, φ′) is another such pair, then there exists a unique isomorphism "k": "A" → "A"′ such that φ′ = "U"("k")φ. This is easily seen by substituting ("A"′, φ′) for ("Y", "f") in the definition of the universal property.

It is the pair ("A", φ) which is essentially unique in this fashion. The object "A" itself is only unique up to isomorphism. Indeed, if ("A", φ) is a universal morphism and "k": "A" → "A"′ is any isomorphism then the pair ("A"′, φ′), where φ′ = "U"("k")φ, is also a universal morphism.

Equivalent formulations

The definition of a universal morphism can be rephrased in a variety of ways. Let "U" be a functor from "D" to "C", and let "X" be an object of "C". Then the following statements are equivalent:
* ("A", φ) is a universal morphism from "X" to "U"
* ("A", φ) is an initial object of the comma category ("X" ↓ "U")
* ("A", φ) is a representation of Hom_"C"("X", "U"—)

The dual statements are also equivalent:
* ("A", φ) is a universal morphism from "U" to "X"
* ("A", φ) is a terminal object of the comma category ("U" ↓ "X")
* ("A", φ) is a representation of Hom_"C"("U"—, "X")

Relation to adjoint functors

Suppose ("A"₁, φ₁) is a universal morphism from "X"₁ to "U" and ("A"₂, φ₂) is a universal morphism from "X"₂ to "U". By the universal property, given any morphism "h": "X"₁ → "X"₂ there exists a unique morphism "g": "A"₁ → "A"₂ such that the following diagram commutes:

If "every" object "X"_"i" of "C" admits a universal morphism to "U", then the assignment $X\_i\; mapsto\; A\_i$ and $h\; mapsto\; g$ defines a functor "V" from "C" to "D". The maps φ_"i" then define a natural transformation from 1_"C" (the identity functor on "C") to "UV". The functors ("V", "U") are then a pair of adjoint functors, with "V" left-adjoint to "U" and "U" right-adjoint to "V".

Similar statements apply to the dual situation of morphisms from "U". If such morphisms exist for every "X" in "C" one obtains a functor "V": "C" → "D" which is right-adjoint to "U" (so "U" is left-adjoint to "V").

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 η and co-unit ε (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in "C" and "D":
*For each object "X" in "C", ("F"("X"), η_"X") is a universal morphism from "X" to "G". That is, for all "f": "X" → "G"("Y") there exists a unique "g": "F"("X") → "Y" for which the following diagrams commute.
*For each object "Y" in "D", ("G"("Y"), ε_"Y") is a universal morphism from "F" to "Y". That is, for all "g": "F"("X") → "Y" there exists a unique "f": "X" → "G"("Y") 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 "C" (equivalently, every object of "D").

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.

ee also

* Free object
* Monad (category theory)
* Variety of algebras
* Cartesian closed category

References

* Cohen, Paul M., "Universal Algebra" (1981), D.Reidel Publishing, Holland. ISBN 90-277-1213-1.
* Mac Lane, Saunders, "Categories for the Working Mathematician" 2nd ed. (1998), Graduate Texts in Mathematics 5. 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 0201006391.