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: 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.

Contents

Motivation

Formal definition

Examples

Tensor algebras

Kernels

Limits and colimits

Properties

Existence and uniqueness

Equivalent formulations

Relation to adjoint functors

History

See also

References

Motivation

What use does a universal property have? Once one recognizes a certain construction as given by a universal property, one gains several benefits:

★ Universal 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 property.

★ 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.

★ If the universal construction can be carried out for every ''X'' in ''C'', then we know that we obtain a functor from ''C'' to ''D''. (So for example, forming kernels is functorial: every morphism (α,β) from the morphism ''f'' to the morphism ''g'' induces a morphism from the kernel of ''f'' to the kernel of ''g''.)

★ Furthermore, this functor is a right or left adjoint to ''U''. But right adjoints commute with limits and left adjoints commute with colimits! So we can for example immediately conclude that the kernel of a product of maps is equal to the product of the kernels.

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 ''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:

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 ''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:

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 couniversal and the latter universal.

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

Given any vector space $V$ over a field ''K'' we can construct the tensor algebra of $V$, which is an algebra containing $V$. This construction is universal in the sense that any linear map $f\; colon\; V\; o\; B$ from $V$ in an algebra $B$, has a unique extension to an algebra morphism $ilde\; f\; colon\; TV\; o\; B$.
The construction works for any vector space $V$, so that $T$ is a functor from the category of vector spaces '''k''-Vect' over ''k'' to the category of algebras '''k''-Alg' over ''k''. The universal morphism is the pair $(TV,\; i)$, where $i\; colon\; V\; o\; TV$ is the inclusion map and the universal property is the unique factorisation
:$V\; xrightarrow\{f\}\; TV\; xrightarrow\{\; ilde\; f\}\; B$
so that $f=\; ilde\{f\}\; circ\; i$
The tensor functor ''T'' is left-adjoint to the forgetful functor ''U'' which assigns to each algebra its underlying vector space. Thus, in terms of the definition given above, one has that ''D'' is '''k''-Alg', and ''C'' is '''k''-Vect', so that $U\; colon\; D\; o\; C$ becomes ''U'': '''k''-Alg'→ '''k''-Vect', the functor that forgets multiplication in '''k''-Alg'.
The algebra was assumed only unital and associative but other universal constructions are possible on richer categories. For example, the symmetric $SV$ (resp. exterior ) algebra is the universal commutative (resp. anticommutative) algebra constructed over the space $V$. One may also start from other "base" categories; for example, the universal enveloping algebra $mathcal\; U\; (mathfrak\; g)$ is the universal associative algebra containing a given Lie algebra $mathfrak\; g$.
Properties of many other algebraic objects, such as kernels, direct sums, quotients, etc., are similar because they are defined by the same universal property.

Kernels

Suppose ''D'' is a category with zero morphisms (such as the category of groups) and ''f'': ''X'' → ''Y'' is a morphism in ''D''. A kernel of ''f'' is any morphism ''k'': ''K'' → ''X'' such that

★ ''fk'' is the zero morphism from ''K'' to ''Y'';

★ Given any morphism ''k''′: ''K''′ → ''X'' such that ''fk''′ is the zero morphism, there is a unique morphism ''u'': ''K''′ → ''K'' such that ''ku'' = ''k''′.
To understand this in the framework of the general setting above, we define the category ''C'' of morphisms in ''D''. The objects of ''C'' are morphisms ''f'': ''X'' → ''Y'' in ''D'', and a morphism from ''f'': ''X'' → ''Y'' to ''g'': ''S'' → ''T'' is given by a pair (α,β) of morphisms α: ''X'' → ''S'' and β: ''Y'' → ''T'' such that β''f'' = gα.
Define a functor ''F'': ''D'' → ''C'' that maps an object ''K'' of ''D'' to the zero morphism 0_''KK'': ''K'' → ''K'' and a morphism ''r'': ''K'' → ''L'' to the pair (''r'',''r'').
Now, given a morphism ''f'': ''X'' → ''Y'' in the category ''D'' (thought of as an object in the category ''C'') and an object ''K'' of ''D'', a morphism from ''F''(''K'') to ''f'' is given by a pair (''k'',''l'') such that ''fk'' = ''l''0_''KK'' = 0_''KY'', which is exactly what shows up in the universal property of kernels given above. The abstract “universal morphism from ''F'' to ''f''” is nothing but the universal property of a kernel.

Limits and colimits

Limits and colimits are important special cases of universal constructions. Let ''J'' and ''C'' be categories with ''J'' small (''J'' is thought of as an 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 unique up to a ''unique'' isomorphism. That is, if (''A''′, φ′) is another such pair, then there exists a unique isomorphism ''g'': ''A'' → ''A''′ such that φ′ = ''U''(''g'')φ. This is easily seen by substituting (''A''′, φ′) for (''Y'', ''f'') in the definition of the universal property.

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.

See also

★ Free object

★ Monad (category theory)

★ Variety of algebras

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.