| Universal Property |
Article Index for Universal |
Website Links For Universal |
Information AboutUniversal Property |
| CATEGORIES ABOUT UNIVERSAL PROPERTY | |
| category theory | |
|
Below, we will give a general treatment of universal properties. It is advisable to study several examples first: Direct Product and Direct Sum , Free Group , Product Topology , Stone-Čech Compactification , Tensor Product , Inverse Limit and Direct Limit , Kernel and Cokernel , Pullback , Pushout and Equalizer . 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:
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:
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. 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:
The dual statements are also equivalent:
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 ''Xi'' ''Ai'' and ''h'' ''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 ''U V''. The functors (''V'', ''U'') are then a pair of Adjoint Functor s, 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'':
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''). EXAMPLES We give 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'' be 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 universal property of the tensor algebra expresses the fact that the pair (''T''(''V''), ''i''), where ''i'' : ''V'' → ''T''(''V'') is the inclusion map, is a universal morphism from ''V'' to ''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 ''U''. Kernels Suppose ''D'' is a category with of ''f'' is any morphism ''k'': ''K'' → ''X'' such that
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 ''f'' ''k'' = ''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. WHAT IS IT GOOD FOR? Once one recognizes a certain construction as given by a universal property, one gains several benefits:
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 . REFERENCES
|
|
|