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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 . MOTIVATION What use does a universal property have? Once one recognizes a certain construction as given by a universal property, one gains several benefits:
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, 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 over a Field ''K'' we can construct the Tensor Algebra of , which is an Algebra containing . This construction is universal in the sense that any linear map from in an algebra , has a unique extension to an algebra morphism . The construction works for any vector space , so that 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 , where is the inclusion map and the universal property is the unique factorisation : so that 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 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 (resp. Exterior ) algebra is the universal Commutative (resp. Anticommutative ) algebra constructed over the space . One may also start from other "base" categories; for example, the Universal Enveloping Algebra is the universal associative algebra containing a given Lie Algebra . 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 Morphism s (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
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:
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 and 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 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''). 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 REFERENCES
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