In the sequel, we will give a general treatment of universal properties. It is advisable to study several examples first: product of groups and direct sum, free group, product topology, StoneČech compactification, tensor product, inverse limit and direct limit, kernel and cokernel[?], pullback[?], pushout[?] and equalizer[?].
Let C and D be categories, F : C > D be a functor, and X an object of D. A universal morphism from F to X consists of an object A_{X} of C and a morphism φ_{X} : F(A_{X}) > X in D, such that the following universal property is satisfied:
The existence of the morphism ψ intuitively expresses the fact that A_{X} is "large enough" or "general enough", while the uniqueness of the morphism ensures that A_{X} is "not too large".
From the definition, it follows directly that the pair (A_{X}, φ_{X}) is determined up to a unique isomorphism by X, in the following sense: if A'_{X} is another object of C and φ'_{X} : F(A'_{X}) > X is another morphism which has the universal property, then there exists a unique isomorphism f : A_{X} > A'_{X} such that φ'_{X} f = φ_{X}.
More generally, if φ_{X1} : F(A_{X1}) > X_{1} and φ_{X2} : F(A_{X2}) > X_{2} are two universal morphisms, and h : X_{1} > X_{2} is a morphism in D, then there exists a unique morphism A_{h}: A_{X1} > A_{X2} such that φ_{X2} F(A_{h}) = φ_{X1}.
Therefore, if every object X of D admits a universal arrow, then the assignment X > A_{X} and h > A_{h} defines a covariant functor from D to C, the rightadjoint of F.
The dual concept of a couniversal construction also exists: it assigns to every object X of D an object B_{X} of C and a morphism ρ_{X}: X > F(B_{X}) in D, such that the following universal property is satisfied:
It is important to realize that not every functor F has a rightadjoint or a left adjoint; in other words: while one may always write down a universal property defining an object A_{X}, that does not mean that such an object also exists.
Search Encyclopedia

Featured Article
