Universal property

In category theory, abstract algebra and other fields of mathematics, constructions are often defined by an abstract property which requires the existence of unique morphisms under certain conditions. These properties are called universal properties.

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, Stone-Čech 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:

Whenever U is an object of C and φ : F(U) -> X is a morphism in D, then there exists a unique morphism ψ : U -> A_X such that φ_X F(ψ) = φ.

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) -> X₁ and φ_X2 : F(A_X2) -> X₂ are two universal morphisms, and h : X₁ -> X₂ 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 right-adjoint of F.

The dual concept of a co-universal 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:

Whenever U is an object of C and ρ : X -> F(U) is a morphism in D, then there exists a unique morphism σ : B_X -> U such that F(σ) ρ_X = ρ. A Co-universal constructions also defines a covariant functor from D to C, the so-called left-adjoint of F.

It is important to realize that not every functor F has a right-adjoint 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.