Universal property

In abstract algebra and category theory, 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? and [direct sum]?, tensor product, inverse limit and [direct limit]?, kernel and cokernel?, pullback?, pushout? and equalizer?.

Let C and D be categories and F : C -> D be a functor. A universal construction assigns to every object X of D 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.

From this definition, it follows 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 univeral property, then there exists a unique isomorphism f : A_X -> A'_X such that φ'_X f = φ_X.

Furthermore, if 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. 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 of 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 for an object A_X, that does not mean that such an object also exists.