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.
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2 Co-universal constructions 3 A worked example: kernels 4 What is it good for? 5 History |
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 AX of C and a morphism φX : F(AX) -> X in D, such that the following universal property is satisfied:
From the definition, it follows directly that the pair (AX, φX) 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 : AX -> A'X such that φ'X F(f) = φX.
More generally, if φX1 : F(AX1) -> X1 and φX2 : F(AX2) -> X2 are two universal morphisms, and h : X1 -> X2 is a morphism in D, then there exists a unique morphism Ah: AX1 -> AX2 such that φX2 F(Ah) = φX1.
Therefore, if every object X of D admits a universal morphism, then the assignment X |-> AX and h |-> Ah defines a covariant functor from D to C, and this functor is 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 BX of C and a morphism ρX: X -> F(BX) in D, such that the following universal property is satisfied:
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 objects AX and BX for every X, that does not mean that such objects also exist.
Suppose C is a category with zero morphisms and f : A → B is a morphism in C. A kernel of f is any morphism k: K → A such that:
The functor F : C → D maps an object K of C to the zero morphism 0KK : K → K and a morphism r : K → L to the pair (r,r).
Now, given a morphism f : A → B in the category C (i.e. an object of the category D) and an object K of C, a morphism from F(K) to f is given by a pair (k,l) such that f k = l 0KK = 0KB, 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.
Once one recognizes a certain construction as given by a universal property, one gains several benefits:
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.Universal constructions
The existence of the morphism ψ intuitively expresses the fact that AX is "large enough" or "general enough", while the uniqueness of the morphism ensures that AX is "not too large".Co-universal constructions
If BX exists for every X in D, then this co-universal constructions also defines a covariant functor from D to C, the so-called left-adjoint of F.A worked example: kernels
To understand this in the framework of the general setting above, we define the category D of all morphisms of C. The objects of D are morphisms φ : R → S in C, and a morphism from φ : R → S to ψ : U → V is given by a pair (r,s) of morphisms r : R → U and s : S → V such that s φ = ψ r. What is it good for?
History