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In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism 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 AX of C and a morphism φX : F(AX) → 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 → AX such that φX F(ψ) = φ.
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".
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) = h 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: Whenever U is an object of C and ρ : X → F(U) is a morphism in D, then there exists a unique morphism σ : BX → U such that F(σ) ρX = ρ.
If BX exists for every X in D, then this co-universal construction 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 objects AX and BX for every X, that does not mean that such objects also exist.
A worked example: kernels
- f k is the zero morphism from K to B;
- Given any morphism k′: K′ → A such that f k′ is the zero morphism, there is a unique morphism u: K′ → K such that k u = k′.
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.
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 (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.
What is it good for?
Once one recognizes a certain construction as given by a universal property, one gains several benefits:
- Universal properties define objects up to isomorphism; one strategy to prove that two objects are isomorphic is therefore to show that they satisfy the same universal property.
- The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details.
- If the universal construction can be carried out for every X in D, then we know that we obtain a functor from D to C. (So for example, forming kernels is functorial: every morphism (u,v) from the morphism f to the morphism g induces a morphism from the kernel of f to the kernel of g.)
- Furthermore, this functor is a right or left adjoint of F, depending on whether the construction is universal or co-universal. But right adjoints commute with limits and left adjoints commute with colimits! (So we can for example immediately conclude that the kernel of a product of maps is equal to the product of the kernels.)
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.
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