nLab
universal construction

Contents

Idea

Universal properties are commonly used in mathematics, often without mentioning the term “universal property”.

For example, if one were asked to give a map ×\mathbb{R} \to \mathbb{R} \times \mathbb{C}, they might write down something like x(x 2,x+ix)x \mapsto (x^2, x + i x). In effect, what is done is that a pair of maps \mathbb{R} \to \mathbb{R} and \mathbb{R} \to \mathbb{C} is given, namely (xx 2,xx+ix)(x \mapsto x^2, x \mapsto x + i x). The universal property of the product says that giving a map to A×BA \times B is the same as giving a map to AA and a map to BB, and moreover this correspondence is natural in some precise sense.

Similarly, given rings RR and SS, if we want to extend a ring homomorphism RSR \to S to a homomorphism from the polynomial ring R[x]SR[x] \to S, all we have to do is to specify an element of SS that we send xx to. In other words, a homomorphism R[x]SR[x] \to S is the same as a homomorphism RSR \to S and an element of SS.

For it to be a universal property, just the existence of such a bijection is not sufficient. We will need some conditions to make sure the bijection is “natural”. Abstractly, this says that the bijection is given by a natural isomorphism of certain functors. More concretely, by the Yoneda lemma, this is equivalent to saying the bijection is “mediated” by some “universal maps”, which is how universal properties are usually formulated. See Concrete examples for more details.

Recall that by the Yoneda lemma, specifying how we can map in or out of an object uniquely determines the object up to isomorphism. So we can use these universal properties as definitions of the constructions! These are known as universal constructions. Of course, these definitions are not actually “constructions”. We still have to do the concrete constructions the good, old way to show that there are objects satisfying the universal property (or apply general theorems such as the adjoint functor theorem).

Example

Concrete examples

We first look at a few concrete examples of universal properties. These are all special cases of the ones described below.

Example (products)

The product of two objects (eg. sets, groups, rings etc.) is specified by the property that maps f:XA×Bf: X \to A \times B biject naturally with pairs of maps (f 1:XA,f 2:XB)(f_1: X \to A, f_2: X \to B).

The naturality condition is that if f:XA×Bf: X \to A \times B corresponds to f 1:XA,f 2:XBf_1: X \to A, f_2: X \to B, and g:YXg: Y \to X is a map, then fgf \circ g corresponds to f 1gf_1 \circ g and f 2gf_2 \circ g, so that the bijection respects composition.

In particular, the identity map id:A×BA×Bid: A \times B \to A \times B corresponds to a pair of projection maps π 1:A×BA\pi_1: A \times B \to A and π 2:A×BB\pi_2: A \times B \to B. Then if f:XA×Bf: X \to A \times B is a map, then by naturality, it corresponds to π 1f:XA\pi_1 \circ f: X \to A and π 2f:XB\pi_2 \circ f: X \to B.

Suppose we are not given a bijection, but just an object PP with maps π 1:PA\pi_1: P \to A and π 2:PB\pi_2: P \to B. Then as above, we obtain a function from maps XPX \to P to pairs of maps XA,XBX \to A, X \to B by composition. This makes PP into the product of AA and BB exactly when this function is a bijection, ie. for any pair of maps f 1:XA,f 2:XBf_1 : X \to A, f_2: X \to B, there is a unique map f:XPf: X \to P whose compositions with π 1,π 2\pi_1, \pi_2 are f 1,f 2f_1, f_2 respectively (naturality is easy to check).

(The experienced reader will notice that this is just a special case of the Yoneda lemma)

Thus, the universal property can be stated as follows: CC is a product of AA and BB if there exists maps π 1:CA\pi_1: C \to A and π 2:CB\pi_2: C \to B such that given any pair of maps f 1:XAf_1: X \to A and f 2:XBf_2: X \to B, there is a unique map f:XCf: X \to C such that the following diagram commutes:

X f 1 f f 2 A π 1 C π 2 B \array{ & & X & & \\ & ^\mathllap{f_1}\swarrow & \downarrow^\mathrlap{f} & \searrow^\mathrlap{f_2}\\ A & \underset{\pi_1}{\leftarrow} & C & \underset{\pi_2}{\rightarrow} & B }

In this case, we tend to write A×BA \times B for CC.

Note that if we are talking about sets, then an element of a set XX is equivalent to a map 1X1 \to X from the singleton set 11. Thus in particular, the above definition says an element of A×BA \times B is the same as a pair of elements (a,b)(a, b), where aAa \in A and bBb \in B.

Example (free groups)

The free group on nn generators is a group F nF_n such that group homomorphisms F nGF_n \to G bijects (naturally) with nn elements of GG (not necessarily distinct).

Similar to the above, the naturality condition says if f:F nGf: F_n \to G corresponds to g 1,...,g nGg_1, ..., g_n \in G, and h:GHh: G \to H is a map, then hfh \circ f corresponds to the elements h(g 1),...,h(g n)h(g_1), ..., h(g_n). In particular, suppose the identity map id:F nF nid: F_n \to F_n corresponds to nn elements x 1,...,x nF nx_1, ..., x_n\in F_n. Then any homomorphism f:F nGf: F_n \to G corresponds to the elements f(x 1),...,f(x n)f(x_1), ..., f(x_n) of GG.

Thus, given the specified elements x 1,...,x nF nx_1, ..., x_n \in F_n, the universal property says given any nn elements of GG, we can find a unique homomorphism f:F nGf: F_n \to G that sends x 1,...,x nx_1, ..., x_n to the nn elements.

Diagrammatically, picking nn elements out of a set XX is the same as a function (of sets) nXn \to X. If we write U(G)U(G) for the underlying set of the group GG (ie. UU is the forgetful functor to SetSet), the universal property of the free group says that there is a specified function ϕ:nU(F n)\phi: n \to U(F_n), such that for every function f:nU(G)f: n \to U(G), we can find a unique group homomorphism f˜:F nG\tilde{f}: F_n \to G such that the following diagram commutes:

U(F n) U(f˜) U(G) ϕ f n \array{ U(F_n) & \overset{U(\tilde{f})}{\to} & U(G)\\ ^\mathllap{\phi}\uparrow & \nearrow_{\mathrlap{f}}\\ n }

In other words, every map f:nU(G)f: n \to U(G) factors through the universal map ϕ:nU(F n)\phi: n \to U(F_n) uniquely.

Example (tensor products)

The tensor product of vector spaces has the universal property that a bilinear map V×WUV \times W \to U bijects naturally with linear maps VWUV \otimes W \to U. The naturality condition is given by the existence of a universal bilinear map ϕ:V×WVW\phi: V \times W \to V \otimes W such that every bilinear map V×WUV \times W \to U factors through ϕ\phi uniquely.

We have more degenerate examples such as the terminal object:

Example (terminal and initial objects)

In the category of sets, the singleton 11 satisfies the property that there is always a unique from any object to 11. So we can say that the maps A1A \to 1 biject (necessarily naturally) with the set 11. More generally, in any category, if an object XX is such that there is always a unique map from any object to XX, then XX is called the terminal object.

Dually, an intial object is an object 00 such that there is a unique map from 00 to any object XX.

Classes of examples

In general, the universal constructions in category theory include

Each of these may be defined by requiring it to satisfy a universal property. A universal property is a property of some construction which boils down to (is manifestly equivalent to) the property that an associated object is a universal initial object of some (auxiliary) category.

In good cases, every single one of these is a special case of every other, so somehow one single concept here comes to us with many different faces.

Some or all of these have analogs in higher category theory, notably in 2-category theory and (∞,1)-category theory:

Revised on September 21, 2016 03:47:45 by David Corfield (51.6.72.106)