Building complicated examples out of simple ones

Contents

Quick description
Products
Quotients
Subobjects
Spaces of functions
Tensor products
Limits
Gluing
Ultraproducts

This article is incomplete. The last four methods need writing about.

Quick description

If you would like to find a mathematical object of a certain type with certain properties, and if none of the basic examples of objects of the given type have the required properties, then consider trying to build a more complicated object using the basic examples as building blocks. Three of the commonest building methods are products, quotients, and passing to subobjects. Another, which could be thought of as "exponentiation", is to look at spaces of functions built out of basic examples, and another very useful one is taking limits of various kinds. This article gives just a brief account of each general method and links to other articles.

There are also many important ways of creating mathematical objects of one type out of mathematical objects of another type. To find out about these, look at the page Making one mathematical object out of another.

Products

If $X$ and $Y$ are two mathematical structures, such as groups, vector spaces or topological spaces, then we can attempt to form a new mathematical structure out of the Cartesian product $X\times Y.$ Often there is a unique sensible way of doing this: sometimes there are interestingly different ways.

Here are two contrasting examples. ( For example, let $G$ and $H$ be groups. How can we put a group structure on the Cartesian product $G\times H$ ? We need to think of a way of multiplying two ordered pairs $(g_1,h_1)$ and $(g_2,h_2).$ There is one obvious idea, which is to let the product be $(g_1g_2,h_1h_2),$ and this does indeed produce a group, known as the direct product of $G$ and $H.$ Note that the maps $g\mapsto (g,e_H)$ and $h\mapsto (e_G,h)$ give us isomorphisms from $G$ and $H$ to subgroups of $G\times H.$

However, the direct product is not always the only good way of turning $G\times H$ into a group. For instance, if $H$ acts on $G$ , meaning that with each element $h\in H$ we can associate an automorphism $\phi_h$ of $G$ in such a way that $\phi_{h_1}\phi_{h_2}=\phi_{h_1h_2}$ for every $h_1$ and $h_2$ , then the following definition of multiplication gives rise to a group structure on $G\times H$ . We let $(g_1,h_1)(g_2,h_2)=(g_1\phi_{h_1}(g_2),h_1h_2).$ This is called a semidirect product of $G$ and $H.$

Click here if you want to know a sense in which the direct product is unique. ( The direct product of two groups $G$ and $H$ has the following universal property. Suppose that $K$ is a group and that $G\rightarrow K$ and $H\rightarrow K$ are two homomorphisms. Let $\alpha$ and $\beta$ be the isomorphic embeddings defined above. Then there is a unique homomorphism $G\times H\rightarrow K$ such that $\phi=\omega\circ\alpha$ and $\psi=\omega\circ\beta.$ This turns out to characterize the direct product. ) )

For more details, look at the article on product constructions.

Quotients

If $X$ is a mathematical structure, and $\sim$ is an equivalence relation on $X,$ then the quotient set $X/\sim$ is the set of all equivalence classes. If the equivalence relation is compatible with the structure on $X$ (in some appropriate sense that varies from example to example), then one can give a structure to the quotient set $X/\sim.$ This is a powerful way of creating examples of mathematical structures.

Here is an important example. ( A basic example of this is when $X$ is the plane $\mathbb{R}^2$ and $\sim$ is the equivalence relation where $(a,b)\sim(c,d)$ if and only if $c-a$ and $d-b$ are integers. The equivalence classes are translates of $\mathbb{Z}^2.$ That is, they are grids of the form $m,n\in\mathbb{Z}\}.$ Let us write $\mathbb{T}$ for the quotient set.

The plane has a great deal of structure on it: it is an Abelian group, a vector space, and a metric space (in several different ways). Moreover, we can identify the plane with the complex numbers and thereby think of it as a Riemann surface. How much of all this structure can we give to $\mathbb{T}?$

The way one gives structure to a quotient set always has the same general flavour: in order to do things to equivalence classes, one picks representatives, does things to the representatives, and takes the equivalence class of the result. How, for example, could one turn $\mathbb{T}$ into a group? Well, given two grids, $m,n\in\mathbb{Z}\}$ and $m,n\in\mathbb{Z}\},$ we pick the two obvious representatives $(a,b)$ and $(c,d)$ (or rather, obvious when the grids are presented to us like this), add them to obtain $(a+c,b+d),$ and take the equivalence class of the result to obtain the grid $m,n\in\mathbb{Z}\}.$

When one does this, one almost always has something important to check: that the operation is well-defined. That is, one must be sure that choosing different representatives would not have led to a different answer. In this example, the grid $m,n\in\mathbb{Z}\}$ is the same as a grid such as $m,n\in\mathbb{Z}\},$ so we need to be sure that if we had chosen $(a-3,b+12)$ as the representative of that grid, we would have ended up with the same answer. More generally, we need to know that whenever we add integers to $a,b,c,d,$ the final answer is unaffected, which is easily seen to be the case.

How about making $\mathbb{T}$ into a metric space? Given two of our grids, we would like to say what we mean by the distance between them.

After a moment's thought, one comes to see that the most natural notion of distance between two grids $G$ and $G'$ is the shortest distance between any point of $G$ and any point of $G'.$ This has the nice property that the quotient map , that is, the map that takes $(a,b)$ to the equivalence class of $(a,b)$ , is continuous, and indeed for small distances is an isometry. To be precise about the second assertion, if the distance between $(a,b)$ and $(c,d)$ is less than $1/2,$ then the distance between the corresponding grids is the same as the distance between the points themselves.

It is not hard to check that this really does define a metric on $\mathbb{T}.$ The proof depends on the fact that if $G$ and $G'$ are two translates of $\mathbb{Z}^2$ and you choose $(a,b)\in G$ and $(c,d)\in G'$ that have the shortest possible distance, then $(a+m,b+n)$ and $(c+m,b+n)$ will also have that distance for any pair of integers $(m,n).$

Can we make $\mathbb{T}$ into a vector space over $\mathbb{R}$ ? The answer is no, and it gives us an example of a structure that is not compatible with a natural equivalence relation. The problem is with scalar multiplication: if $(a,b)\sim(c,d)$ and $\lambda$ is a real number, it is not necessarily the case that $(\lambda a,\lambda b)\sim(\lambda c,\lambda d),$ and indeed they are usually not equivalent. (For example, $(0,0)\sim(0,1)$ but $(0,0)\not\sim(0,1/2).$ ) Therefore, there is no natural way of defining scalar multiples of translates of $\mathbb{Z}^2$ (or rather, there is, but the result is not a translate of $\mathbb{Z}^2$ unless the scalar is $\pm 1$ ).

Finally, we mention briefly that the Riemann-surface structure of $\mathbb{R}^2$ does make $\mathbb{T}$ into a Riemann surface. This turns out to be a very important construction, especially when one applies it not just to the quotient by $\mathbb{Z}^2$ but also to the quotient by other lattices: the resulting Riemann surfaces are different, and together they form a central example of a moduli space. )

Further examples are discussed in the article on quotient constructions.

Subobjects

Sometimes, if one is trying to find a mathematical object $X$ with given properties, the best approach is to start with a simpler object $Z,$ choose a property $P,$ and prove that the set $X$ of all $x\in Z$ such that $P(x)$ holds is an object with the required properties.

For example, a convenient way of defining a sphere $(x,y,z)$ is to take the set of all points in $\mathbb{R}^3$ such that $x^2+y^2+z^2=1.$

This article is in need of attention. Is it worth saying any more about this? There will be separate articles about things like finding clever subsets of the plane.

Spaces of functions

If $V$ and $W$ are vector spaces, then $L(V,W),$ the set of all linear maps from $V$ to $W,$ can also be made into a vector space very easily. Indeed, if $T$ and $U$ are linear maps from $V$ to $W,$ then we define $(T+U)(v)$ to be $Tv+Uv$ and we define $(\lambda T)(v)$ to be $\lambda(Tv).$

If $G$ is a group and $X$ is a set, then the set $G^X$ of all functions from $X$ to $G$ is a group under the operation $(f_1f_2)(x)=f_1(x)f_2(x).$

More generally, there are many examples where a set of functions from one object to another can be given a structure related to the structures of the objects.