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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 and
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
Often there is a unique sensible way of doing this: sometimes there are interestingly different ways.
Here are two contrasting examples. ( For example, let and
be groups. How can we put a group structure on the Cartesian product
? We need to think of a way of multiplying two ordered pairs
and
There is one obvious idea, which is to let the product be
and this does indeed produce a group, known as the direct product of
and
Note that the maps
and
give us isomorphisms from
and
to subgroups of
However, the direct product is not always the only good way of turning into a group. For instance, if
acts on
, meaning that with each element
we can associate an automorphism
of
in such a way that
for every
and
, then the following definition of multiplication gives rise to a group structure on
. We let
This is called a semidirect product of
and
Click here if you want to know a sense in which the direct product is unique. ( The direct product of two groups and
has the following universal property. Suppose that
is a group and that
and
are two homomorphisms. Let
and
be the isomorphic embeddings defined above. Then there is a unique homomorphism
such that
and
This turns out to characterize the direct product. ) )
For more details, look at the article on product constructions.
Quotients
If is a mathematical structure, and
is an equivalence relation on
then the quotient set
is the set of all equivalence classes. If the equivalence relation is compatible with the structure on
(in some appropriate sense that varies from example to example), then one can give a structure to the quotient set
This is a powerful way of creating examples of mathematical structures.
Here is an important example. ( A basic example of this is when is the plane
and
is the equivalence relation where
if and only if
and
are integers. The equivalence classes are translates of
That is, they are grids of the form
Let us write
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
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 into a group? Well, given two grids,
and
we pick the two obvious representatives
and
(or rather, obvious when the grids are presented to us like this), add them to obtain
and take the equivalence class of the result to obtain the grid
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 is the same as a grid such as
so we need to be sure that if we had chosen
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
the final answer is unaffected, which is easily seen to be the case.
How about making 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 and
is the shortest distance between any point of
and any point of
This has the nice property that the quotient map , that is, the map that takes
to the equivalence class of
, is continuous, and indeed for small distances is an isometry. To be precise about the second assertion, if the distance between
and
is less than
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 The proof depends on the fact that if
and
are two translates of
and you choose
and
that have the shortest possible distance, then
and
will also have that distance for any pair of integers
Can we make into a vector space over
? 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
and
is a real number, it is not necessarily the case that
and indeed they are usually not equivalent. (For example,
but
) Therefore, there is no natural way of defining scalar multiples of translates of
(or rather, there is, but the result is not a translate of
unless the scalar is
).
Finally, we mention briefly that the Riemann-surface structure of does make
into a Riemann surface. This turns out to be a very important construction, especially when one applies it not just to the quotient by
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 with given properties, the best approach is to start with a simpler object
choose a property
and prove that the set
of all
such that
holds is an object with the required properties.
For example, a convenient way of defining a sphere is to take the set of all points in
such that
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Spaces of functions
If and
are vector spaces, then
the set of all linear maps from
to
can also be made into a vector space very easily. Indeed, if
and
are linear maps from
to
then we define
to be
and we define
to be
If is a group and
is a set, then the set
of all functions from
to
is a group under the operation
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.
Tensor products
Limits
Gluing
Ultraproducts
For now, take a look at the article on how to use ultrafilters.