To start our journey, we discuss the basic concept of

representation from a formal point of view. This is the

key concept underlying the number-theoretic methods of

Galois representations that are our goal. To ﬂesh out

the abstract formalism, we go through an example: The

ordinary act of counting can be viewed as a representation

of sets. So we give (or review) mathematical deﬁnitions

of sets, functions, morphisms,andrepresentations,which

will be with us for the whole book.

The Bare Notion of Representation

Before we narrow our focus to mathematical concepts, we start by

discussing the general concept of a representation. In philosophy,

the concept of one thing representing or misrepresenting another

thing is a central concern. The distinction between truth and

appearance, the thing-in-itself and its representation, is a keynote

of philosophy. It plays a critical role in the works of such ﬁgures

as Plato, Kant, Schopenhauer, and Nietzsche. Generally speaking,

for these philosophers the “appearance” of something is thought

to be an impediment or veil, which we wish to penetrate through

to the reality acting behind it. But in mathematics, matters stand

somewhat differently.

Consider, in an abstract way, the relationship that occurs when

one thing represents another. Say B represents A.Wehavethree

4 CHAPTER 1

terms that stand together in some kind of relationship: A, B,and

the fact that B represents A. We can call this fact X.Itisimportant

to remember that, in a representation, the three terms A, B,andX

are usually distinct.

For example, A may be a citizen of Massachusetts, B her state

representative, and X the legal fact that B represents A by voting

in the legislature on her behalf. Or, to jump ahead, A may be an

abstract group, B a group of matrices, and X a morphism from A to

B. (We will deﬁne these terms later.)

It can happen, though, that A = B. For instance, B may be said to

(also) represent herself in the state legislature. Or A may be a group

of matrices and B the same group of matrices. But whether A = B

or A = B, we call these relationships “representations.”

1

Note that

the fact of representation, X, is always going to be different from A

and B,becauseA and B are objects and X is a fact of representation.

Now, what would be a good picture of A, B,andX?WecanviewX

as an arrow going from A to B. This captures the one-way quality of

the relationship, showing that B is representing A, not vice versa:

A −→ B.

2

We can abstract even further, if we do not want to name A and

B and we just want to visualize their relationship. We can picture

them with dots. Then the picture of a representation becomes

•−→•

which is the ultimate in abstraction. The dots are just placeholders

for the names of the objects. The two dots can stand for two different

objects or the same object. The dot or object from which the arrow

emanates is called the source of that arrow, and the dot or object to

which the arrow goes is called the target of that arrow.

In normal life, if A represents B, B and A can be very different

kinds of things. For instance, a ﬂag can represent a country, a

1

It may not seem to make sense for an object to represent itself, or it may seem like

the best, most exact possible representation. Mathematicians do not take sides in this

debate. We just agree to call it a representation even when A represents itself.

2

It could happen that, at the same time, A also represents B, and we would picture

that as B −→ A. But this is a different representation from the previous one. Its “fact of

representation” Y is not equal to X.

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