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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.”
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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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