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.”
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.
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
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.
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.