2

ANDREJ S£EDROV

limits (pointwise). Each object X of (C gives rise to the represent-

able functor h

x

, defined by hx(Y) = Hom^^X) = the set of all morphisms

Y • X in (C . On morphisms, h is given by composition. (We also men-

X X

tion the covariant representable functor h given by h (Y) = Horn (X,Y)).

In this way, to each object X of (E , we associate a uniquely

Eop

a

given object hv of 5 . Moreover, each morphism X -* - X' in (C

ha

defines a natural transformation h

x

-* • h , by composition. The reader

can check that this gives a functor (C -* • S , which is furthermore

k° t n full (meaning that each morphism hy -» • h.., in S comes from a

morphism X -» • X1 in (C ) and faithful (meaning that any two morphisms

ha • CEop a'

h h , in S are equal iff X X' are equal as mor-

h a

a

phisms in (C ). This functor is called the Yoneda embedding. One has the

following:

Cop

Yoneda Lemma. For objects X in CC , F in S , there is a 1-1 cor-

respondence (natural in both X and F) between morphisms h

x

-* F in

Eop

S , and elements of the set F(X) .

The above facts are sometimes rephrased as:

Eop

Lemma. Any object of S can be expressed as a colimit of a diagram

whose vertices are representable functors.

More detailed discussion is e.g. in [Fre 0] or in [ML],

0.3. Elementary Topoi.

A category E is called an elementary topos if:

(i) E has finite limits (it suffices to require pullbacks and

a terminal object).

(ii) E is cartesian closed, i.e. for any objects X,Y , there is

an object Y so that there is a 1-1 correspondence

Y

x

Z*X