limits (pointwise). Each object X of (C gives rise to the represent-
able functor h
, 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-
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
given object hv of 5 . Moreover, each morphism X -* - X' in (C
defines a natural transformation h
-* • 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-
phisms in (C ). This functor is called the Yoneda embedding. One has the
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
-* F in
S , and elements of the set F(X) .
The above facts are sometimes rephrased as:
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