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
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
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