Many properties of morphisms can be restated in terms of points. For example, a map
f is said to be a
monomorphism if : For all maps
g,
h, if then . Suppose and in
C. Then
g and
h are
A-valued points of
B, and therefore monomorphism is equivalent to the more familiar statement :
f is a monomorphism if it is an
injective function on points of
B. Some care is necessary.
f is an
epimorphism if the
dual condition holds: : For all maps
g,
h (of some suitable type), implies . In set theory, the term "epimorphism" is synonymous with "
surjection", i.e. : Every point of
C is the
image, under
f, of some point of
B. This is clearly not the translation of the first statement into the language of points, and in fact these statements are
not equivalent in general. However, in some contexts, such as
abelian categories, "monomorphism" and "epimorphism" are backed by sufficiently strong conditions that in fact they do allow such a reinterpretation on points. Similarly,
categorical constructions such as the
product have pointed analogues. Recall that if
A,
B are two objects of
C, their product is an object such that : There exist maps , , and for any
T and maps , , there exists a unique map such that and . In this definition,
f and
g are
T-valued points of
A and
B, respectively, while
h is a
T-valued point of
A ×
B. An alternative definition of the product is therefore: :
A ×
B is an object of
C, together with projection maps and , such that
p and
q furnish a
bijection between points of and
pairs of points of
A and
B. This is the more familiar definition of the
product of two sets. == Geometric origin ==