Fibre product of objects in a category
A special case of the concept of an (inverse or projective) limit. Let be a category and let
and
be given morphisms in
. An object
, together with morphisms
,
, is called a fibre product of the objects
and
(over
and
) if
, and if for any pair of morphisms
,
for which
there exists a unique morphism
such that
,
. The commutative square
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is often called a universal or Cartesian square. The object , together with the morphisms
and
, is a limit of the diagram
![]() |
The fibre product of and
over
and
is written as
![]() |
If it exists, the fibre product is uniquely defined up to an isomorphism.
In a category with finite products and kernels of pairs of morphisms, the fibre product of and
over
and
is formed as follows. Let
be the product of
and
with projections
and
and let
be the kernel of the pair of morphisms
. Then
, together with the morphisms
and
, is a fibre product of
and
over
and
. In many categories of structured sets,
is the subset of
consisting of all those pairs
for which
.
Comments
In the literature on category theory, fibre products are most commonly called pullbacks, and examples of the dual notion (i.e. fibre products in the opposite of the category under consideration) are called pushouts. The name "fibre product" derives from the fact that, in the category of sets (and hence, in any concrete category whose underlying-set functor preserves pullbacks), the fibre of over an element
(i.e. the inverse image of
under the mapping
) is the Cartesian product of the fibres
and
. Note also that (binary) products (cf. Product of a family of objects in a category) are a special case of pullbacks, in which the object
is taken to be a final object of the category.
References
[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |
[a2] | J. Adámek, "Theory of mathematical structures" , Reidel (1983) |
Fibre product of objects in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibre_product_of_objects_in_a_category&oldid=30737