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

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)