PULLBACK (CATEGORY THEORY)

In category theory, a branch of mathematics, a 'pullback' (also called a 'fibered product' or 'Cartesian square') is the limit of a diagram consisting of two morphisms ''f'' : ''X'' → ''Z'' and ''g'' : ''Y'' → ''Z'' with a common codomain. The pullback is often written
:''P'' = ''X'' ×_''Z'' ''Y''.

Contents

Universal property

Weak pullbacks

Examples

Properties

See also

References

Universal property

Explicitly, the pullback of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''p''₁ : ''P'' → ''X'' and ''p''₂ : ''P'' → ''Y'' for which the diagram

commutes. Moreover, the pullback (''P'', ''p''₁, ''p''₂) must be universal with respect to this diagram. That is, for any other such set (''Q'', ''q''₁, ''q''₂) there must exist a unique ''u'' : ''Q'' → ''P'' making the following diagram commute:

As with all universal constructions, the pullback, if it exists, is unique up to a unique isomorphism.

Weak pullbacks

A pullback (''P'',''p''₁, ''p''₂) is called 'weak' if it is not universal, i.e. for any such ''Q'', ''u'' as given above is not unique.

Examples

In the category of sets the pullback of ''f'' and ''g'' is the set: $X\; imes\_Z\; Y\; =\; \{(x,\; y)\; in:X\; imes\; Y|\; f(x)\; =\; g(y)\}$ , together with the restrictions of the projection maps $pi\_1$ and $pi\_2$ to ''X'' ×_''Z'' ''Y'' .

★ This example motivates another way of characterizing the pullback: as the equalizer of the morphisms ''f'' o ''p''₁, ''g'' o ''p''₂ : ''X'' × ''Y'' → ''Z'' where ''X'' × ''Y'' is the binary product of ''X'' and ''Y'' and ''p''_1,2 are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with a terminal object, binary products and equalizers.
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : ''E'' → ''B'' and a continuous map ''f'' : ''X'' → ''B'', the pullback ''X'' ×_''B'' ''E'' is a fiber bundle over ''X'' called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
In any category with a terminal object ''Z'', the pullback ''X'' ×_''Z'' ''Y'' is just the ordinary product ''X'' × ''Y''.

Properties

★ Whenever ''X''×_''Z''''Y'' exists, then so does ''Y''×_''Z''''X'' and there is an isomorphism ''X''×_''Z''''Y'' $cong$ ''Y''×_''Z''''X''.

★ Monomorphisms are stable under pullback: if the arrow ''f'' above is monic, then so is the arrow ''p''₂. For example, in the category of sets, if ''X'' is a subset of ''Z'', then, for any ''g'':''Y'' → ''Z'', the pullback ''X''×_''Z''''Y'' is the inverse image of ''X'' under ''g''.

★ Isomorphisms are also stable, and hence, for example, ''X''×_''X''''Y'' $cong$''Y'' for any map ''Y'' → ''X''.

See also

★ The categorical dual of a pullback is a called a 'pushout'.

★ Pullbacks in differential geometry

★ Equijoin in relational algebra.

References

★ Paul M.Cohen, ''Universal Algebra'' (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 ''(Originally published in 1965, by Harper & Row)''.