Pullback (category theory): Difference between revisions

Revision as of 03:11, 23 April 2005

Revision as of 20:22, 10 August 2005

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:''P'' = ''X'' ×_''Z'' ''Y''.

The notation comes from the following example. In the [[category of sets]] the pullback of ''f'' and ''g'' is the set

:''X'' ×_''Z'' ''Y'' = {(x, y) ∈ ''X'' × ''Y'' | ''f''(''x'') = ''g''(''y'')},

together with the restrictions of the [[projection map]]s <math>\pi_1</math> and <math>\pi_2</math> to ''X'' &times;_''Z'' ''Y'' .

The maps ''p''₁ and ''p''₂ are just the projections onto the first and second factors.

This example motivates another way of characterizing the pullback: as the [[equalizer]] of the morphisms ''f'' <small>Oo</small> ''p''₁, ''g'' <small>Oo</small> ''p''₂ : ''X'' &times; ''Y'' &rarr; ''Z'' where ''X'' &times; ''Y'' is the [[product (category theory)|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.

The [[dual (category theory)|categorical dual]] of a pullback is a called a '''[[pushout (category theory)|pushout]]'''.