Editing Pullback (category theory) (section)

{{Short description|Most general completion of a commutative square given two morphisms with same codomain}}
{{other|pullback}}
{{Redirect|Fiber product|the case of schemes|Fiber product of schemes}}

In [[category theory]], a branch of [[mathematics]], a '''pullback''' (also called a '''fiber product''', '''fibre product''', '''fibered product''' or '''Cartesian square''') is the [[limit (category theory)|limit]] of a [[diagram (category theory)|diagram]] consisting of two [[morphism]]s {{math|''f'' : ''X''&nbsp;→&nbsp;''Z''}} and {{math|''g''&nbsp;:&nbsp;''Y''&nbsp;→&nbsp;''Z''}} with a common codomain. The pullback is written

:{{math|''P'' {{=}} ''X'' ×<sub>''f'', ''Z'', ''g''</sub> ''Y''}}.

Usually the morphisms {{mvar|f}} and {{mvar|g}} are omitted from the notation, and then the pullback is written

:{{math|''P'' {{=}} ''X'' ×<sub>''Z''</sub> ''Y''}}.

The pullback comes equipped with two natural morphisms {{math|''P''&nbsp;→&nbsp;''X''}} and {{math|''P''&nbsp;→&nbsp;''Y''}}.  The pullback of two morphisms {{math|''f''}} and {{math|''g''}} need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, {{math|''X'' ×<sub>''Z''</sub> ''Y''}} may intuitively be thought of as consisting of pairs of elements {{math|(''x'', ''y'')}} with {{math|''x''}} in {{math|''X''}}, {{math|''y''}} in {{math|''Y''}}, and {{math|''f''(''x'')&nbsp; {{=}} &nbsp;''g''(''y'')}}. For the general definition, a [[universal property]] is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a [[commutative diagram|commutative square]].

The [[Dual (category theory)|dual concept]] of the pullback is the ''[[Pushout (category theory)|pushout]]''.