Editing Pullback (category theory) (section)

==Properties==
*In any category with a [[terminal object]] {{mvar|T}}, the pullback {{math|''X''&nbsp;×<sub>''T''</sub>&nbsp;''Y''}} is just the ordinary [[product (category theory)|product]] {{math|''X''&nbsp;×&nbsp;''Y''}}.<ref>Adámek, p. 197.</ref>
*[[Monomorphism]]s are stable under pullback: if the arrow {{mvar|f}} in the diagram is monic, then so is the arrow {{math|''p''<sub>2</sub>}}. Similarly,  if {{mvar|g}} is monic, then so is {{math|''p''<sub>1</sub>}}.<ref>Mitchell, p. 9</ref>
*[[Isomorphism]]s are also stable, and hence, for example, {{math|''X''&nbsp;×<sub>''X''</sub>&nbsp;''Y'' ≅ ''Y''}} for any map {{math|''Y''&nbsp;→&nbsp;''X''}} (where the implied map {{math|''X''&nbsp;→&nbsp;''X''}} is the identity).
* In an [[abelian category]] all pullbacks exist,<ref>Mitchell, p. 32</ref> and they preserve [[kernel (category theory)|kernels]], in the following sense: if
::[[File:Categorical pullback.svg|125px|class=skin-invert]]
::
:is a pullback diagram, then the induced morphism {{math|ker(''p''<sub>2</sub>)&nbsp;→&nbsp;ker(''f'')}} is an isomorphism,<ref>Mitchell, p. 15</ref> and so is the induced morphism {{math|ker(''p''<sub>1</sub>)&nbsp;→&nbsp;ker(''g'')}}. Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are [[exact sequence|exact]]:
<div class="center"><math>
\begin{array}{ccccccc} 
&&&&0&&0\\
&&&&\downarrow&&\downarrow\\
&&&&L&=&L\\
&&&&\downarrow&&\downarrow\\
0&\rightarrow&K&\rightarrow&P&\rightarrow&Y \\
&&\parallel&&\downarrow&  & \downarrow\\
0&\rightarrow&K&\rightarrow&X&\rightarrow&Z
\end{array}
</math></div>
:Furthermore, in an abelian category, if {{math|''X''&nbsp;→&nbsp;''Z''}} is an epimorphism, then so is its pullback {{math|''P''&nbsp;→&nbsp;''Y''}}, and symmetrically: if {{math|''Y''&nbsp;→&nbsp;''Z''}} is an epimorphism, then so is its pullback {{math|''P''&nbsp;→&nbsp;''X''}}.<ref>Mitchell, p. 34</ref> In these situations, the pullback square is also a pushout square.<ref>Mitchell, p. 39</ref>

*There is a natural isomorphism (''A''×<sub>''C''</sub>''B'')×<sub>''B''</sub> ''D'' &cong; ''A''×<sub>''C''</sub>''D''. Explicitly, this means: 
** if maps ''f'' : ''A'' &rarr; ''C'', ''g'' : ''B'' &rarr; ''C'' and ''h'' : ''D'' &rarr; ''B'' are given and 
** the pullback of ''f'' and ''g'' is given by ''r'' : ''P'' &rarr; ''A'' and ''s'' : ''P'' &rarr; ''B'', and
** the pullback of ''s'' and ''h'' is given by  ''t'' : ''Q'' &rarr; ''P'' and ''u'' : ''Q'' &rarr; ''D'' , 
** then the pullback of ''f'' and ''gh'' is given by ''rt'' : ''Q'' &rarr; ''A'' and  ''u'' : ''Q'' &rarr; ''D''. 
:Graphically this means that two pullback squares, placed side by side and sharing one morphism, form a larger pullback square when ignoring the inner shared morphism.
<div class="center"><math>
\begin{array}{ccccc} 
Q&\xrightarrow{t}&P& \xrightarrow{r} & A \\
\downarrow_{u} &  & \downarrow_{s} & &\downarrow_{f}\\
D & \xrightarrow{h} & B &\xrightarrow{g} & C
\end{array}
</math></div>

* Any category with pullbacks and products has equalizers.