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''co-universal square, pull-back square, in a category''
 
''co-universal square, pull-back square, in a category''
  
 
The diagram
 
The diagram
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206401.png" /></td> </tr></table>
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$$
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\begin{array}{ccl}
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A \prod _ {S} B  &\  \mathop \rightarrow \limits ^ { {p _ A}}  \  & A  \\
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p _ {B} \downarrow \  &{}  &\downarrow \  \alpha  \\
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B  &\  \mathop \rightarrow \limits _  \beta  \  &S . \\
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\end{array}
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$$
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Here  $  A \prod _ {S} B $(
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the notation  $  A \times _ {S} B $
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is also used) is the fibred product of the objects  $  A $
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and  $  B $,
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which is associated with
 +
 
 +
$$
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\begin{array}{l}
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{} \\
 +
{} \\
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B
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\end{array}
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\  \begin{array}{l}
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{} \\
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{} \\
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  \mathop \rightarrow \limits _  \beta 
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\end{array}
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\  \begin{array}{l}
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A \\
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\downarrow \alpha \\
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S ,
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\end{array}
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 +
$$
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 +
and  $  p _ {A} $
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and  $  p _ {B} $
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are the canonical projections. The diagram
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206402.png" /> (the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206403.png" /> is also used) is the fibred product of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206405.png" />, which is associated with
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$$
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\begin{array}{r}
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P \\
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\gamma \downarrow \\
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B
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\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206406.png" /></td> </tr></table>
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\  \begin{array}{l}
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\mathop \rightarrow \limits ^  \delta  \\
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{} \\
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  \mathop \rightarrow \limits _  \beta 
 +
\end{array}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206408.png" /> are the canonical projections. The diagram
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\  \begin{array}{l}
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A \\
 +
\downarrow \alpha \\
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S
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c0206409.png" /></td> </tr></table>
+
$$
  
is a Cartesian square if and only if it is commutative and if for any pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064012.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064013.png" /> which satisfies the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020640/c02064015.png" />.
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is a Cartesian square if and only if it is commutative and if for any pair of morphisms $  \mu : \  V \rightarrow A $,  
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$  \nu : \  V \rightarrow B $
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such that $  \alpha \mu = \beta \nu $
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there exists a unique morphism $  \lambda : \  V \rightarrow P $
 +
which satisfies the conditions $  \mu = \delta \lambda $,  
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$  \nu = \gamma \lambda $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR></table>

Latest revision as of 11:42, 8 February 2020


co-universal square, pull-back square, in a category

The diagram

$$ \begin{array}{ccl} A \prod _ {S} B &\ \mathop \rightarrow \limits ^ { {p _ A}} \ & A \\ p _ {B} \downarrow \ &{} &\downarrow \ \alpha \\ B &\ \mathop \rightarrow \limits _ \beta \ &S . \\ \end{array} $$

Here $ A \prod _ {S} B $( the notation $ A \times _ {S} B $ is also used) is the fibred product of the objects $ A $ and $ B $, which is associated with

$$ \begin{array}{l} {} \\ {} \\ B \end{array} \ \begin{array}{l} {} \\ {} \\ \mathop \rightarrow \limits _ \beta \end{array} \ \begin{array}{l} A \\ \downarrow \alpha \\ S , \end{array} $$

and $ p _ {A} $ and $ p _ {B} $ are the canonical projections. The diagram

$$ \begin{array}{r} P \\ \gamma \downarrow \\ B \end{array} \ \begin{array}{l} \mathop \rightarrow \limits ^ \delta \\ {} \\ \mathop \rightarrow \limits _ \beta \end{array} \ \begin{array}{l} A \\ \downarrow \alpha \\ S \end{array} $$

is a Cartesian square if and only if it is commutative and if for any pair of morphisms $ \mu : \ V \rightarrow A $, $ \nu : \ V \rightarrow B $ such that $ \alpha \mu = \beta \nu $ there exists a unique morphism $ \lambda : \ V \rightarrow P $ which satisfies the conditions $ \mu = \delta \lambda $, $ \nu = \gamma \lambda $.

References

[1] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
How to Cite This Entry:
Cartesian square. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian_square&oldid=13918
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article