Cartesian square
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) |
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