# 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)
How to Cite This Entry:
Cartesian square. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian_square&oldid=44389
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article