Category:Definitions/Set Products

This category contains definitions related to Set Products.
Related results can be found in Category:Set Products.

Let $S$ and $T$ be sets.

Let $P$ be a set.

Let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that:

For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that:

$\phi_1 \circ h = f_1$

$\phi_2 \circ h = f_2$

that is, such that:

$\begin{xy}\xymatrix@+1em@L+3px{ & X \ar[ld]_*+{f_1} \ar@{-->}[d]^*+{h} \ar[rd]^*+{f_2} \\ S & P \ar[l]^*+{\phi_1} \ar[r]_*+{\phi_2} & T }\end{xy}$

is a commutative diagram.

Then the ordered triple $\struct {P, \phi_1, \phi_2}$ is called a product of $S$ and $T$.