Name of a map coupled with the identity

Question

Given a map $\tau:\Omega\longrightarrow \mathbb{R}$, is there a standard name for the map $\tilde{\tau}:\Omega\longrightarrow\Omega\times\mathbb{R}$ that maps $\omega$ to $(\omega,\tau(\omega))$? It is somewhat useful in Probability, but I do not know of any standard terminology nor if there is some standard way to construct it besides just defining it as such. Thank you for any knowledge you might be able to share.

Answer 1

People call it the 'graph' map, because it is the natural map from $\Omega$ to the graph of $\tau$. It can be constructed using the universal property of products: consider the maps $id_\Omega: \Omega \to \Omega$ (the identity map) and $\tau: \Omega \to \mathbb{R}$. Then $\tilde{\tau}$ is the unique map $\Omega \to \Omega \times \mathbb{R}$ such that $p \circ \tilde{\tau} = id_\Omega$ and $q \circ \tilde{\tau} = \tau$, where $p: \Omega \times \mathbb{R} \to \Omega$ is the first projection and $q: \Omega \times \mathbb{R} \to \mathbb{R}$ is the second projection, satisfying the universal property of products (that is "terminal" with this property). It can be defined in any category with products.