Question

For every product in a category, there exists an operation $\langle\cdot, \cdot\rangle$ that turns morphisms $f : C \to A$ and $g : C \to B$ into morphisms $\langle f, g\rangle : C \to A \times B$. How are this operation and its results usually called? Calling $\langle f, g\rangle$ the product of $f$ and $g$ does not make sense to me, since I would expect $f \times g$ to be the product of the two morphisms.

Furthermore there is the generalisation of binary products in the form of arbitrary products $\prod_{i \in I} A_i$ for families $\{A_i\}_{i \in I}$ of objects. These products have, of course, variants of the $\langle\cdot, \cdot\rangle$-operator. What is the usual notation for these variants? So far, I wrote $\langle\{f_i\}_{i \in I}\rangle : C \to \prod_{i \in I} A_i$ for families of morphisms $f_i : C \to A_i$. Is this standard?

Answer 1

Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the operation have called it the "pairing arrow."

I would write $\langle f_i\rangle_{i \in I} : C \to \prod_{i \in I} A_i$ without curly brackets in the angle brackets and i would call it the arrow to the product. A more specific name for the operation could be "tupling arrow."