What does "$\times_i$" mean in function notation? As in "$\phi:\times_iM_i^\phi\to A$"

Question

What does the $\times_i$ mean? For reference, $M_i$ is a set.

Answer 1

Direct product. Think $M_1 \times M_2 \times \dots \times M_n$. The indexing set does not necessarily need to be finite. The more common notation for this is a capital pi: $$ \prod_{i \in I} M_i. $$

If the indexing set is understood from the context, e.g. $I = \{1,\dots,n\}$ is a common indexing set, then the "$\in I$" part of the notation may be dropped. We might also write

$$ \prod_{i = 1}^n M_i. $$

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If you haven't seen direct products before, they're just tuples. For instance $\mathbb{R} \times \mathbb{R} \times \mathbb{C}$ is the set of all tuples $(x,y,z)$ where $x$ and $y$ are real and $z$ is complex.

For example, a function $f : \mathbb{R} \times \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ would be declared in a programming language like

double f(double x, int y, int z);