If $J$ is a set and $T$ is a set of vectors, what might $T^J = \times_{k \in J}T$ symbolize?

Question

If $J$ is a set and $T$ is a set of vectors, I am wondering what the notation $T^J = \times_{k \in J}T$ might mean. Specifically, I am wondering what a cross product symbol might refer to in mathematical literature. Thanks.

Answer 1

For any sets $X$ and $Y$, the set $Y^X$ is the set of all functions $X\to Y$. The notation is motivated at least in part by the fact that $$|Y^X|=|Y|^{|X|}$$ for finite sets $X$ and $Y$. The set $Y^X$ can be identified with the cartesian product $\prod_{x\in X} Y$, with $f:X\to Y$ corresponding to $(f(x))_{x\in X}$. The cartesian product is sometimes written $\times_{x\in X}Y$.

Answer 2

A simple example is probably easiest to understand. Let $J = \{1,2,3,4\}$. Then

$$T^J = T \times T \times T \times T = \{(t_1,t_2,t_3,t_4) : t_k \in T \text{ for } k \in J\}.$$