# Why does each component of the tuple that represents an action have a categorical distribution in the TRPO paper?

I was going through the TRPO paper, and there was a line under Appendix D "Approximating Factored Policies with Neural Networks" in the last paragraph which I am unable to understand

The action consists of a tuple $$(a_1, a_2..... , a_K)$$ of integers $$a_k\in\{1, 2,......,N_k\}$$ and each of these components is assumed to have a categorical distribution.

I can't seem to get how each component has a categorical distribution. I think it should be the tuple that has a categorical distribution. I think I am getting something wrong.

As an example, imagine a game in which you can both move and jump at the same time. In that case, you might have a 4-dimensional discrete action space for moving (NSWE), and a 2-dimensional discrete action space for jumping (yes/no jump), both of which will require a Categorical distribution which has the size of that factor ($$N_k$$ in the paper).