My dynamical systems understanding is somewhat basic, so I wanted to ask the possible following question.
Consider a two sided shift over a finite alphabet $\mathcal{A}$, denoted $\mathcal{A}^{\mathbb{Z}}$, and the shift map $T:\mathcal{A}^\mathbb{Z}\to \mathcal{A}^\mathbb{Z}$. Let $\Omega_1\supseteq \Omega_2$ be two closed shift-invariant subsets, also known as subshifts. I would want to think that $\Omega_2$ is a factor of $\Omega_1$. For that, I formally need a surjective map commuting with the restricted shift action on $\mathcal{A}^{\mathbb{Z}^2}$. Is there naturally such a map?
For example if there is some $\omega_0\in \Omega_1$ such that $T(\omega_0)=\omega_0$ then the map
$$ i(\omega):= \begin{cases} \omega_0 &; \omega\notin \Omega_2\\ \omega & ;\omega\in \Omega_2 \end{cases}, $$
should be a factor map. But obviously there is no need for such an $\omega_0$ to be in $\Omega_1$. Is there perhaps a counter example to the statement saying inclusion between subshifts implies one is a factor of the other?
I would also apperciate if someone can try to give a short description of the factor relation between subsets of the same dynamical systems.