So we were given the homework of formulating and proving the Fundamental Theorem of Homomorphisms for G-Sets, which factors equivariance maps. (this is not the homework question though, but a follow up).
So take two G-Sets $M$,$N$ and an equivariance map ($\varphi:N\rightarrow M$ with $\varphi(gn)=g\varphi(n)$). It's pretty straightforward to do this with the equivalence classes which arise from the relation $m\sim m':\Leftrightarrow \varphi(m)=\varphi(m')$.
But as this is not a very new result and doesn't use equivariance anywhere, I was wondering if those equivalence classes might have a more straightforward representation in Group theory (maybe, if we assume both $M$ and $N$ to be transitive?).
So my question is basically, if the sets of the form $\{m\in M\vert\varphi(m)=n\}$ for some $n\in N$ have any significance.
As all the stabilizers of all m in any of those sets have to be contained in the stabilizer of $n$ (if $gm=m$ and $\varphi(m)=n$, then $n=\varphi(m)=\varphi(gm)=g\varphi(m)=gn$) my first thought was to take all elements of m with the same stabilizer as n, but as elements don't necessarily are equal, just because their stabilizers are equal, this does not work in general. Is there any significance to these sets/can they be represented in a nice group theoretic way?