Suppose I have $n+1$ sets of objects, $T_0,T_1,\dots,T_n$ and that I have $n$ mappings, $\alpha_1,\dots,\alpha_n$ such that $\alpha_i(T_j) \subseteq T_{j+1}$ for all $i=1,\dots,n$ and $j=0,\dots,n-1.$
What extra conditions are required for the following to hold: $T_n = (\alpha_1+\alpha_2+\dots + \alpha_n)T_{n-1} - (\alpha_1 \alpha_2 + \alpha_1 \alpha_3 + \dots + \alpha_{n-1}\alpha_n)T_{n-2} \cdots +(-1)^{n-1} (\alpha_1 \alpha_2 \cdots \alpha_n) T_0$ interpreted such that the objects on the left hand side appears exactly one time on the right hand side after cancellation.
Some natural conditions would be some or all of the following:
(1) $\alpha_i(\alpha_j(T_k)) = \alpha_j(\alpha_i(T_k)) =: \alpha_i \alpha_j T_k.$
(2) $T_n = (\alpha_1+\alpha_2+\dots + \alpha_n)T_{n-1}$ AS SETS, (but some objects appear multiple times on the right hand side).
(3) $t_1 \in T_k$, $\alpha_j(t_1) = \alpha_i(t_1) \Rightarrow i=j.$
(4) $t_1,t_2 \in T_k$, $\alpha_j(t_1) = \alpha_j(t_2) \Rightarrow t_1=t_2.$
(5) $t_1,t_2 \in T_k$, $\alpha_i(t_1) = \alpha_j(t_2) \Rightarrow \exists! t \in T_{k-1}: \alpha_i \alpha_j(t) = \alpha_i(t_1) = \alpha_j(t_2).$
All of the above holds for regular inclusion/exclusion where the $T_i$ are the subsets of $n$ of size $i,$ and $\alpha_i$ is interpreted as adding element $i.$