Notation for Permuting Sets If I have some arbitrary sets $A_i : i \in I$ and I want to permute their intersections pairwise, how would I write such a permutation? Would I use some permutation tensor? Essentially I want to permute $A_i \cap A_k \, \, \forall i,k \in I$. How would I notate this formally?

You want $$\left\\{A_i\cap A_k:\\{i,k\\}\in[I]^2\right\\}\;.$$ The notation $[X]^\kappa$, where $X$ is any set and $\kappa$ is any cardinal number, is defined to be $$[X]^\kappa=\\{S\subseteq X:|S|=\kappa\\}\;,$$ the family of subsets of $X$ of cardinality $\kappa$. This is a standard notation, but it’s not universally familiar, so you’d probably want to define it first.