How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Solution 1:

After more intense search, I've found the answer from this mathoverflow and math.se questions.

The relation of the subject statement to the axiom of choice seems poor. So, the subject statement is consistent with $\sf ZF + \neg\sf AC$, as in final remark in this paper. More, its negation also may hold in models of $\sf ZF$ without choice, as in this paper!

For reference, permutation models of $\sf ZF$ with atoms (or urelements) used in the above papers are described in book "The Axiom of Choice" by Thomas J. Jech.

I hope there may be another interesting contributions to the question.