  This paper focuses on a mean-field optimal stopping problem with non-Markov
dynamics and common noise, inspired by Talbi, Touzi, and Zhang
\cite{TalbiTouziZhang1,TalbiTouziZhang3}. The goal is to establish the limit
theory and demonstrate the equivalence of the value functions between weak and
strong formulations. The difference between the strong and weak formulations
lies in the source of randomness determining the stopping time on a canonical
space. In the strong formulation, the randomness of the stopping time
originates from Brownian motions. In contrast, this may not necessarily be the
case in the weak formulation. Additionally, a $(H)$-Hypothesis-type condition
is introduced to guarantee the equivalence of the value functions. The limit
theory encompasses the convergence of the value functions and solutions of the
large population optimal stopping problem towards those of the mean-field
limit, and it shows that every solution of the mean field optimal stopping
problem can be approximated by solutions of the large population optimal
stopping problem.
