According to my understanding in 2023, there is no specific statement by physicist Edward Witten suggesting that a universe must have higher-dimensional categories to support itself as the foundation of mathematics. Edward Witten is a prominent theoretical physicist who has made significant contributions to various areas of physics, particularly in the field of string theory.
Witten's work has primarily focused on theoretical physics and its mathematical aspects, particularly in relation to string theory, supersymmetry, and quantum field theory. While he has explored connections between physics and mathematics, including the role of higher-dimensional concepts in string theory, I am not aware of any explicit claims by Witten regarding the necessity of higher-dimensional categories for a universe to support itself as the foundation of mathematics.
It's important to note that scientific and mathematical ideas can evolve over time, and new insights may have emerged since my last knowledge update. Therefore, it's always a good idea to consult the most recent research and publications to obtain the most up-to-date understanding on a specific topic.