Ultra-products are a fundamental construction in Model theory. One of their applications is as a way to construct limiting objects. This allows identifying generic phenomena in a family of mathematical objects. A renowned example is The Ax-Kochen Theorem which states the isomorphism of the ultraproducts $\Prod^U Q_p = \Prod^U F_p((t))$ for along any non-principle ultrafilter U on the set of primes.

In the talk, I will present how to construct a homotopical version of ultraproduct - suitable to be used in modern algebraic topology and homological algebra. I will present "Ax-Kochen" type theorems in this realm.

This is joint work with Tobias Barthel and Nathaniel Stapleton.