Model theory of limits II

I will give an update on a proposed model theory for directed limits and colimits of first-order structures, originally motivated by applications to commutative algebra and the model theory of valued fields. To illustrate the usefulness of the formalism, I will prove a new general AKE theorem in mixed characteristic in a language with a cross-section of the value group and a lift of the residue field.

I will also discuss connections with other approaches to this topic, including pro- and ind-definable sets, infinitary logic, Feferman's local functors, accessible functors, and ultraproducts, some of which I have not discussed previously.