# Functors and infinitary interpretations of structures

## Russell Miller

### City University of New York

It has long been recognized that the existence of an interpretation of one countable structure B in another one A yields a homomorphism from the automorphism group Aut(A) into Aut(B). Indeed, it yields a functor from the category Iso(A) of all isomorphic copies of A (under isomorphisms) into the category Iso(B). In traditional model theory, the converse is false. However, when we extend the concept of interpretation to allow interpretations by L_{ω1ω} formulas, we find that now the converse essentially holds: every Borel functor arises from an infinitary interpretation of B in A, and likewise every Borel-measurable homomorphism from Aut(A) into Aut(B) arises from such an interpretation. Moreover, the complexity of an interpretation matches the complexities of the corresponding functor and homomorphism. We will discuss the concepts and the forcing necessary to prove these results and other corollaries.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center. He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.