Hilbert spaces admit no finitary discrete imaginaries

Abstract

We prove that every functor from the category of Hilbert spaces and linear isometric embeddings to the category of sets which preserves directed colimits must be essentially constant on all infinite-dimensional spaces. In other words, every finitary set-valued imaginary over the theory of Hilbert spaces, in a broad signature-independent sense, must be essentially trivial. This extends a result and answers a question by Lieberman--Rosick\'y--Vasey, who showed that no such functor on the supercategory of Hilbert spaces and injective linear contractions can be faithful.