Resolvent embeddings

Abstract

Many ex/reg completions $J:\mathcal C\to\mathcal C_{\rm ex/reg}$ that arise in categorical realizability and tripos theory admit left Kan extensions of arbitrary finitely continuous functors to arbitrary exact categories. This paper identifies the property which is responsible for these extensions: the functors are *resolvent*. Resolvency is characteristic of toposes that are ex/reg completions of regular categories with (weak) dependent products and generic monomorphisms. It also helps to characterize the toposes that the tripos-to-topos construction produces.