A Type Theory for Comprehension Categories with Applications to Subtyping

Abstract: In this paper we develop a type theory that we show is an internal language for comprehension categories. This type theory is closely related to Martin-L\"of type theory (MLTT). Indeed, semantics of MLTT are often given in comprehension categories, albeit usually only in discrete or full ones. As we explain, requiring a comprehension category to be full or discrete can be understood as removing one `dimension' of morphisms. Thus, in our syntax, we recover this extra dimension. We show that this extra dimension can be used to encode subtyping in a natural way. Important instances of non-full comprehension categories include ones used for constructive or univalent intensional models of MLTT and directed type theory, and so our syntax is a more faithful internal language for these than is MLTT.