Directed type theory, with a twist

Abstract: In recent years, Homotopy Type Theory (HoTT) has had great success both as a foundation of mathematics and as internal language to reason about $\infty$-groupoids (a.k.a. spaces). However, in many areas of mathematics and computer science, it is often the case that it is categories, not groupoids, which are the more important structures to consider. For this reason, multiple directed type theories have been proposed, i.e., theories whose semantics are based on categories. In this paper, we present a new such type theory, Twisted Type Theory (TTT). It features a novel ``twisting'' operation on types: given a type that depends both contravariantly and covariantly on some variables, its twist is a new type that depends only covariantly on the same variables. To provide the semantics of this operation, we introduce the notion of dependent 2-sided fibrations (D2SFibs), which generalize Street's notion of 2-sided fibrations. We develop the basic theory of D2SFibs, as well as characterize them through a straightening-unstraightening theorem. With these results in hand, we introduce a new elimination rule for Hom-types. We argue that our syntax and semantics satisfy key features that allow reasoning in a HoTT-like style, which allows us to mimic the proof techniques of that setting. We end the paper by exemplifying this, and use TTT to reason about categories, giving a syntactic proof of Yoneda's lemma.