Fascinating links between the semantics of concurrent programs and algebraic topology
have been discovered and developed since the 1990s, motivated by the hope that each field …

Directed type theory is an analogue of homotopy type theory where types represent
categories, generalizing groupoids. A bisimplicial approach to directed type theory …

Reasoning about weak higher categorical structures constitutes a challenging task, even to
the experts. One principal reason is that the language of set theory is not invariant under the …

In this paper, we present a directed homotopy type theory for reasoning synthetically about
(higher) categories and directed homotopy theory. We specify a new'homomorphism'type …

Polymorphic type systems such as System F enjoy the parametricity property: polymorphic
functions cannot inspect their type argument and will therefore apply the same algorithm to …

Dependent type theory allows us to write programs and to prove properties about those
programs in the same language. However, some properties do not require much proof, as …

Programming with dependent types is a blessing and a curse. It is a blessing to be able to
bake invariants into the definition of datatypes: we can finally write correct-by-construction …

We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter
prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial …

Within the framework of Riehl–Shulman's synthetic (∞, 1)-category theory, we present a
theory of two-sided cartesian fibrations. Central results are several characterizations of the …

We give structural results about bifibrations of (internal)(∞, 1)-categories with internal sums.
This includes a higher version of Moens' Theorem, characterizing cartesian bifibrations with …