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 …

We present a cubical type theory based on the Cartesian cube category (faces,
degeneracies, symmetries, diagonals, but no connections or reversal) with univalent …

We define a computational type theory combining the contentful equality structure of
cartesian cubical type theory with internal parametricity primitives. The combined theory …

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 …

We present a domain-specific type theory for constructions and proofs in category theory.
The type theory axiomatizes notions of category, functor, profunctor and a generalized form …

arXiv:2105.01724v6 [math.CT] 12 Aug 2022 Page 1 Synthetic fibered (∞,1)-category theory Ulrik
Buchholtza and Jonathan Weinbergerb aFunctional Programming Lab, School of Computer …

One idiosyncratic framing of type theory is as the study of operations invariant under
substitution. Modal type theory, by contrast, concerns the controlled integration of operations …

Formalized 1-category theory forms a core component of various libraries of mathematical
proofs. However, more sophisticated results in fields from algebraic topology to theoretical …

We show that the extension types occurring in Riehl--Shulman's work on synthetic $(\infty, 1)
$-categories can be interpreted in the intended semantics in a way so that they are strictly …