7 papers · 1 filter
Day convolution for algebraic patterns
Thomas Blom, Félix Loubaton, Jaco Ruit
We characterize the exponentiable objects for a wide range of structures prevalent in -categorical algebra, extending the construction of Day convolution to more general st…
Formal category theory in -equipments I
Jaco Ruit
We generalize proarrow equipments from strict category theory to the -categorical setting, introducing the concept of -equipments. These are specific double $\infty…
On the squares functor and the Gaitsgory-Rozenblyum conjectures
Félix Loubaton, Jaco Ruit
In the seminal work of Gaitsgory and Rozenblyum on derived algebraic geometry, eight conjectures regarding the theory of -categories are stated. This paper aims to clar…
Homotopy coherent companionships and conjunctions
Jaco Ruit
We demonstrate that companionships and conjunctions in double -categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living compani…
A short proof of the straightening theorem
Fabian Hebestreit, Gijs Heuts, Jaco Ruit
We provide a short and reasonably self-contained proof of Lurie's straightening equivalence, relating cartesian fibrations over a given -category with contravariant fun…
Formal category theory in -equipments II: Lax functors, monoidality and fibrations
Jaco Ruit
We study the framework of -equipments which is designed to produce well-behaved theories for different generalizations of -categories in a synthetic and uniform fas…