Related papers: Lax colimits and free fibrations in $\infty$-categ…

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the…

In this work, we study oplax normalised functors of $(\infty,2)$-categories. Our main theorem is a comparison between the notion of oplax normalised functor of scaled simplicial sets due to Gagna-Harpaz-Lanari and the corresponding notion…

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The…

We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories…

We give a topological description of Ext groups between simple representations of categories via a nerve type construction. We use it to show that the Koszulity of indiscretely based category algebras is equivalent to the locally bouquet…

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…

We present some results on (co)limits of diagrams in $\infty$-categories, as well as those in $(n, 1)$-categories. In particular, we deduce a way to reshape colimit diagrams into simplicial ones, and a characterisations of $n$-cofinality…

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose…

We prove a universal property for $\infty$-categories of spans in the generality of Barwick's adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a…

We introduce the notion of a lax monoidal fibration and we show how it can be conveniently used to deal with various algebraic structures that play an important role in some definitions of the opetopic sets (Baez-Dolan,…

In this note we introduce a notion of free cofibrations of permutative categories. We show that each cofibration of permutative categories is a retract of a free cofibration.

We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the…

Lax extensions of set functors play a key role in various areas including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between…

In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the constructions these fibrations make available to end-users of higher…

We study lax epimorphisms in 2-categories, with special attention to $\mathsf{Cat}$ and $\mathcal{V}$-$\mathsf{Cat}$. We show that any 2-category with convenient colimits has an orthogonal $LaxEpi$-factorization system, and we give a…

We investigate the properties of lax comma categories over a base category $X$, focusing on topologicity, extensivity, cartesian closedness, and descent. We establish that the forgetful functor from $\mathsf{Cat}//X$ to $\mathsf{Cat}$ is…

We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural…

This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of…

We define the notion of an indexed profunctor over a 2-category, and use it to develop an abstract theory of limits. The theory subsumes (conical) limits, weighted limits, ends and Kan extensions. Results include an abstract version of the…

A notion of a coring extension is defined and it is related to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable…