相关论文: 2-Cartesian fibrations II: A Grothendieck construc…
We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $\mathcal{B}$, and prove that they encode the four variance flavors of $\mathcal{B}$-indexed diagrams of $\infty$-categories. We then use this…
We produce an indexed version of the Grothendieck construction. This gives an equivalence of categories between opfibrations over a fixed base in the 2-category of 2-copresheaves and 2-copresheaves on the Grothendieck construction of the…
We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction for categorical $\mathbb{O}$-operads, $\mathbb{O}$ being a unary operadic category. The construction is a fully faithful functor…
We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty…
Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction,…
Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided…
We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We make use of a specific instance of the Grothendieck construction to provide, for every bicategory equipped with a possible…
The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories of fractions allow one to invert a collection of arrows in a…
A standard result from the theory of Grothendieck fibrations states that if $p : E \to B$ is a fibration, then $E$ has limits of shape $\mathcal{J}$ if $B$ has limits of shape $\mathcal{J}$ the fibers of $\mathcal{E}$ have limits of shape…
In this work, we study the notion of cofinal functor of $\infty$-bicategories with respect to the theory of partially lax colimits. The main result of this paper is a characterization of cofinal functors of $\infty$-bicategories via…
We give a direct proof of the fact that Lurie's Unstraightening functor induces an equivalence between the strict $(\infty,2)$-category of indexed quasi-categories and the strict $(\infty,2)$-category of fibered quasi-categories over any…
The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of $(\infty,1)$-categories, namely quasi-categories. In this work we generalize this result by proving…
We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category $\mathcal{C}$. For the discrete case, we build a model structure on the slice…
This paper introduces and studies split two-sided 2-fibrations and locally discrete split two-sided 2-fibrations, using a formal categorical approach. We generalise Street's notion of split two-sided fibration internal to a 2-category to…
Within the framework of Riehl-Shulman's synthetic $(\infty,1)$-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition \`{a} la Chevalley, Gray,…
We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two…
Given an operad $\mathcal{O}$, we define a notion of weak $\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category $\mathsf{Cat}$ of…
Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very…
The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's…
This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate…