Related papers: Higher geometric sheaf theories

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and…

A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…

In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved…

We develop foundations for oriented category theory, an extension of $(\infty,\infty)$-category theory obtained by systematic usage of the Gray tensor product, in order to study lax phenomena in higher category theory. As categorical…

We give a detailed and self-contained introduction to the theory of $\lambda $-toposes and prove the following: 1) A $\lambda $-separable $\lambda $-topos has enough $\lambda $-points. 2) The classifying $\lambda $-topos of a $\kappa $-site…

We define a covering of a profinite graph to be a projective limit of a system of covering maps of finite graphs. With this notion of covering, we develop a covering theory for profinite graphs which is in many ways analogous to the…

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0 they see the…

The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of…

We discuss a systematic procedure for categorifying presentable six-functor formalisms. Our main result produces, given the input of a representation of the $\infty$-category of correspondences of an $\infty$-category with finite limits…

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes…

We provide a functorial presentation of the $(\infty, 1)$-category of sheaves of $(n, r)$-categories for all $-2 \leq n\leq\infty$ and $0 \leq r\leq n+2$ based on complete Segal space objects. In this definition, the equivalences of sheaves…

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of…

In this paper we present a new semidefinite programming hierarchy for covering problems in compact metric spaces. Over the last years, these kind of hierarchies were developed primarily for geometric packing and for energy minimization…

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular…

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical…

We introduce a pro-cdh topology on formal schemes and prove that the $\infty$-topos of pro-cdh sheaves of spaces has an optimal bound of homotopy dimension. This remedies a defect for a pro-cdh topology on schemes introduced in [KS23]. As…

In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal…

The goal of this article is to develop the theory of presentable categories and topoi internal to an arbitrary $\infty$-topos $\mathcal{B}$. Our main results are internal analogues of Lurie's and Lurie-Simpson's characterisations of…