Jacob Lurie
In this paper we investigate an infinitely categorical analogue of the theory of Grothendieck topoi. In particular, we define infinity topoi and prove an analogue of Giraud's theorem, expressing the equivalence of ``intrinsic'' and ``extrinsic'' definitions. We also discuss the relationship between the theory of infinity topoi and classical topics in homotopy theory and dimension theory.
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Left-exact Localizations of $\infty$-Topoi II: Grothendieck Topologies
本文在任意 ∞-拓扑斯上引入扩展 Grothendieck 拓扑，证明其与多种局部化和拓扑概念构成的偏序集同构且具有 frame 结构，并提出 forcing 工具用于计算 ∞-拓扑斯的局部化。
Solution to Lawvere's first problem: a Grothendieck topos that has proper class many quotient topoi
本文通过构造自由单oid的投射上函子，解决Lawvere提出的关于存在具有proper class许多商上函子的Grothendieck上函子的问题。
Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor
本文提出了一种Grothendieck上同调的完全连通性定义，并展示了其在不同上同调中的应用与理论构建。
We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor, categorifying an analog 1-categorical result. We also describe further laxness conditions, giving some properties of the already known lax familial pseudofunctors. We also describe 2-dimensional analogs of orthogonality and factorization systems, and use them to construct examples of bistable pseudofunctors through inclusion of left objects and left maps. W…
Strict universes for Grothendieck topoi
该研究调整Shulman的论证，在任意Grothendieck拓扑斯中构造出满足重排性质的累积宇宙层级，实现带累积宇宙的Martin-Löf类型论的直接解释，并拓展了相关合成方法的适用范围。
We give necessary and sufficient conditions on a presentable infinity-category C so that families of objects of C form an infinity-topos. In particular, we prove a conjecture of Joyal that this is the case whenever C is stable.
Classifying Infinity Topoi via Weighted Limits
本文通过加权极限构造分类$\infty$-拓扑，并展示若干预堆具有分类拓扑，包括谱的预堆。
Gabriel-Ulmer duality for topoi and its relation with site presentations
该文研究正则基数下限制到κ-可展示Grothendieck拓扑斯时的Gabriel-Ulmer对偶，刻画了对应表示2-范畴的结构，分析其与场表示的关联并证明了相关反射子双范畴结论。
Reconstruction of schemes from their étale topoi
本文对任意特征的无限域证明了Grothendieck关于将 $k$-概形映射至其étale topos构成完全忠实函子的猜想，实现了概形的重构并推广了Voevodsky的工作。
We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the étale topos of a scheme, to its refinement …
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 description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived al…
From Torsors to Topoi: An Introduction with a View Toward $Σ$-Protocols in Cryptography
本文介绍sheaves和topoi，为密码学中Σ-协议的框架奠定基础，涵盖Grothendieck拓扑、sheaves、site上的torsors、descent、sheaf topoi等核心概念，帮助理解密码安全性的结构基础。
Lecture Notes on Modular Infinity Operads and Grothendieck-Teichmueller Theory
本讲义整理自2021年巴塞罗那CRM研讨会上的三场讲座，通过模无穷操作子在Grothendieck-Teichmueller理论中的应用，介绍该概念并阐释其研究动机。
We construct a Grothendieck-Witt space for any stable infinity category with duality. If we apply our construction to perfect complexes over a commutative ring in which 2 is invertible we recover the classical Grothendieck-Witt space. Our Grothendieck-Witt space is a grouplike E-infinity space which is part of a genuine C_2-spectrum, the connective real K-theory spectrum.
We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category of weak infinity groupoids, whose objects are all fibrant and which is in a precise sense "freely generated by an object". We show that all those semi model categories are Quillen equivalent together and Quillen to the model category of spaces. A general procedure is given to produce such coherator, and several explicit examples are presented: one…
The Inductive Coherator For Grothendieck Infinity Groupoids
本文推广单子的分配级数理论，构造两个可完备分配级数，分别诱导出归纳余生成器和严格∞-群胚理论，为Grothendieck∞-群胚提供归纳构造。
There appeared not long ago a Reduction Formula for derived Hochschild cohomology, that has been useful e.g., in the study of Gorenstein maps and of rigidity w.r.t. semidualizing complexes. The formula involves the relative dualizing complex of a ring homomorphism, so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via a number of noncanonical projective and injective resolutions of differential graded objects. Recent efforts aim at more intrinsic approaches, hopefully upgradable to "higher" contexts--lik…
Brave new categorical spectral positive Schubert geometry and the categorical Dual Amplituhedron
本文通过谱代数几何和微分凝聚方法重写正格拉斯曼流形及其positroid簇的奇异粘合，构造了振幅多面体的对偶，为高能粒子散射振幅计算提供新数学框架。
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