A higher Grothendieck construction

Higher Equipments, Double Colimits and Homotopy Colimits

This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically meaningful. The most well-known two-fold structures are double categories, typically used to organize bimodules in various contexts. However there is no double category of spaces even though notions of bimodule are conceivable. We first remedy this defect of double category theory by constructing a meaningful simplicial category of spaces. Then we develop the a…

内部高阶范畴论中的余极限与余完备化

Colimits and cocompletions in internal higher category theory

本文发展了以 $\infty$-topos 为内部的范畴理论，讨论了内部范畴的伴随、极限、余极限与 Kan 扩张，证明了内部预层范畴的泛性质，并构造了由任意图形形状类索引的自由余完备化。

Grothendieck 构造的幺半闭性：基于 $Σ$-可处理幺半结构与 Dialectica 公式

Monoidal closure of Grothendieck constructions via $Σ$-tractable monoidal structures and Dialectica formulas

本文通过引入 $Σ$-可处理幺半结构，刻画 Grothendieck 构造的纤维极限与余极限，建立全范畴幺半闭的充分条件，推广 Gödel 的 Dialectica 解释，并讨论闭结构的纤维性。

Marked colimits and higher cofinality

Given a marked $\infty$-category $\mathcal{D}^{\dagger}$ (i.e. an $\infty$-category equipped with a specified collection of morphisms) and a functor $F: \mathcal{D} \to \mathbb{B}$ with values in an $\infty$-bicategory, we define $\operatorname{colim}^{\dagger} F$, the marked colimit of $F$. We provide a definition of weighted colimits in $\infty$-bicategories when the indexing diagram is an $\infty$-category and show that they can be computed in terms of marked colimits. In the maximally marked case $\mathcal{D}^{\sharp}$, our construction retrieves the $\infty$-categorical colimit of $F$ in …

二维切片中的余极限

Colimits in 2-dimensional slices

本研究在增强范畴论框架下将一维切片余极限理论推广到二维，定义lax切片与相关2-函子，证明其2-余极限的保持、反射与提升性质，并将基变换2-函子的余极限保持结果推广到带稠密生成子的有限完全2-范畴。

A Theory of Elementary Higher Toposes

We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, locality and classification of univalent morphisms, generalizing results by Lurie and Gepner-Kock. We also define $\infty$-logical functors and show the resulting $\infty$-category is closed under limits and filtered colimits, generalizing the analogous result for elementary toposes and Grothendieck $\infty$-toposes. Moreover, we give an alternative characterizati…

$(\infty,1)$-范畴的双范畴模型

Double categorical model of $(\infty,1)$-categories

基于已有工作，本文在双范畴上构造了四种模型结构，分别建模单纯空间、Segal空间、$(\infty,1)$-范畴和$\infty$-群胚，并利用Grothendieck构造给出了计算同伦余极限的显式公式。

加权余极限的高阶代数

The higher algebra of weighted colimits

本文在弱双充实∞-范畴框架下建立加权余极限理论，证明存在性、余端表达、自由余完备化等结果，并构造了多种高阶范畴的张量积结构。

有界LB空间与幂等伴随

Bornological LB-spaces and idempotent adjunctions

本文研究两种“有界LB空间”概念，指出其不同但密切相关，并深入探讨有界化和拓扑化函子的幂等伴随。

参数化高阶范畴论

Parametrized higher category theory

本文为参数化∞-范畴建立范畴论基础，提出索引同伦极限与余极限理论，并证明G-空间范畴由可缩G-空间自由生成，证实Hill猜想。

Adjoining colimits

This paper develops a theory of colimit sketches "with constructions" in higher category theory, formalising the input to the ubiquitous procedure of adjoining specified "constructible" colimits to a category such that specified "relation" colimits are enforced (or preserved). From a more technical standpoint, sketches are a way to describe dense functors using techniques from the homotopy theory of diagrams. We establish basic properties of diagrams in an infinity-category C as a model for presheaves on C and Bousfield localisations thereof, discuss extensions of functors and adjunctions, and…

Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several…

Euler Characteristics of Categories and Homotopy Colimits

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to highe…

稳定导子上的$t$-结构与Grothendieck心

$t$-Structures on stable derivators and Grothendieck hearts

本文证明了强稳定导子上的$t$-结构可典范提升至图表范畴，并利用紧生成性质推导出其心为Grothendieck范畴。

覆盖与格罗滕迪克拓扑斯

Coverages and Grothendieck Toposes

本文从覆盖角度系统阐述格罗滕迪克拓扑斯理论，证明饱和覆盖与格罗滕迪克拓扑的双射，并给出层化、站点态射、吉罗定理等核心结果的详细证明。

Higher categories, colimits, and the blob complex

We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when W=S^1 they coincide). The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold W. We outline the important properties of the blob complex, and sketch the pro…

Grothendieck ∞-群胚的归纳余生成器

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。

On the Galois Theory of Grothendieck

In this paper we deal with Grothendieck's interpretation of Artin's interpretation of Galois's Galois Theory (and its natural relation with the fundamental group and the theory of coverings) as he developed it in Expose V, section 4, ``Conditions axiomatiques d'une theorie de Galois'' in the SGA1 1960/61. This is a beautiful piece of mathematics very rich in categorical concepts, and goes much beyond the original Galois's scope (just as Galois went much further than the non resubility of the quintic equation). We show explicitly how Grothendieck's abstraction corresponds to Galois work. We int…

分类双刃剑

Categorical Ambidexterity

本文证明了对于允许一组余极限的∞-范畴的双刃剑结果，统一并扩展了可呈现∞-范畴的极限和余极限识别以及∞-范畴的∞-半可加性。

Parametrized higher category theory II: Universal constructions

We develop parametrized generalizations of a number of fundamental concepts in the theory of $\infty$-categories, including factorization systems, free fibrations, exponentiable fibrations, relative colimits and relative Kan extensions, filtered and sifted diagrams, and the universal constructions Ind and $P^Σ$.

同调类型理论中的余极限

Coslice Colimits in Homotopy Type Theory

本文研究了同调类型理论中余极限的性质，通过构造余切余极限，证明了其在树结构上的创建性，并探讨了其与正交因子系统和上同调理论的相互作用。

Enriched infinity categories I: enriched presheaves

This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.

拓扑斯的Gabriel-Ulmer对偶及其与场表示的关系

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。

在 Cat 中形式化余极限

Formalizing colimits in Cat

本文在 Lean 中形式化了范畴的 1-范畴的基本构造，构造神经嵌入的左伴随并证明该伴随是反射的，从而首次形式化证明 Cat 具有余极限且范畴的范畴是余完备的。

Monad的Eilenberg-Moore范畴上的微分挠理论

Differential torsion theories on Eilenberg-Moore categories of monads

本文证明了在正合且保持余极限的monad的Eilenberg-Moore范畴上，所有遗传挠理论均为微分的，且U-模上的δ-导子可唯一扩张至其商模。

高阶范畴论中的极限图重塑与共尾性

Reshaping limit diagrams and cofinality in higher category theory

该研究在高阶范畴论框架下，探讨∞-范畴与(n,1)-范畴中的（余）极限问题，提出余极限图的单纯形重塑方法、∞-范畴函子的n-共尾性刻画，并简要介绍n-筛选性基础内容。

Exponentiable Higher Toposes

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the $\infty$-category of $\mathcal C$-valued sheaves on $\mathcal X$ as an $\infty$-category of functors that satisfy finite limits conditions as well as filtered colimits conditions (instead of limits conditions purely); we call such functors $ω$-continuous sheaves. As an application, we show that when $\mathcal X$ is exponentiable, its $\infty$-category of stable sheaves $\m…

Higher dualizability and singly-generated Grothendieck categories

Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints $f_i$ from $\mathcal{C}$ to $\mathrm{Vect}_k$ are products of copies of $\mathrm{Vect}_k$. This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object $x$ with the property that every object is a copower of $x$: they are precisely the categories o…

曲面上的伽罗瓦作用与高亏及格罗滕迪克-泰希米勒群

Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group

本文通过构造群胚中的模操作子$\mathbf{S}$，建立了高亏格泰希米勒塔的操作子模型，并证明了$\widehat\Gamma$子群在$\widehat{\mathbf{S}}$上的忠实作用，从而给出了$\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$的作用。

Abstract cubical homotopy theory

Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to develop some aspects of a rich cubical calculus. Applied to the models in the background, this enhances the typical examples of triangulated and tensor-triangulated categories. The main players are the cardinality filtration of n-cubes, the induced interpolation between cocartesian and strongly cocartesian n-cubes, and the yoga of iterated cone construction…

Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience

We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as a potential yoga for dealing with processes and processes of processes.

Higher Topos Theory

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

$(\u221e,1)$-位点的概念及相关形式结构

Notions of $(\infty,1)$-sites and related formal structures

本研究围绕$(\u221e,1)$-位点展开，通过形式$(\u221e,2)$-范畴论定义高等Lawvere-Tierney算子，借超学说思路引入几何核概念，还推广了典范Grothendieck拓扑并证明相关比较引理的成立与失效情况，丰富了$\u221e$-拓扑斯理论。

可草图化无穷范畴

Sketchable infinity categories

本文将草图理论同伦相干地推广到$\infty$-范畴，证明了可展示$\infty$-范畴是极限草图模型，可及$\infty$-范畴是任意草图模型，并为多种$\infty$-范畴给出了具体草图。

Higher stacks as diagrams

Several possible presentations for the homotopy theory of (non-hypercomplete) $\infty$-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists, similar to Cisinski's presentation, based on work of Quillen, Thomason and Grothendieck, of usual homotopy theory by small categories and their smallest (basic) localizer. As an application it is shown that any (local) fibered (a.k.a. algebraic) derivator over S with stable fibers extends to $\infty$-stacks in a well-defined way under mild assumptions.

Higher Theories and Monads

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.

Grothendieck groups and Auslander-Reiten (d+2)-angles

Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.

Higher fundamental functors for simplicial sets

This is a sequel to a previous paper, developing an intrinsic, combinatorial homotopy theory for simplicial complexes; the latter form the cartesian closed subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial sets, or presheaves on the category of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself, !Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a groupoid, and preserves all colimits - a strong van Kampen property. Sim…

定向范畴论中的纤维化

Fibrations in Oriented Category Theory

从定向范畴论角度研究高阶范畴纤维化，给出其等价特征，表明纤维化范畴构成定向范畴，研究其与定向拉回相互作用并构建相关纤维化版本及格罗滕迪克构造。

Higher homotopy categories, higher derivators, and K-theory

For every $\infty$-category $\mathscr{C}$, there is a homotopy $n$-category $\mathrm{h}_n \mathscr{C}$ and a canonical functor $γ_n \colon \mathscr{C} \to \mathrm{h}_n \mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy $n$-categories, we introduce the notion of an $n$-derivator and study the main examples arising from $\infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\infty$-derivators and prov…

The Eckman-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its $Hom$-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is `the same' as a braided monoidal category. In this p…

$d$-外角化范畴的 Grothendieck 群与修正的 Caldero-Chapoton 映射

Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map

本文在更高同调背景下，证明了 $d$-外角化范畴的 Grothendieck 群与簇倾斜子范畴商群的同构关系，并据此定义了可恢复 Holm-Jørgensen 工作的修正 Caldero-Chapoton 映射。

对运算子和代数的模型结构从全局视角研究

Model structures on operads and algebras from a global perspective

本文从全局视角研究运算子和其代数、模的同伦理论，通过模型范畴和半模型范畴提供统一框架，并在新环境中产生模型结构、校正结果和适当性结果。

All $(\infty,1)$-toposes have strict univalent universes

We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.

Homotopical interpretation of globular complex by multipointed d-space

Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows introduced in arXiv:math/0308054. The underlying category of this model category is a variant of M. Grandis'…

Topology and Higher-Dimensional Category Theory: the Rough Idea

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the higher-dimensional diagrams one draws to represent these structures can be taken quite literally as pieces of topology. Examples of this are the braids in a braided monoidal category, and the pentagon which appears in the definitions of both monoidal category and A_infinity space. I will try to give a Friday-afternoonish description of some of the dreams peo…

高阶向量丛

Higher Vector Bundles

本文在(Kan)单纯纤维化背景下引入分裂的高阶类比，建立了同伦表示与带分裂的单纯向量丛的等价关系，作为Higher Grothendieck Correspondence的体现并扩展了VB-groupoids的早期工作。

高阶几何层论

Higher geometric sheaf theories

本文引入基础∞-范畴中的高阶覆盖图概念，构建高阶几何层论以统一恢复各类下降条件，并证明其通过余拓扑局部化区别于无穷余凝聚层论，进而推广了Lurie的∞-拓扑斯层定义。

Schur函子与范畴化Plethysm

Schur Functors and Categorified Plethysm

本文证明Schur函子范畴构成一个“2-plethory”，其下降诱导了对称函数环的plethory结构，从而揭示对称函数的丰富结构在更高范畴层次上的存在。