Brice Le Grignou
In this article we introduce the notion of a square structure on a model category, that generalises cubical model categories. We then show that under some homotopical conditions on this square structure the induced cubical category is a localisation of the model category.
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The goal of this article is to emphasize the role of cubical sets in enriched categories theory and infinity-categories theory. We show in particular that categories enriched in cubical sets provide a convenient way to describe many infinity-categories appearing in the context of homological algebra.
Multiple categories: the equivalence of a globular and a cubical approach
我们证明了两类严格重范畴，即著名的球形Ω范畴和带联络的立方Ω范畴的等价性。
Rigidification of cubical quasi-categories
本文构造了从cubical sets到simplicial categories的刚性化函子，通过适配Dugger和Spivak的necklaces框架，证明其在Joyal模型结构与Bergner模型结构间建立了Quillen等价。
Thin Elements and Commutative Shells in Cubical omega-categories
薄元素之间的关系，交换壳和连接在立方Ω范畴探索的方法，不涉及使用粘贴理论或神经的Ω范畴（这两个都是以前需要为这个目的;见以前的工作由Al-Agl/布朗/施泰纳。它表明，复合材料的交换壳是交换的，薄结构是等价于适当的连接集;这项工作扩展到所有维度的结果证明在2和3维布朗/莫萨，布朗/坎普斯/波特。
In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $Θ^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $Θ_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connec…
We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK…
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is consequence of the other axioms.
We prove that each of the model structures for ($n$-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure for ($n$-trivial, saturated) complicial sets on the category of marked simplicial sets, as well as to the corresponding comical model structures on cubical sets with connections. As a consequence, we show that the cubical Joyal model structure on cubical sets without connections is equivalent to its analogues on cubical sets with connections and to the Joyal…
Symmetry in the cubical Joyal model structure
本文研究立方体Joyal模型结构的对称性，证明其在笛卡尔单oidal性和弱等价下的对称性，并诱导出带有对称性的$(\infty,1)$-范畴模型结构。
Cartesian cubical model categories
该研究定义笛卡尔立方集范畴，并利用一元类型理论立方系统构造的思想建立其上的 Quillen 模型结构。
We show that various cube categories (without diagonals, but with symmetries / connections / reversals) are Eilenberg-Zilber categories. This generalizes a result of Isaacson for one particular cubical site. Our method does not involve direct verification of any absolute pushout diagrams. While we are at it, we record some folklore descriptions of cube categories with diagonals and determine exactly which of these are EZ categories. Beforehand, we develop some general theory of Eilenberg-Zilber categories. We show that a mild generalization of the EZ categories of Berger and Moerdijk are in fa…
Cubical models of $\infty$-presheaves and the Bousfield-Kan formula
本文分别在立方体集、带标记立方体集的切片范畴上构造协变与余笛卡尔模型结构，并据此为满足Muro公理的余纤维生成幺半模型范畴推导Bousfield-Kan公式。
Equivalence of cubical and simplicial approaches to $(\infty,n)$-categories
本文证明了立方体与单纯形方法在$(\infty,n)$-范畴中的等价性，基于锥形理论的理论发展。
In this article we introduce the notion of cubical $(ω,p)$-categories, for $p \in \mathbb N \cup \{ω\}$. We show that the equivalence between globular and groupoid $ω$-categories proven by Al-Agl, Brown and Steiner induces an equivalence between globular and cubical $(ω,p)$-categories for all $p \geq 0$. In particular we recover in a more explicit fashion the equivalence between globular and cubical groupoids proven by Brown and Higgins. We also define the notion of $(ω,p)$-augmented directed complexes, and show that Steiner's adjunction between augmented directed complexes and globular $ω$-ca…
A cubical model for $(\infty, n)$-categories
本文提出了一种新的$(\infty,n)$-范畴模型，基于标记立方集，并展示了其在Gray张量积下的单向性和三角化 functor 的左Quillen性质。
Involutive Weak Cubical $ω$-categories
本文采用Penon的单子代数方法定义并研究对合弱立方 $ω$-范畴，并提供了若干具体例子。
This note informally describes a way to build certain cubical n-categories by iterating a process of taking models of certain finite limits theories. We base this discussion on a construction of "double bicategories" as bicategories internal to Bicat, and see how to extend this to n-tuple bicategories (and similarly for tricategories etc.) We briefly consider how to reproduce "simpler" definitions of weak cubical n-category from these.
Cubical Approximation for Directed Topology II
本文通过立方逼近方法，建立了定向拓扑与立方集之间的等价关系，并探讨了其在同调单形和分类空间中的应用。
Thin fillers in the cubical nerves of omega-categories
证明了严格Ω范畴的立方神经是具有立方面运算的集合序列，且满足一定的薄填充条件的薄元的区别子类。它还表明，这种类型的序列是一个严格的欧米茄范畴唯一同构的立方神经;立方神经函子因此是范畴的等价。所涉及的集合序列实际上是适合于严格ω范畴的三次T-复形的类似物。在这些序列的定义中不需要简并，但实际上可以构造为薄填料。薄填充条件的证明使用链复形和链同伦。
Single-set cubical categories and their formalisation with a proof assistant (extended version)
本文提出含连接与逆的立方 ω-范畴单集合公理化，通过范畴等价证明其合理性，并借助 Isabelle/HOL 完成形式化。
We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.
Localisation theorems for the connective K-theory of exact categories
该研究证明了正合∞-范畴过滤子范畴K-理论的局部化定理，涵盖了稳定∞-范畴、阿贝尔范畴及Schlichting的s-过滤子范畴的相关局部化定理。
We introduce a new cubical model for homotopy types. More precisely, we'll define a category Qs with the following features: Qs is a PROP containing the classical box category as a subcategory, the category Qs-Set of presheaves of sets on Qs models the homotopy category, and combinatorial symmetric monoidal model categories with cofibrant unit all have homotopically well behaved Qs-Set enrichments.
Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)
在分别讨论了弱立方范畴和{ \it collarable}余跨距的两篇文章之后，我们综合起来构造了拓扑空间的立方{ \it collared}余跨距的{\it weak}立方范畴.我们还建立了第二个结构，称为{ \it quasi}立方范畴，形成的任意立方cospans连接同伦推出。这种结构更简单但更弱，具有{ \it lax}恒等式。它包含了一个类似的框架，cobordisms的流形与角落，因此可能是基础，以扩大研究的TQFT的第二部分，以更高的立方度。
We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.
Koszul Feynman Categories
本文证明了所有立方 Feynman 范畴都是 Koszul 的，并由此构造出显式极小的余纤维化解，为基于图和非基于图的广义 operad 提供 $\infty$ 版本模型。
We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. The methods to construct this localisation are similar to the Ore localisation for a 2-arrow calculus; in particular, we do not have to use zigzags of arbitrary length. Applications include the localisation of an arbitrary model category with respect to its weak equivalences as well as the localisation of it…
Parametrised noncommutative motives and equivariant cubical descent in algebraic K-theory
本文引入参数化完美稳定范畴，构建参数化非交换动机，研究其在有限2群上的共变结构，揭示其与Mackey函子的关系。
Consider a Quillen adjunction of two variables between combinatorial model categories from $\mathcal{C}\times\mathcal{D}$ to $\mathcal{E}$, and a set $\mathcal{S}$ of morphisms in $\mathcal{C}$. We prove that there is a localised model structure $L_{\mathcal{S}}\mathcal{E}$ on $\mathcal{E}$, where the local objects are the $\mathcal{S}$-local objects in $\mathcal{E}$ described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations.…
In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on S.
Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.
We introduce pseudocubical objects with pseudoconnections in an arbitrary category, obtained from the Brown-Higgins structure of a cubical object with connections by suitably relaxing their identities, and construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors. The crucial point in the construction is that projective precubical resolutions which are naturally used to define our cubical derived functors possess pseudodegeneracies and pseudoconnections. The same fact is essentially used for proving that in the case of an additive functor between abelian categories,…
We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2-Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.
Cartan calculus of cubical forms in tangent categories
本文在笛卡尔切触范畴中构造微分立方形式的Cartan演算，证明其构成CDGA，并在无2-挠条件下得到介于Kähler与Lie-Rinehart形式之间的演算，恢复de Rham复形。
Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness.
We give an account of Bousfield localisation and colocalisation for one-dimensional model categories---ones enriched over the model category of $0$-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith's theorem.
We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.
We give a new criterion guaranteeing existence of model structures left-induced along a functor admitting both adjoints. This works under the hypothesis that the functor induces idempotent adjunctions at the homotopy category level. As an application, we construct new model structures on cubical sets, prederivators, marked simplicial sets and simplicial spaces modeling $\infty$-categories and $\infty$-groupoids.
A notion of semi-cubical tribe
本文提出了一种基于$J$-frames的$J$-部落构造方法，用于处理具有对称性和反向的半立方体图示的范畴。
Nerves of enriched categories via necklaces
本文提出项链神经函子统一了同伦相干神经、微分分次神经和立方神经，证明其可提升为templicial对象，并给出左伴随的显式描述条件，从而得到dg-神经与立方神经左伴随的新简洁表达式。
Derived mapping spaces of $\infty$-categories
本文证明了导出映射空间引理，推广了 Cisinski 关于 $\infty$-范畴分数演算的核心定理，为分析范畴局部化中的映射空间提供了统一框架，并给出了立方或单纯范畴是底层范畴同伦等价局部化的充分条件。
In this paper, we introduce a method to construct new categories which look like "cubes", and discuss model structures on the presheaf categories over them. First, we introduce a notion of thin-powered structure on small categories, which provides a generalized notion of "power-sets" on categories. Next, we see that if a small category $\mathcal{R}$ admits a good thin-powered structure, we can construct a new category $\square(\mathcal{R})$ called the cubicalization of the category. We also see that $\square(\mathcal{R})$ is equipped with enough structures so that many arguments made for the c…
Cartesian Gray-Monoidal Double Categories
本文通过引入局部立方Gray范畴和双松弛函子，为对称Gray-幺半群双范畴建立了笛卡尔结构，并利用曲面图给出其简洁表示。
For each $n \geq -1$, a quasi-category is said to be $n$-truncated if its hom-spaces are $(n-1)$-types. In this paper we study the model structure for $n$-truncated quasi-categories, which we prove can be constructed as the Bousfield localisation of Joyal's model structure for quasi-categories with respect to the boundary inclusion of the $(n+2)$-simplex. Furthermore, we prove the expected Quillen equivalences between categories and $1$-truncated quasi-categories and between $n$-truncated quasi-categories and Rezk's $(n,1)$-$Θ$-spaces.
We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for $\mathtt{W}=\mathtt{W}_{\infty}$ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebra…
The purpose of this text is to prove all technical aspects of our model for dependent type theory with parametric quantifiers [Nuyts, Vezzosi and Devriese, 2017]. It is well-known that any presheaf category constitutes a model of dependent type theory [Hofmann, 1997], including a hierarchy of universes if the metatheory has one [Hofmann and Streicher, 1997]. We construct our model by defining the base category BPCube of bridge/path cubes and adapting the general presheaf model over BPCube to suit our needs. Our model is heavily based on the models by Atkey, Ghani and Johann [2014], Huber [2015…
This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is use…
Arrow Categories of Monoidal Model Categories
本文证明幺半模型范畴的箭头范畴在推出积幺半结构与投射模型结构下仍是幺半模型范畴，回答了霍维的问题，并推动了立方同伦理论中的幺半积应用，附有多个非余生成实例。
We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.
On Multi-Determinant Functors for Triangulated Categories
本文推广Deligne行列式函子至张量三角范畴，通过2-多范畴和立方体复形刻画多正合性，并证明在Verdier结构下取值于范畴环。
We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories. Variations on the notion of span provide further examples of interest, an important one being the intercategory of sets. We consider the three kinds of morphism of intercategory as well as the cells binding them with applications to the above structures. In particular hom functors are studied.
Sheaves of $(\infty, \infty)$-categories
本文基于 complete Segal space 对象给出了 $(n, r)$-范畴层的函子式表示，证明了 $(\infty, \infty)$-范畴层与 $\omega$-范畴层在底层 topos 上定义了 distributors，并得出了相应的层化结果。
Groupoidal and truncated $n$-quasi-categories
本文定义了群胚性与截断的 $n$-拟范畴，证明其为特定模型结构的纤维对象，并建立其与单纯集上空间模型的 Quillen 等价关系，同时构造了柱对象。
Abelian versus triangulated quotients
本文证明了三角范畴的局部化可诱导阿贝尔范畴的局部化，并由此构造了仅在更大格罗滕迪克宇宙中存在的阿贝尔商范畴的例子，同时将结果推广到任意范畴的阿贝尔包。
The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary monads on certain presheaf categories, generalising a well-known result due to Lawvere, Bénabou and Linton for ordinary multisorted algebraic theories. We use this to recognise dependently typed algebraic theories for a number of classes of algebraic structures, such as small categories, n-categories, strict and weak omega-categories, planar coloured operads and …
The diagrammatic presentation of equations in categories
本文定义了精确表示“同解”的范畴图等价，并证明图范畴沿该等价的局部化与切片范畴沿初始函子的局部化同构，且推广至局部可展示 2-范畴。
Parametrised Presentability over Orbital Categories
本文在轨道范畴上的参数化同伦理论框架下发展了可表现性概念，通过相关拉直刻画参数化可表现范畴，推导出参数化伴随函子定理与局部化结果，并应用于等变代数K-理论中参数化非交换动机范畴的构造。
We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of Toën [26]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [21]. Given a regular cardinal alpha, we define and construct a tensor product of homotopically alpha-cocomplete dg categories and prove that the well generated tensor…
The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$Ω$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properti…
Familial Monads as Higher Category Theories
本文明确描述了预层范畴上族可表示单子代数所定义的高阶范畴结构所需信息，利用范畴多项式理论避免繁琐自然性论证，并给出了分裂opfibration拉回等运算的分类函子描述。
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