Correction to "Simplicial monoids and Segal categories"

Quasi-Categories vs. Segal Spaces: Cartesian Edition

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie.

拟2-Segal集

Quasi-2-Segal sets

本文定义单纯集中的拟2-Segal集，构造其模型结构并与完全2-Segal空间 Quillen 等价，同时给出路径空间与逐边细分判据。

Dendroidal Segal spaces and infinity-operads

We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for infinity-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal's quasi-categories, Rezk's complete Segal spaces, and…

Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture

We show that if $(M,\tensor,I)$ is a monoidal model category then $\REnd_M(I)$ is a (weak) 2-monoid in $\sSet$. This applies in particular when $M$ is the category of $A$-bimodules over a simplicial monoid $A$: the derived endomorphisms of $A$ then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.

Equivariant complete Segal spaces

In this paper we give a model for equivariant $(\infty, 1)$-categories. We modify an approach of Shimakawa for equivariant $Γ$-spaces to the setting of simplicial spaces. We then adapt Rezk's Segal and completeness conditions to fit with this setting.

Degeneracies in quasi-categories

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent quasi-categories. The method is purely combinatorial and extends to semisimplicial objects in other categories; in particular to semi-simplicial spaces satisfying the Segal condition (semi-Segal spaces).

Dendroidal sets and simplicial operads

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by …

Adding inverses to diagrams encoding algebraic structures

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal's model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups.

Partial model categories and their simplicial nerves

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus. We then show that Charles Rezk's result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories. We also note that conversely ever…

单纯幺半群模型范畴的可容许替换

Admissible replacements for simplicial monoidal model categories

本文利用Dugger万有模型范畴构造，为单纯与组合对称幺半群模型范畴构造了具有更好operadic性质的替换，并证明稳定情形下当幺半群单位是紧生成元时，此类范畴可由交换环谱分类，且与唯一确定的交换环谱上的模强幺半群Quillen等价。

On a simplicial monoid whose underlying simplicial set is not a quasi-category

It is well known that the underlying simplicial set of any simplicial group is a Kan complex. Roughly speaking, Kan complex is an infinite-dimensional analogue of groupoid, and the relation between groupoids and categories resembles that between groups and monoids. Thus one may ask if the underlying simplicial set of each simplicial monoid is a quasi-category. In this short note, we construct a simplicial monoid whose underlying simplicial set is not a quasi-category.

Segal Topoi and Natural Phenomena: Universality of Physical Laws

J. Lurie proved in Higher Topos Theory that for $K$ a simplicial set, $\mathcal{C}$ a simplicial category, $f: \mathfrak{C}[K] \rightarrow \mathcal{C}^{\text{op}}$ an equivalence of simplicial categories, we have a Quillen equivalence $(\text{Set}^+_Δ)_{/K} \rightleftarrows (\text{Set}^+_Δ)^{\mathcal{C}}$. We prove a partial converse to this theorem at the level of Segal categories, namely that if $L(\text{Set}^+_Δ)_{/K}$ is isomorphic to $L(\text{Set}^+_Δ)^{\mathcal{C}}$ in Ho(SePC), then $L \mathfrak{C}[K]^{\text{op}}$ and $L\mathcal{C}$ are equivalent as Segal pre-categories relative to Seg…

识别半Segel空间映射的同伦纤维

Identifying the homotopy fiber of a map of semi-Segal spaces

本文针对半Segel空间映射证明了Quillen定理版本，构造了双半单纯分解，并将该映射的同伦纤维表示为特定半单纯空间的几何实现。

与带余纤维化的范畴相关的2-Segal映射

2-Segal maps associated to a category with cofibrations

该文研究带余纤维化范畴的Waldhausen $S_\bullet$-构造的2-Segal性质，确定了几类单纯对象中恒为等价的2-Segal映射，约化了相关判定问题并给出充分条件，还引入新范畴概念与反例。

A Model for the Higher Category of Higher Categories

We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations. Finally, we use equivalences between quasi-categories and complete Segal spaces to present analogous constructions using fibrations of quasi-categories.

塞格尔空间的扭曲箭头构造

Twisted Arrow Construction for Segal Spaces

该研究给出单纯形空间扭曲箭头构造的显式描述，证明其保持完全塞格尔空间的定义性质，且对塞格尔空间而言，该构造的自然投影是左纤维化。

A characterization of fibrant Segal categories

In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure SeCat_c. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories which are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure SeCat_f.

Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-S…

Higher Segal structures in algebraic $K$-theory

We introduce higher dimensional analogues of simplicial constructions due to Segal and Waldhausen, respectively producing the direct sum and algebraic $K$-theory spectra of an exact category. We then investigate their fibrancy properties, based on the formalism of higher Segal spaces by Dyckerhoff-Kapranov.

Segal Group Actions

We define a model category structure on a slice category of simplicial spaces, called the "Segal group action" structure whose fibrant-cofibrant objects may be viewed as representing spaces $X$ with a coherent action of a given Segal group (i.e. a group-like, reduced Segal space). We show that this model structure is Quillen equivalent to the projective model structure on $G$-spaces, $\mathcal{S}^{\mathbb{B}G}$, where $G$ is a simplicial group represented by this Segal group. Since Segal group actions are invariant under weak monoidal endofunctors of spaces they enable to construct, for an arb…

Simplicial Monoid Actions and The Associated Universal Simplicial Monoid Construction

The reduced universal monoid on the action category associated to a pointed simplicial M-set has appeared in the guise of various simplicial monoid and group constructions. These include the classical constructions of Milnor and James, as well as their later generalizations by Carlsson and Wu. We prove that, if any two n-simplices in the same orbit differ by the action of an invertible monoid element, then the classifying space of this reduced universal monoid is the homotopy cofiber of the inclusion from the pointed simplicial set into its reduced Borel construction. The known formulae for th…

范畴化等变幺半群

Categorifying equivariant monoids

本文结合 PROPs、PROBs 与交叉单纯群理论，构造出编码等变幺半群结构的范畴，其代数范畴等价于带群作用的各类幺半群，并扩展至平衡辫张范畴和对合幺半群。

Segal Enriched Categories I

We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal M_W -categories. Our motivation was to generalize the notion of "up-to-homotopy monoids" in a monoidal category M, introduced by Leinster. The formalism adopted generalizes the classical Segal categories and extends the theory of enriched category over a bicategory. In particular we have a linear version of Segal categories which did not exist so far. Our goal in this paper is to present the theory and provide some examples. Applications ar…

高阶Segal空间与偏群

Higher Segal spaces and partial groups

本研究探讨d-Segal条件与partial groups的相互作用，定义partial群的度并开发基于其作用离散几何的度计算工具，在穿孔外尔群等实例中验证了度与半单李代数阿贝尔子代数最大维数的关联。

Cartesian Fibrations of Complete Segal Spaces

Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences. This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In addition to that, it allows us to define…

Lax Diagrams and Enrichment

We introduce a new type of weakly enriched categories over a given symmetric monoidal model category M; these are called Co-Segal categories. Their definition derives from the philosophy of classical (enriched) Segal categories. We study their homotopy theory by giving a model structure on them. One of the motivations of introducing these structure was to have an alternative definition of higher linear categories following Segal-like methods.

Segal's multisimplicial spaces

Some sufficient conditions on a simplicial space $X$ guaranteeing that $X_1\simeq Ω|X|$ were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar construction, providing an $n$-fold delooping of the classifying space of a category.

Segal K-理论通过Waldhausen范畴

Segal K-theory factors through Waldhausen categories

本文通过构造Waldhausen范畴Γ(C)，证明Segal K理论可通过Waldhausen范畴因子化，并展示连通谱可通过Waldhausen K理论获得。

On complicial homotopy monoids

For a Kan complex with a vertex, we have the notion of its simplicial homotopy groups. In this paper, for a weak complicial set in the sense of Verity with a vertex, we construct monoids which are a generalization of simplicial homotopy groups.

2-Segal sets and the Waldhausen construction

It is known by results of Dyckerhoff-Kapranov and of Gálvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.

Simplicial distributions, convex categories and contextuality

The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell. Such distributions are called contextual. Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces. Contextuality can be defined in this generalized setting. This paper introduces the notio…

dagger $(\infty,1)$-范畴的模型 II：dagger 完备 Segal 空间与 Joyal--Tierney 等价

Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence

本文为dagger (∞,1)-范畴发展单纯空间模型，构造酉完备dagger Segal空间模型结构，并证明Joyal--Tierney等价的dagger类比，建立dagger Bergner、Joyal与Rezk模型间的Quillen等价链。

$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories

The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicial ly enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; (ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the o…

分解空间视角

The decomposition space perspective

本文介绍了分解空间和2-Segal空间，统一了两种视角。通过simplicial范畴的active-inert因子分解系统定义分解空间，并展示其等价性。关键结果包括路径空间准则和边细分准则，还引入了来自外面复形的自由分解空间。

Monoidal Pull-Push I: Cocartesian Fibrations and Categories of Spans

We develop a basic theory of cocartesian fibrations between Segal spaces (in line with that of arxiv:2102.05190), and use it to provide a proof of a theorem of Barwick (the main result of arxiv:1404.0108). Note: This work was originally the author's master's thesis, submitted 21.09.2020.

A survey of (\infty, 1)-categories

In this paper we give a summary of the comparisons between different definitions of so-called (\infty,1)-categories, which are considered to be models for \infty-categories whose n-morphisms are all invertible for n>1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.

Relative categories: Another model for the homotopy theory of homotopy theories

We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

用Segal空间建模$(\infty,1)$-范畴

Modeling $(\infty,1)$-categories with Segal spaces

本文提出了一种用Segal空间建模$(\infty,1)$-范畴的模型结构，并证明其与完整Segal空间和Segal范畴模型等价，同时具有笛卡尔闭和左正则等优良性质。

单纯 $*$-模与温和作用

Simplicial $*$-modules and mild actions

本文在单纯集合范畴中发展了 $*$-模理论，基于单纯独异点 $E\mathcal M$ 的作用，证明了严格交换独异点可产生等变及全局相干交换独异点的模型，并通过温和条件刻画了单纯 $*$-模。

Moment categories and operads

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment…

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a "homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to …

In the category of relative categories the Rezk equivalences are exactly the DK-equivalences

In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more explicit description of the weak equivalences in this model structure by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.

A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi

Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small 1-category $C$ and a morphism $g:C\to A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A\to \hat{C}$, such that $i$ admits …

On loop spaces with marking

An analogous construction of simplicial homotopy group for Kan complex can be applied to saturated complicial sets to give monoids. In this paper, we investigate how the construction of loop spaces of Kan complexes lifts to the complicial setting and relates to such monoids.

余模的幺半 Dold-Kan 对应

A monoidal Dold-Kan correspondence for comodules

本文构造了余模范畴中基于 Postnikov 塔的归纳纤维性替换，证明单纯余模与连通微分分次余模 Quillen 等价，并推出单连通空间的有理 A-理论等价于相应余模结构完美链复形的 K-理论。

On the equivalence between Lurie's model and the dendroidal model for infinity-operads

We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Gamma. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.

On the construction of limits and colimits in $\infty$-categories

In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an $\infty$-category - for instance, a quasi-category, a complete Segal space, or a Segal category - is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly th…

dagger (∞,1)-范畴的模型 III：万有带标记 dagger (∞,1)-范畴与酉完备 dagger Segal 动画

Models for dagger $(\infty,1)$-categories III: Univalent flagged dagger $(\infty,1)$-categories and unitarily complete dagger Segal anima

该文将 dagger 模型的 (∞,1)-范畴与万有带标记 dagger (∞,1)-范畴等同，构造伴随对证明其等价性，将万有性对应酉完备性，还等同了万有化与酉完备化。

关于对称幺半范畴的整体同伦论

On the global homotopy theory of symmetric monoidal categories

本文证明可和范畴的整体同伦论可由对称幺半范畴建模，并等价于可和单纯集的同伦论，为整体相干交换幺半群提供桥梁。

McDuff-Segal群完备化定理的等变推广

An equivariant generalisation of McDuff-Segal's group-completion theorem

本文证明了McDuff-Segal群完备化定理关于有限群G的G-等变推广，通过在G-谱中的E-infinity-环同伦群上分离出简单条件，解决了真正等变局部化的新复杂情况，并借助乘性范数结构验证了条件，为相关G-球面群环给出了局部化公式。

Composing PROBs

A PROB is a "product and braid" category. Such categories can be used to encode the structure borne by an object in a braided monoidal category. In this paper we provide PROBs whose categories of algebras in a braided monoidal category are equivalent to the categories of monoids and comonoids using the category associated to the braid crossed simplicial group of Fiedorowicz and Loday. We show that PROBs can be composed by generalizing the machinery introduced by Lack for PROPs. We use this to define a PROB for bimonoids in a braided monoidal category as a composite of the PROBs for monoids and…

Non-Simplicial Nerves for Two-Dimensional Categorical Structures

The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it is a quasicategory which has unique fillers for inner horns of dimension $3$ and greater. Using Duskin's technique, we show how his nerve applies to $(2,1)$-category functors, making it a fully faithful inclusion of $(2,1)$-categories into simplicial sets. Then we consider analogues of this extension of Duskin's result for several different two-dimensional …

Fibrations and Yoneda's lemma in an $\infty$-cosmos

We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. Previous work in this series shows that the basic category theory of $\infty$-categories and $\infty$-functors can be developed only in reference to the axioms …

高阶范畴的同调

Homology of higher categories

本文基于Eilenberg-Steenrod公理定义高阶范畴的同调，通过范畴论类比公理建立Dold-Kan对应，利用Street神经网络作为奇异复形，推导出具有乘法结构的高阶同调计算。