Mapping spaces in homotopy coherent nerves

Geometric Realizations of Tricategories

Any tricategory characteristically has associated various simplicial or pseudo-simplicial objects. This paper explores the relationship amongst three of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the tricategory, the simplicial bicategory termed its Segal nerve, and the simplicial set called its Street geometric nerve, and it proves the fact that the geometric realizations of all of these possible candidate 'nerves of the tricategory' are homotopy equivalent. Our results provide coherence for all reasonable extensions to tricategories of Quillen's definition of the …

On the homotopy type of a cofibred category

In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The "fibred nerve" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the "cleaved nerve", a smaller variant which emerges from a distinguished closed cleavage. We interpret some classical theorems by Thomason and Quillen in terms of our constructions, and use the fibred and cleaved nerve to establish new results on homotopy and homo…

Finite Homotopy Limits of Nerves of Categories

Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the geometric realization of a category. This categorical model for the homotopy limit is defined explicitly, as a category of natural transformations of diagrams. For the poset $\bullet\to\bullet\leftarrow\bullet$ we recover the model for homotopy pullbacks provided by Quillen's Theorem $B$ (specifically Barwick and Kan's version of Quillen's Theorem $B_2$). For …

Nerves of 2-categories and 2-categorification of $(\infty,2)$-categories

We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's model structure for 2-categories and Riehl-Verity's model structure for 2-complicial sets. Furthermore, we show that Lack's model structure is transferred along this nerve and that the nerve is homotopically fully faithful.

Homotopy coherent structures

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such "up to homotopy" settings. These notes provide a three-part introduction to homotopy coherent category theory. The first part surveys the classical theory of homotopy coherent diagrams of topological spaces. The second part introduces the…

Rigidification and the Coherent Nerve for Enriched Quasicategories

We introduce, for \(\C\) a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on \(C\). We then develop a coherent realization and nerve for this model structure and demonstrate using an enriched version of the necklaces of Dugger and Spivak that our model category is Quillen-equivalent to the category of categories enriched in simplicial presheaves on \(\C\). We then show that for any Cartesian-closed left-Bousfield localization of the category of simplicial presheaves on \(\C\), the coherent nerve and r…

Strong homotopy types, nerves and collapses

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial $G$-actions …

Nerve complexes of circular arcs

We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to …

From homotopy operads to infinity-operads

The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk--Weiss in order to generalize the simplicial notion of infinity-category of Joyal--Lurie. This functor extends in two directions the simplicial nerve of Faonte--Lurie for A-infinity-…

Topology of nerves and formal concepts

The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen-…

Homotopy theory of monoids and derived localization

We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology. This includes a recent generalization of Adams' cobar-construction to the non-simply connected case, and a new model for the homotopy theory of connected topological spaces using an infinity category of discrete monoids.

Classification spaces of maps in model categories

This paper corrects a small mistake in a paper of Dwyer-Kan, and uses this to identify homotopy function complexes in a model category with the nerves of certain categories of zig-zags.

Categories and orbispaces

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types …

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales…

开半球与闭半球覆盖

Coverings by open and closed hemispheres

本文研究球面开半球与闭半球对径覆盖的神经复形同伦类型，给出闭半球覆盖神经复形的楔和分解公式，结合根系$A_d$情形关联图空间同伦结果，探索DAG空间在AutoML中的应用。

Homotopy coherent centers versus centers of homotopy categories

Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a Bousfield…

Strictifying Homotopy Coherent Actions on Hochschild Complexes

If P is a dg-operad acting on a dg-algebra A via algebra homomorphisms, then P acts on the Hochschild complex of A. In the more general case when P is a dg-prop, we show that P still acts on the Hochschild complex, but only up to coherent homotopy. We moreover give a functorial dg-replacement of P that strictifies the action. As an application, we obtain an explicit strictification of the homotopy coherent commutative Hopf algebra structure on the Hochschild complex of a commutative Hopf algebra.

Homotopy coherent adjunctions and the formal theory of monads

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contr…

富化的同伦相干结构

Enriched homotopy-coherent structures

本文基于“代数模式”框架，为Segal条件描述的同伦相干代数结构引入一般性富化概念，恢复了已知的富化∞-范畴等实例，并提出了富化模∞-代数操作等新实例。

从旧范畴模型到新范畴模型：通过开覆盖

From old categorical models to new ones through open covers

本文通过用相容范畴模型替代开覆盖的交集，并利用Grothendieck构造组装局部模型，推广了神经定理，得到有限无圈范畴，其分类空间保持原空间的弱同伦型，且构造是递归的。

On the structure of simplicial categories associated to quasi-categories

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (infinity,1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor C also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category CX arising from a quasi-category X are not well understood. We show that when X is…

Homotopy coherent representations

Homotopy coherence has a considerable history, albeit also by other names. For this volume highlighting symmetries, the appropriate use is: Homotopy coherence of representations, at one time known as strong homotopy representations. We provide a brief semi-historical survey providing some links that may not be common knowledge.

Equivariant Nerve Lemma, simplicial difference, and models for configuration spaces on simplicial complexes

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variati…

有限空间的关系多重柱与强-好覆盖的神经定理

Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover

本文提出关系多重柱概念以推广映射柱并捕捉有限空间链的复杂同伦结构，同时引入强-好覆盖，证明了具有该覆盖的单纯复形与有限空间及其神经保持相同简单同伦型，强化了经典神经定理。

Homotopy coherent theorems of Dold-Kan type

We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.

Representations up to coherent homotopy

Homotopy coherence has a considerable history, albeit also by other names. We provide a brief semi-historical survey providing some links that may not be common knowledge.

同伦相干伴侣与联结

Homotopy coherent companionships and conjunctions

本文研究了双∞-范畴中伴侣和联结的扩展性质，并通过2-细胞表征其特征，应用于(∞,2)-范畴理论。

球面上一些同伦相干有限群作用

Some homotopy coherent finite group actions on spheres

研究球面上同伦相干有限群作用，给出非严格作用的例子作为贡献，核心方法未提及。

Ribbon Complexes & their Approximate Descriptive Proximities. Ribbon & Vortex Nerves, Betti Numbers and Planar Divisions

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not include the interior of the inner cycle. Each planar ribbon has its own distinctive shape determined by its outer and inner boundaries and the interior within its boundaries. A Vergili ribbon complex (briefly, ribbon complex) in a CW space is a non-void collection of countable pl…

等变同伦交错与持久Whitehead定理

Equivariant Homotopy Interleaving and Persistent Whitehead Theorem

本文针对有限群作用的持久空间，定义了$G$-同伦交错距离并证明其普适性，建立了等变持久Whitehead定理与神经引理，最终导出等变弱大数定律。

Configuration spaces and directed paths on the final precubical set

The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the "total" configuration space of points on the plane; by "total" we mean that any finite number of points in a configuration is allowed. We also provide several applications: we define new invariants of precubical sets, prove that directed path spaces on any precubical complex have the homotopy types of CW-complexes and construct certain presentations of configuration spaces of points on the plane as nerves of categories.

Homotopical Quantum Field Theory

Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy algebraic quantum field theories and homotopy prefactorization algebras and investigate their homotopy coherent structures. Homotopy coherent diagrams, homotopy inverses, A-infinity-algebras, E-infinity-algebras, and E-infinity-modules arise naturally in this context. In particular, each homotopy algebraic quantum field theory has the structure of a homotopy …

Spaces of directed paths on pre-cubical sets II

For a given pre-cubical set ($\square$--set) $K$ with two distinguished vertices $\bO$, $\bI$, we prove that the space $\vP(K)_\bO^\bI$ of d-paths on the geometric realization of $K$ with source $\bO$ and target $\bI$ is homotopy equivalent to its subspace $\vP^t(K)_\bO^\bI$ of tame d-paths. When $K$ is the underlying $\square$--set of a Higher Dimensional Automaton $A$, tame d-paths on $K$ represent step executions of $A$. Then, we define the cube chain category of $K$ and prove that its nerve is weakly homotopy equivalent to $\vP(K)_\bO^\bI$.

Homotopy coherent diagrams and approximate fibrations

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy type of the total space and give an application to the fibering of Hilbert cube manifolds.

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be though…

有限偏序集的拓扑实现

Topological realizations of finite posets

本文提出CL-复形为有限偏序集提供替代拓扑实现，可简短证明偏序集同伦的多个标准结果，其定义源于Walker的Carrier引理推广，建立了范畴等价关系。

Brane actions for coherent $\infty$-operads

We prove that Mann-Robalo's construction of the brane action extends to general coherent $\infty$-operads, with possibly multiple colors and non-contractible spaces of unary operations. This requires to establish two results regarding spaces of extensions that were left unproven in the aforementioned construction. First, we show that Lurie's and Mann-Robalo's models for such spaces are equivalent. Second, we prove that the space of extensions in the sense of Lurie is not in general equivalent to the homotopy fiber of the associated forgetful morphism, but rather to its homotopy quotient by the…

On the classification of fibrations

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we prove that this is possible for the category of functors indexed a small category I which assign to any morphism in I a weak equivalence in a given model category. We identify the homotopy type of this category with the mapping space out of the nerve of I into the classifying space of the space of weak equivalences of values of these functors. We use it to re…

Homotopy-coherent algebra via Segal conditions

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", bywhich we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects. Examples of structures that occur as such "Segal $\mathcal{O}$-spaces" for an algebraic pattern $\mathcal{O}$ include $\infty$-categories, $(\infty,n)$-categories, $\infty$-operads, $\infty$-properads, and algebras for an $\infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal obj…

四维流形上的非动力学同伦相干作用

Non-kinetic homotopy coherent actions on four-manifolds

该研究给出闭单连通光滑四维流形上二阶非动力学光滑同伦相干作用的首个例子，通过限制离散圆周群在稳定化\(K3\)曲面上的作用获得，还构造相关扩张并展示特殊边界对合，证明了关于整同调球上自由对合光滑扩张的定理。

Adams' cobar construction revisited

We give a short and streamlined proof of the following statement recently proven by the author and M. Zeinalian: the cobar construction of the dg coassociative coalgebra of normalized singular chains on a path-connected pointed space is naturally quasi-isomorphic as a dg associative algebra to the singular chains on the based loop space. This extends a classical theorem of F. Adams originally proven for simply connected spaces. Our proof is based on relating the cobar functor to the left adjoint of the homotopy coherent nerve functor.

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it …

针对同伦凝聚余代数的点集模型

Point-set models for homotopy coherent coalgebras

本文通过归纳法证明，当操作子为cofibrant时，通过局部化微分格子代数得到的∞-范畴与基于富集∞-操作子的余代数∞-范畴等价，从而为E_n和E_∞余代数提供显式点集模型。

Morse 流范畴作为出口路径范畴

Morse flow categories as exit path categories

本文证明光滑闭流形上 Morse-Smale 对 $(f,ξ)$ 诱导的拓扑流范畴 $\mathcal{M}$ 作为 $\infty$-范畴等价于 Lurie 的关于稳定流形分层的出口路径 $\infty$-范畴 $\mathrm{Sing}_A(X)$。

Smooth constructions of homotopy-coherent actions

We prove that, for nice classes of infinite-dimensional smooth groups G, natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of G. This yields a bridge between infinite-dimensional smooth groups and homotopy theory. The result relies on two computations: One showing that the diffeological homotopy groups of the Milnor classifying space BG are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of BG. We then prove a result in symplectic geome…

同伦连贯的庞特里亚金-托姆同构

A homotopy coherent Pontryagin-Thom isomorphism

构造了光滑流形上具有传递的同伦不变层的一个可表现对称幺半稳定∞-范畴，其单位恰为（结构化）几何配边，并证明单位层的自同态环可典范地等同于关联的E_∞-托姆环谱，给出了庞特里亚金-托姆同构的E_∞-提升。

面向同伦论研究者的环面镜像对称

Toric Mirror Symmetry for Homotopy Theorists

本研究构建了将球面谱上环面概形的等变拟凝聚层映射为实向量空间可构造谱层的函子，实现环面同调镜像对称定理的谱提升，得到新的对称幺模结构与函子性结果，并给出Beilinson相关结论的新证明。

Descent for quasi-coherent sheaves on stacks

We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site C. We use this to prove homotopy invariance and generalized descent statements for categories of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a generalized change of rings theorem of Hovey.

细分双过滤的神经模型

Nerve Models of Subdivision Bifiltrations

研究Sheehy细分双过滤的同伦规模，提出基于神经的单纯模型，并为度量空间上的subdivision-Rips双过滤构建了更鲁棒的$\sqrt{2}$-近似及特定条件下的精确模型。

On the Goodwillie derivatives of the identity in structured ring spectra

The aim of this paper is three-fold: (i) we construct a naturally occurring highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad $\mathcal{O}$ in spectra, (ii) we prove that every connected $\mathcal{O}$-algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on $\mathcal{O}$-algebras and the operad $\mathcal{O}$. Al…

Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy

Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sens…

Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories

Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with coefficients in $M_g$ carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus $g+1$. The ordinary Hochschild complex of $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor $\mathfra…

Pair component categories for directed spaces

The notion of a homotopy flow on a directed space was introduced in \cite{Raussen:07} as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy type of path spaces, such a flow (and these parameter maps) are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points and whose morphisms may be induced by these inessential d-maps. Localization with respect to subcategories with these inessential d-maps as morphisms can be combi…

包裹Fukaya范畴上的连续且相干作用

Continuous and coherent actions on wrapped Fukaya categories

本文建立了包裹Fukaya范畴关于Liouville自同构的连续函子性，证明了Teleman猜想，并在余切丛情形下将作用与微分同胚作用交织，给出非平凡的有理同伦群映射。