W. Chacholski, J. Scherer
We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces.
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$G\operatorname{-CGWH}$ spaces and their (homotopy) colimits
本文研究带紧李群$G$作用的紧生成弱豪斯多夫空间及其余极限、同伦余极限，探讨（同伦）余极限与映射空间及同伦群的关系。
We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). Along the way, we obtain relative-homological-algebraic generalizations of classical …
On sifted homotopy colimits of algebras over an $N_{\infty}$-operad
本文证明了从 $N_{\infty}$-操作代数到等变空间的遗忘函子保持筛选同调余极限。
We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the categor…
We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z_k X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections, and that any nilpotent space for which the space of pointed self-maps map_*(X,X) is "canonically" discrete must be aspher…
In this short note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by a finite category, but the behavior is more uniform when we index by finite quasicategories instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.
We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence.
The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces $X_{g,n}$, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes $Δ_g$, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each $X_{g,n}$. Furthermore, we use…
We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere.
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…
Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups
本文研究了群图的非阿贝尔同调群，证明了其与类ifying空间图的同调极限的同调群同构，并发展了寻找同调极限最小维数非零同调群的方法。
We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and seems to be of independent interest.
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E-2 spaces categorically.
Building on a previous definition of homotopy limit of model categories, we give a definition of homotopy colimit of model categories. Using the complete Segal space model for homotopy theories, we verify that this definition corresponds to the model-category-theoretic definition in that setting.
We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets.
The main objective of this paper is to show that the homotopy colimit of a diagram of quasi-categories and indexed by a small category is a localization of Lurie's higher Grothendieck construction of the diagram. We thereby generalize Thomason's classical result which states that the homotopy colimit of a diagram of categories has the homotopy type of (the classifying space of) the Grothendieck construction of the diagram of categories.
Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend to homotopy colimits of 2-functors lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the Homotopy Invariance Theorem (by Bousfield and Kan), the Homotopy Colimit Theorem (Thomason), Theorems A and B (Quillen), or the Homotopy Cofinality Theorem (Hirschhorn).
We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirm…
The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces.…
On the Cohomology of the Total Space of Classifying Space for Commutativity in $U(3)$
本文将 $E_{com} U(3)$ 描述为空间图表的同伦余极限，利用相关谱序列计算了其模 $2$ 和模 $3$ 上同调，并给出了有理上同调的环结构。
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…
We describe several equivalent models for the infinity-category of infinity-local systems of chain complexes over a space using the framework of quasi-categories. We prove that the given models are equivalent as infinity-categories by exploiting the relationship between the differential graded nerve functor and the cobar construction. We use one of these models to calculate the quasi-categorical colimit of an infinity-local system in terms of a twisted tensor product.
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an étale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical extended and generalized powers, which refine the categorical versions of these constructions as special cases. We also offer more computationally tractable models of these constructions using equivariant motivic homotopy theory. This is the first in a series of papers on power operations in motivic stable homotopy theory.
We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundle…
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…
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。
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…
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。
In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. Our key result says that if a category admits a model approximation then so does any diagram category with values in this category. From the homotopy theoretical point of view categories with model approximations have similar properties to those of model categories. They admit homotopy categories (localizations with respect to weak equivalences). They als…
How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds -- the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.
For $\Cc$ a $G$-category, we give a condition on a diagram of simplicial sets indexed on $\Cc$ that allows us to define a natural $G$-action on its homotopy colimit, and in some other simplicial sets and categories defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to an equivariant version.
We show that the classifying space functor $B: Mon \to Top*$ from the category of topological monoids to the category of based spaces is left adjoint to the Moore loop space functor $Ω': Top*\to Mon$ after we have localized $Mon$ with respect to all homomorphisms whose underlying maps are homotopy equivalences and $Top*$ with respect to all based maps which are (not necessarily based) homotopy equivalences. It is well-known that this localization of $Top*$ exists, and we show that the localization of $Mon$ is the category of monoids and homotopy classes of homotopy homomorphisms. To make this …
We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as …
To every homotopy n-nilpotent group, defined in earlier work by Dwyer and the author, we associate an endofunctor of pointed spaces and prove that it is looped and n-excisive. As a tool we prove that $ΩP_n({\rm id})$ commutes with sifted colimits of connected spaces.
In this paper we prove that various quasi-categories whose objects are $\infty$-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit…
Let $G$ be a compact connected Lie group and let $ξ,ν$ be complex vector bundles over the classifying space $BG$. The problem we consider is whether $ξ$ contains a subbundle which is isomorphic to $ν$. The necessary condition is that for every prime $p$ the restriction $ξ|_{BN_p^G}$, where $N_p^G$ is a maximal $p$-toral subgroup of $G$, contains a subbundle isomorphic to $ν|_{BN_p^G}$. We provide a criterion when this condition is sufficient, expressed in terms of $Λ^*$-functors of Jackowski, McClure \& Oliver and we prove that this criterion applies if $ν$ is a universal bundle over $BU(n)$. …
This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate `nerves of C' are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason's `Homotopy Colimit Theorem' to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying spa…
Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on hom…
In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.
Given a strong homotopy pushout cube of spaces A, we measure how far it is from also being a homotopy pullback cube. Explicitly, letting P be the homotopy colimit of the diagram obtained from A by forgetting the initial vertex $A_\varnothing$, we study the homotopy fibre of the double suspension of the comparison map $A_\varnothing \to P$. This difference is expressible in terms of the homotopy fibres of the original maps in A.
Detecting isomorphisms in the homotopy category
该研究通过构造大对称群反例证明非基点空间同伦范畴无共同反射同构的对象集，同时指出同伦2-范畴中球面可共同反射等价，并证伪了Heller论证所依赖的超限序列图弱余极限相关命题。
We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery.
In this article we consider a space B_{com}G assembled from commuting elements in a Lie group G first defined in [Adem, Cohen, Torres-Giese 2012]. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that ZxB_{com}U is a loop space and define a notion of commutative K-theory for bundles over a finite complex X which is isomorphic to [X,ZxB_{com}U]. We compute the rational cohomology of B_{com}G for G equal to any of the classical groups U(n), SU(n) and Sp(n), and exhibit the rat…
Let $\mathcal C$ be a $\mathcal V$-enriched model category. We say that an object $x$ of $\mathcal C$ is homotopy tiny if the total right derived functor of $\mathcal C(x, -) : \mathcal{C} \rightarrow {\mathcal V}$ preserves homotopy weighted colimits. Let $\mathcal C_0$ be a full subcategory of $\mathcal C$ all of whose objects are homotopy tiny. Our main result says that the homotopy category of the category generated by $\mathcal C_0$ under weak equivalences and homotopy weighted colimits is equivalent to the homotopy category of the category $\mathcal V^{\mathcal C_0^{op}}$ of $\mathcal V$…
A Dwyer-Rezk classification for polynomial functors in Weiss calculus
本文通过不同方法证明了Weiss微分学中多项式函子的分类，并给出了同质函子分类的新证明。
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.
This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's theory of the localization of spaces and spectra. 3) Homotopy limits and colimits and their applications. 4) The James construction, symmetric products, and the Dold$-$Thom theorem. 5) Brown and Adams representability in the setting of triangulated categories 6) Operads and the May$-$Thomason theorem on the uniqueness of infinite loop space machines. 7) The plus …
We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime. As an application, we show that the sp…
In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the pr…
Cubical approximation for directed topology I
本文研究有向拓扑中的立方逼近理论，证明了立方集和有向空间的几何实现诱导等价，并探讨了有向映射同伦关系的相等条件。
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on…
The goal of this article is to extend a theorem of Lurie \[ \mathsf{Sh}_A (X) = \mathsf{Fun}(\mathsf{Exit}_A (X), \mathsf{S}) \] representing constructible sheaves with values in $ \mathsf{S} $, the $ \infty $-category of spaces, on a stratified space $ X $ with poset of strata $ A $, as functors from the exit paths $ \infty $-category $ \mathsf{Exit}_A (X) $ to $ \mathsf{S} $. Lurie's representation theorem works provided $ A $ satisfy the ascending chain condition. This typically rules out infinite dimensional examples of stratified space. Building on it and with the help of a stratified hom…
We study the mod-$\ell$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z}, G)$, at a prime $\ell$. We show that the mod-$\ell$ homology of $B(\mathbb{Z}, G)$ depends on the mod-$\ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\ell$ homology isomorphism $BG \to BH$ for such groups induces a mod-$\ell$ homology isomorphism $B(\mathbb{Z}, G) \to B(\mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an …
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。
In this paper, we show that for reduced homotopy endofunctors of spaces, F, and for all $n \geq 1$ there are adjoint functors $R_n, L_n$ with $T_n F \simeq R_n F L_n$, where $P_n F$ is the $n$-excisive approximation to $F$, constructed by taking the homotopy colimit over iterations of $T_n F$. This then endows $T_n$ of the identity with the structure of a monad and the $T_n F$'s are the functor version of bimodules over that monad. It follows that each $T_n F$ (and $P_nF$) takes values in spaces of symmetric Lusternik-Schnirelman cocategory $n$, as defined by Hopkins. This also recovers recent…
Nonstandard diffeology and generalized functions
本文引入非标准微分几何空间范畴，用于研究广义函数，实现分布的乘法与复合，并证明同伦扩张性质。
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。
Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations. This allows for 'directed homotopies', generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the 'fundamental category' of a d-space is introduced and a 'Seifert…
This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it st…
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