Sylvain Douteau
We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filtered CW-complexes, providing a nice model for the homotopy category of stratified spaces.
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By careful analysis of the comparison map from a simplicial set to its image under Kan's ex-infinity functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we are able to quickly deduce some basic facts about ex-infinity and hence provide a new construction of the Kan-Quillen model structure on simplicial sets, which avoids the use of topological spaces or minimal fibrations.
A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces involves invariants such as intersection cohomology which are only invariants under stratification-preserving homotopies. In this thesis, we study the homotopy theory of stratified spaces with respect to those stratified homotopies. To do so, we construct model categories for stratified spaces and we introduce new invariants to characterize them, the filtered homo…
We present two new proofs of Simon Henry's result that the category of simplicial sets admits a constructive counterpart of the classical Kan-Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial arguments on anodyne extensions. We also give new constructive proofs of the left and right properness of the model structure.
We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization.
This brief note elaborates on a result by Gepner and Henriques. They have shown that a Dwyer-Kan equivalence between two small, topological categories gives rise to a Quillen equivalence of the associated categories of topologically enriched presheaves. We present a more detailed account of their proof.
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak-equivalences are characterized by stratified homotopy groups.
Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras an…
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…
There are two main approaches to the problem of realizing a $Π$-algebra (a graded group $Λ$ equipped with an action of the primary homotopy operations) as the homotopy groups of a space $X$. Both involve trying to realize an algebraic free simplicial resolution $G_\bullet$ of $Λ$ by a simplicial space $W_\bullet$ and proceed by induction on the simplicial dimension. The first provides a sequence of André-Quillen cohomology classes in $H_{AQ}^{n+2}(Λ;Ω^{n}Λ)$ for $n \geq 1$ as obstructions to the existence of successive Postnikov sections for $W_\bullet$ by work of Dwyer, Kan and Stover. The se…
Enriched model categories and the Dold-Kan correspondence
研究通过Dold-Kan对应改变富集结构对模型范畴性质的影响，揭示保留在弱单向Quillen对下的属性变化。
A Lie characterization of the Bousfield-Kan ${\mathbb{Q}}$-completion and ${\mathbb{Q}}$-good spaces
本文证明由模型和实现函子给出的Quillen对的单位，在同伦意义下即为Bousfield-Kan Q-完备，从而给出了Q-完备和Q-好空间的Lie刻画。
Combinatorial models for stratified homotopy theory
本文证明分层单纯集上存在组合单纯模型结构，其一与Joyal模型结构的左Bousfield局部化Quillen等价，用于研究分层同伦假设。
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
A variant of a Dwyer-Kan theorem for model categories
本文证明了单纯组合模型范畴在小函子范畴上的同伦模型结构存在性或其替代方案，并以此推广了关于同伦函子范畴 Quillen 等价的 Dwyer-Kan 定理，同时给出了 Goodwillie calculus 的应用。
On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, t…
We define a notion of groupoidal 2-quasi-categories and show that they are the fibrant objects of a model structure on the category of $Θ_2$-sets. We show that this model category is Quillen equivalent to the Kan-Quillen model category of simplicial sets and that 2-truncated groupoidal 2-quasi-categories are models for homotopy 2-types.
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential g…
We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction.
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…
Presenting the topological stratified homotopy hypothesis
本文证明分层拓扑空间上存在呈现三种分层同伦理论的（半）模型结构，并利用Lurie函子建立Quillen等价，从而给出分层同伦假说的模型范畴解释。
Formal weakly enriched category theory
本文构造了编码弱充实的形式范畴论，证明其与拟范畴的无穷宇宙等价，并导出充实版Quillen定理A。
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 …
We introduce a notion of "weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantages of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems f…
A homotopical Dold-Kan correspondence for Joyal's category $Θ$ and other test categories
本文通过引入模型结构的Quillen等价，证明了在特定兼容条件下，测试范畴上的abelian预示子范畴的同调范畴与abelian群的非负导出范畴等价，并应用于Joyal的范畴Θ。
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick--Kan model structure.
André-Quillen Cohomology and $k$-invariants of simplicial categories
本文利用 Dwyer-Kan-Smith 上同调的解释，构建了单纯范畴 $k$-不变量取值的上链复形，并基于同伦余极限分解研究了映射提升至 Postnikov section 的障碍，从而恢复高阶同伦不变量。
Relative elegance and cartesian cubes with one connection
本文在带一种连接的笛卡尔立方体集合上建立Quillen等价，得到第二个同时构造性模拟同伦类型论并呈现无穷群胚的模型结构，首个实例为等变笛卡尔模型。
In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow up of this work. We show that when X is a finite connected simplicial set, then the realization of the model of X is the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on the category of complete differential graded algebras making the two previo…
There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan's Ex functor act on graphs and on nerves of small categories.
If $\mathscr{M}$ is a model category and $\mathcal{U}: \mathscr{A} \rightarrow \mathscr{M}$ is a functor, we defined a Quillen-Segal $\mathcal{U}$-object as a weak equivalence $\mathscr{F}: s(\mathscr{F}) \xrightarrow{\sim} t(\mathscr{F})$ such that $t(\mathscr{F})=\mathcal{U}(b)$ for some $b\in \mathscr{A}$. If $\mathcal{U}$ is the nerve functor $\mathcal{U}: \mathbf{Cat} \rightarrow \mathbf{sSet}_J$, with the Joyal model structure on $\mathbf{sSet}$, then studying the comma category $(\mathbf{sSet}_J \downarrow \mathcal{U})$ leads naturally to concepts, such as Lurie's $\infty$-operad. It al…
A modern perspective on rational homotopy theory
本文通过现代方法扩展了Quillen的模型结构，将理性同调等价和理性空间的概念推广至更广泛的空间范畴。
We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.
Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (…
We generalize Quillen's $F$-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over $C^*(B\mathcal{G},\mathbb{F}_p)$ is stratified and costratified for a large class of $p$-local compact groups $\mathcal{G}$ including compact Lie groups, connected $p$-compact groups, and $p$-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, w…
Profinite completions of topological operads
本文将拓扑算畴的特定有限完备纳入左 Quillen 函子框架，构建了有限同伦算畴的模型范畴，刻画了其底层 $\infty$-范畴与弱等价，并在 Quillen 等价背景下获得了类似结果。
We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model categories provided by the Dwyer-Kan hammock localization. As one particular application, we are able to associate to every object of a stable, combinatorial model category a "homotopy endomorphism ring spectrum". The homotopy types of these ring spectra are preserved by Quillen equivalences, and so are an invariant for stable model categories.
Generalizing a result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding categories of algebras.
In this paper, we show that the Thomason model structure restricts to a Quillen equivalent cofibrantly generated model structure on the category of acyclic categories, whose generating cofibrations are the same as those generating the Thomason model structure. To understand the Thomason model structure, we need to have a closer look at the (barycentric) subdivision endofunctor on the category of simplicial sets. This functor has a well known right adjoint, called Kan's Ex functor. Taking the subdivision twice and then the fundamental category yields a left adjoint of an adjunction between the …
Homotopical recognition of diagram categories
本文通过引入同伦原子概念，刻画了模型范畴与预层范畴Quillen等价的判定条件，并将其应用于分类从有限带点单纯集到谱的多项式函子，推广了Dwyer-Kan轨道模型结构等经典结果。
Diagonal Lemma for Presheaves on Eilenberg-Zilber Categories
本文研究了Eilenberg-Zilber范畴上的对角引理，证明了在Kan-Quillen模型结构下，双层简复形的逐层弱等价可诱导对角简复形的弱等价。
In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natura…
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。
The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^Δ$ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of $K$ to $DG$-rings can be equipped with an associative product and that the resulting functor $DGR^*\to\ass^Δ$, although not itself an equivalence, does induce one at the level of homotopy categories. The dual of this result for chain $DG$ and simplicial rings was obtained independently by S. Schwede and B. Shipley through different method…
For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty$-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in general.…
Arboreal Objects and Their Homotopy Theory
本文构建了树状扩展的范畴OrdFor，并展示其与半单纯同调的关系，通过右伴随扩展得到同伦等价。
The equivariant model structure on cartesian cubical sets
本文基于笛卡尔立方范畴的预层发展了构造性同调类型论模型，通过引入等变条件改进立方Kan纤维化，主要贡献是用计算机证明助手形式化了关键技术结果。
This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) omega-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict omega-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to s…
A new model structure on dendroidal spaces for the theory of $\infty$-operads
本文提出了一种新的 dendroidal 空间模型结构，用于 $\infty$-操作符理论，并证明其与 Cisinski-Moerdijk 模型结构的 Quillen 等价性。
The Yoneda embedding in simplicial type theory
本文在 simplicial 类型理论中构建 Yoneda 入射，发展预层范畴理论，并证明 Quillen 定理 A。
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends to a monad on the homotopy category and defines a category of homotopy $T$-algebras. In this setting there is a forgetful functor from the homotopy category of $T$-algebras to the category of homotopy $T$-algebras. Under suitable hypotheses we provide an obstruction theory, in t…
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