Daniel Dugger
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.
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We give an explicit way of calculating the set of homotopy classes of morphisms from a Tamsamani n-category A to another one B. This calculation uses a Reedy-cofibrant cosimplicial resolution of A, using a new notion of ``free cofibration'' of n-precats. The free cofibrations of n-precats seem to be the analogue for n-categories of the Bousfield-Kan cofibrations in the theory of diagrams.
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of …
Classification diagrams of simplicial categories
本文研究相对单纯范畴所诱导相对 ∞-范畴的分类图，证明其等价于逐层神经，并据此联系同伦一致神经及单纯范畴的逐层局部化。
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categorie…
This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace higher homotopy groups in quasi-category homotopy theory.
Dimitri Ara's 2-quasi-categories, which are certain presheaves over André Joyal's 2-cell category $Θ_2$, are an example of a concrete model that realises the abstract notion of $(\infty,2)$-category. In this paper, we prove that the 2-quasi-categories and the fibrations into them can be characterised using the inner horn inclusions and the equivalence extensions introduced by David Oury. These maps are more tractable than the maps that Ara originally used and therefore our result can serve as a combinatorial foundation for the study of 2-quasi-categories.
Fully Triangulated Categories
本文修订三角化范畴公理，引入高阶三角形及其 distinguished 态射，用有向截断单形给出几何表述，并证明 3×3 引理等关键定理。
Pushouts of Dwyer maps are $(\infty,1)$-categorical
本文证明在特定函子类下，1-范畴的推出在包含进 $(\infty,1)$-范畴时得以保持，并将 Thomason 的弱同伦等价改进为弱范畴等价，同时推广了该函子类。
Duality pairs, phantom maps, and definability in triangulated categories
本文在紧生成三角范畴中定义对偶三元组和对偶对，通过幻影映射的新刻画研究类的纯子对象、纯商和纯扩张封闭性，并引入公理化的Auslander-Gruson-Jensen对偶以定义对偶可定义范畴，应用于倾斜理论和分层张量三角范畴。
A classification is provided of functors, in particular polynomial ones, from a category with a zero object in which every object is a finite sum of copies of a generating object, into an abelian category. This classification is extended to include functors from a category with sums and a zero object, carrying a small, regular projective generator.
In this paper, we study three relative LS categories of a map and study some of their properties. Then we introduce the `higher topological complexity' and `weak higher topological complexity' of a map. Each of them are homotopy invariants. We discuss some lower and upper bounds of these in invariants and compare them with previously known `topological complexities' of a map.
Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -- describing ho…
Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants. We also assign …
Homotopy types of topological stacks of categories
本文将奎伦定理A推广到拓扑空间内范畴，证明全忠实本质满函子诱导分类空间同伦等价，并由此为范畴的拓扑栈构造2函子性同伦类型，推广了Noohi和Ebert关于群胚拓扑栈的结果。
We give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension $p^3$ for $p$ any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension $p^3$ and we determine which of these equivalence classes have equivalent categories of modules.
In this paper we give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules. This classificaction permits to classify the equivalence classes of braded tensor equivalences of twisted Drinfeld doubles of finite groups of order 8.
Classification of fully dualizable linear categories
本文在全可对偶线性范畴的分类研究中，证明了全可对偶的 $R$-线性余完备范畴等价于有限 étale 代数模范畴的 gerbe 扭曲，并将结果推广至任意交换环、分次范畴及 $\infty$-范畴。
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…
Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces. This model category is not cofibrantly generated. We give conditions on a class of maps which…
A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `r…
Convexity in tensor triangular geometry
本文对上同调分层的刚性紧生成tt-$\infty$-范畴中的对偶化局部化理想进行分类，证明其对应于Balmer谱的凸子集，推广了Efimov关于交换noetherian环导出范畴的定理。
A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ are tensor equivalent. We use the Lyndon-Hochschild-Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result may permit to classify the e…
We study representations of wreath product analogues of categories of finite sets. This includes the category of finite sets and injections (studied by Church, Ellenberg, and Farb) and the opposite of the category of finite sets and surjections (studied by the authors in previous work). We prove noetherian properties for the injective version when the group in question is polycyclic-by-finite and use it to deduce general twisted homological stability results for such wreath products and indicate some applications to representation stability. We introduce a new class of formal languages (quasi-…
We construct an unwrapped Floer theory for bundles of Liouville sectors. In particular, we construct a compatible collection of unwrapped Fukaya categories of fibers of a Liouville bundle, and prove that the two natural constructions of continuation maps in this setting behave compatibly. These constructions are exploited in [OT19] to construct homotopically coherent actions of Lie groups on wrapped Fukaya categories, thereby proving a conjecture from Teleman's 2014 ICM address.
Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds
本文引入融合2-范畴的3-群分次扩张，构造了4维流形的非平凡量子同伦不变量，推广了Douglas-Reutter不变量，且在该语境下推广了Pachner定理。
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