DG-category and simplicial bar complex

Model topoi and motivic homotopy theory

Given a small simplicial category $\C$ whose underlying ordinary category is equipped with a Grothendieck topology $τ$, we construct a model structure on the category of simplicially enriched presheaves on $\C$ where the weak equivalences are the local weak equivalences of the underlying (non-enriched) simplicial presheaves. We show that this model category is a $t$-complete model topos and describe the Grothendieck topology $[τ]$ on the homotopy category of $\C$ that corresponds to this model topos. After we first review a proof showing that the motivic homotopy theory is not a model topos, w…

A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [HaSc] for the study of obstructions t…

关于$n$ -范畴和$n$ -广群的非严格概念

On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets

在本文中，我们首先给出了一个简单的方法来定义一个非严格的$n$ -范畴，我们称之为一个$n$ -神经的想法，一个范畴可以解释为一个单纯集，我们证明了我们的建设推广的情况下，通常的非严格2-范畴。其次，我们给出了非严格$n$ -广群的一个单纯定义。然后，我们将任何空间$X$关联到一个$n$ -广群$Π_{_{n}}(X)$，它推广了著名的Poincaré广群$Π_{_{1}}(X)$，并体现了$X$的$n$ -截断同伦类型.我们还给出了$n$ -广群胚的几何实现的自然构造，并猜想从$n$ -截断拓扑空间范畴到$n$ -广群胚范畴$n$ -Gr，函子几何实现是函子$Π_{_{n}}(\ )$的逆等价.

Quasi-coherent sheaves in differential geometry

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particular, weak equivalences between these monoids induce Quillen equivalences between the corresponding categories of modules. On the other hand, it is also proved that the functor of pre-compact bornology applied to simplicial $C^\infty$-rings preserves and reflects weak equivalences…

Homotopy of profinite groups

We study simplicial profinite groups with a view towards applications in profinite combinatorial group theory. This approach provides a natural framework to the concept of pro-$\mathfrak{C}$-presentation of a pro-$\mathfrak{C}$-group $G$ as a 1-truncation of its free simplicial pro-$\mathfrak{C}$-resolution. The category of simplicial pro-$\mathfrak{C}$-groups has a closed simplicial model category structure. This yields a possibility to define some old and new derived functors as left Quillen derived functors from this simplicial model category. When $\mathfrak{C}$ is L-groups, than may const…

Twisted complexes and simplicial homotopies

In this paper we consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map $f: (\mathcal{U},\mathcal{R})\to (\mathcal{V}, \mathcal{S})$ between simplicial ringed spaces induces a dg-functor $f^*: \text{Tw}(\mathcal{V}, \mathcal{S})\to \text{Tw}(\mathcal{U}, \mathcal{R})$ where $\text{Tw}(\mathcal{U}, \mathcal{R})$ denotes the dg-category of twisted complexes on $(\mathcal{U},\mathcal{R})$. In this paper we prove that for simplicial homotopic maps $f$ and $g$, there exists an $A_{\infty}$-natural transformation $Φ: f^*\Rightarrow g^*$ between…

Exodromy for stacks

In this short note we extend the Exodromy Theorem of arXiv:1807.03281 to a large class of stacks and higher stacks. We accomplish this by extending the Galois category construction to simplicial schemes. We also deduce that the nerve of the Galois category of a simplicial scheme is equivalent to its étale topological type in the sense of Friedlander.

关于K理论和S范畴的一个注记

A remark on K-theory and S-categories

现在众所周知，K理论的瓦尔德豪森类别取决于不仅仅是它的（三角）同伦类别（见[Schlichting]）。本文的目的是证明（好）Waldhausen范畴的K理论谱完全由它的Dwyer-Kan单纯局部化决定，而不需要任何附加结构.由于单纯局部化是同伦范畴的一个改进版本，它也决定了三角结构，我们的结果是对一般问题的一个可能的回答："在多大程度上$K$ -理论不是三角导出范畴的不变量？''

Equivariant Functors and Sheaves

In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group $G$ acting on a variety $X$ are in fact equivalent. In the first part of this thesis we introduce and study equivariant categories on a quasi-projective variety $X$. These are a generalization of the equivariant derived category of Lusztig and are indexed by certain pseudofunctors that take values in the 2-category of categories. This 2-categorical generalization allow us to prove rigorously and carefully when such categories are addi…