Related papers: Higher stacks as diagrams
We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the…
We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological…
We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…
The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features…
Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or…
If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…
We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical…
We introduce the notion of a higher covering diagram in a base $\infty$-category $\mathcal{C}$. The theory of higher covering diagrams in $\mathcal{C}$ will be shown to recover various descent conditions known from the $\infty$-categorical…
We give a systematic construction of semiorthogonal decompositions of derived categories of coherent sheaves on quasi-smooth derived algebraic stacks over $\mathbb{C}$, where the summands are subcategories defined by weight conditions, and…
In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can…
It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory.…
We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a…
In \cite{CompTheo} we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy…
We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial…
We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is…
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to…
We generalize and clarify Gerstenhaber and Schack's "Special Cohomology Comparison Theorem". More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category U and the…
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for…