A Convenient Category for The Homotopy Theory of Concurrency

Homotopy theory in a quasi-abelian category

We prove that the category of dg-modules and dg-algebras in a Grothendieck quasi-abelian category are endowed with a Quillen model structure. This allows some flexibility in setting up a theory of derived algebraic geometry in the infinite dimensional setting. For example, the category of complete bornological vector spaces, or equivalently, convenient vector spaces, is a Grothendieck quasi-abelian category. Closely related is the Grothendieck quasi-abelian category of ind-Banach spaces whose associated model category is shown to be Quillen equivalent. Applications include the Chevalley-Eilenb…

Chromatic fracture cubes

In this note, we construct a general form of the chromatic fracture cube, using a convenient characterization of the total homotopy fiber, and deduce a decomposition of the E(n)-local stable homotopy category.

笛卡尔闭范畴中的朴素同伦理论

Naive homotopy theories in cartesian closed categories

本文在笛卡尔闭范畴中利用保持有限积的自函子和满射自然变换构造朴素同伦范畴，并在预内聚2-值topos条件下用显式同伦给出其与经典同伦理论的联系。

Equivariant Motivic Homotopy Theory

In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by Voevodsky and Hu-Kriz-Ormsby. We show that it allows to detect equivariant motivic weak equivalences on fixed points and how this property leads to a topologically convenient behavior of stable equivalences. We also prove a negative result concerning descent for equivariant algebraic K-theory.

Homotopy Theory for C^{*}-algebras

Category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C^{*}-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C^{*}-algebras, KK-theory and E-theory, as all of these can be expressed as the homotopy category of a category of fibrant objects.

A convenient category of locally preordered 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.

On the homotopy groups of symmetric spectra

The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model structure. About the only disadvantage of this model is that the stable equivalences cannot be defined by inverting those morphisms which induce isomorphisms on homotopy groups, because this would leave too many homotopy types. In this sense the naively defined homotopy groups are often `wrong`, and then their precise relationship to the `true` homotopy groups (i.e.…

一个方便的参数化谱范畴

A convenient category of parametrized spectra

本文描述了一个“方便的”参数化正交谱点集范畴及其模型结构，将复杂的同伦范畴陈述简化为点集层面的直接断言，并借此直接构造了参数化谱的双范畴，在具体指标理论与形式双范畴理论间架起桥梁。

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…

On very effective hermitian $K$-theory

We argue that the very effective cover of hermitian $K$-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological $K$-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.

关于截面类别和拓扑复杂性的近似值

Up to one approximations of sectional category and topological complexity

本文在统一框架下研究了James的截面类别和Farber的拓扑复杂性，引入了映射的相对和强相对类别概念，证明两者与截面类别相差不超过1，并利用这些工具确定了Ganea纤维的截面和相对类别。