Explicit fibrant replacement for discrete G-spectra

对称谱上稳定模型结构的一个简洁证明

A concise proof of the stable model structure on symmetric spectra

利用纤维转移定理，在纤维对象间转移稳定模型结构，给出对称谱上稳定模型结构存在性的简洁概念性证明。

The choice of cofibrations of higher dimensional transition systems

It is proved that there exists a left determined model structure of weak transition systems with respect to the class of monomorphisms and that it restricts to left determined model structures on cubical and regular transition systems. Then it is proved that, in these three model structures, for any higher dimensional transition system containing at least one transition, the fibrant replacement contains a transition between each pair of states. This means that the fibrant replacement functor does not preserve the causal structure. As a conclusion, we explain why working with star-shaped transi…

Homotopy theory of spectral sequences

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.

Continuous functors as a model for the equivariant stable homotopy category

In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category W_G of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of Mandell, May, Schwede, and Shipley, we show that there is a "stable model structure" on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second "absolute stable model structure" which is Quillen equivalent to the "stable model structure". Our main result is a concrete identification of the…

Transfer of algebras over operads along derived Quillen adjunctions

Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectr…

Cofibrance and Completion

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple…

动机Thom谱的纤维状消解

Fibrant resolutions for motivic Thom spectra

本研究利用Voevodsky的框架对应理论与框架动机工具，构造动机Thom谱的多种显式纤维状消解，证明相关双谱的表示性，并给出拓扑应用及代数配边谱的计算。

Algebraic Kasparov K-theory. II

A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the $S^1$-spectrum and $(S^1,\mathbb G)$-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is the introduction and study of the symmetric monoidal compactly generated triangulated category of $K$-motives. It i…

The E_2-term of the descent spectral sequence for continuous G-spectra

Let {X_i} be a tower of discrete G-spectra, each of which is fibrant as a spectrum, so that X=holim_i X_i is a continuous G-spectrum, with homotopy fixed point spectrum X^{hG}. The E_2-term of the descent spectral sequence for π_*(X^{hG}) cannot always be expressed as continuous cohomology. However, we show that the E_2-term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in lim_i M_i, where {M_i} is a tower of discrete G-modules.

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。

Partial model categories and their simplicial nerves

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus. We then show that Charles Rezk's result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories. We also note that conversely ever…

函子的弱和强纤维化

Weak and Strong Fibrations of Functors

本文提出了一种针对小范畴的同调框架，扩展了代数拓扑的经典不变量到范畴设置。通过构造真正的路径范畴，定义了函子的强和弱纤维化，并引入了纤维化替代和扩展了如Svarc genus和sectional category等同调不变量到小范畴。最后，将该框架应用于小范畴中的运动规划，提供了Farber拓扑复杂度的范畴类比，并消除了现有方法中常见的有限性约束。

Duality and small functors

The homotopy theory of small functors is a useful tool for studying various questions in homotopy theory. In this paper, we develop the homotopy theory of small functors from spectra to spectra, and study its interplay with Spanier-Whitehead duality and enriched representability in the dual category of spectra. We note that the Spanier-Whitehead duality functor $D\colon \mathrm{Sp}\rightarrow \mathrm{Sp}^{\mathrm{op}}$ factors through the category of small functors from spectra to spectra and construct a new model structure on the category of small functors, which is Quillen equivalent to $\ma…

单纯幺半群模型范畴的可容许替换

Admissible replacements for simplicial monoidal model categories

本文利用Dugger万有模型范畴构造，为单纯与组合对称幺半群模型范畴构造了具有更好operadic性质的替换，并证明稳定情形下当幺半群单位是紧生成元时，此类范畴可由交换环谱分类，且与唯一确定的交换环谱上的模强幺半群Quillen等价。

通过线性预层的谱序列

Spectral sequences via linear presheaves

本文研究谱序列的同调理论，引入扩展谱序列并证明其双完备性，同时定义多种模型结构，揭示谱序列作为∞范畴的模型。