Manfred Stelzer
A group completion functor Q is constructed in the category of algebras in simplicial sets over a cofibrant E_n-operad M. It is shown that Q defines a Bousfield-Friedander simplicial model category on M-algebras.
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Derived $\infty$-categories as exact completions
本文发展了正则 $\infty$-范畴的正合完备化理论，证明其可恢复阿贝尔范畴导出 $\infty$-范畴的连通部分，并扩展了阿贝尔范畴的经典刻画及建立了相关定理的 $\infty$-范畴版本。
For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages.
We prove that the second homology group of the completion of an infinite dimensional free Lie algebra is uncountable.
This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the derived $\infty$-category of a commutative ring, before turning to our main example, the $\infty$-category of spectra. We then go on to consider ring spectra and their $\infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and …
$\infty$-categorical group quotients via skew group algebras
研究通过斜群代数构造∞-范畴群商的方法，探讨dg-范畴和稳定∞-范畴的群商关系及斜群代数的应用。
It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomor…
These are extended notes for a series of lectures on Hall algebras given at the CRM Barcelona in February 2015. The basic idea of the theory of Hall algebras is that the collection of flags in an exact category encodes an associative multiplication law. While introduced by Steinitz and Hall for the category of abelian p-groups, it has since become clear that the original construction can be applied in much greater generality and admits numerous useful variations. These notes focus on higher categorical aspects based on the relation between Hall algebras and Waldhausen's S-construction.
The Algebra of Categorical Spectra
本文为范畴谱建立基础，构造其对应松弛Gray张量积稳定类比的张量积，用于研究稳定现象，并严谨推导带奇异性的配边假说。
A higher categorical approach to graded bivariant $K$-theories for Real $C^*$-algebras
该研究针对局部紧第二可数群G作用下的分次实C*代数，给出经典KK理论与E理论的∞范畴细化，得到连接K理论谱的纤维序列，分析K0群自同态并解答民间问题，证明两类范畴非分次C*代数的Dwyer-Kan局部化。
Categorical structures of Kuranishi spaces with $L_{\infty}[1]$-algebras
本文通过将每个图表与 Kuranishi 截面零点邻域上的 $L_{\infty}[1]$-代数关联，引入 $L_{\infty}$-Kuranishi 空间，并证明它们构成一个范畴，光滑流形范畴自然嵌入其中。
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…
Derived D-Geometry is considered as a convenient language for a coordinate-free investigation of nonlinear partial differential equations up to symmetries. One of the first issues one meets in the functor of points approach to derived D-Geometry, is the question of a model structure on the category C of differential non-negatively graded quasi-coherent commutative algebras over the sheaf D of differential operators of an appropriate underlying variety. In [BPP15a], we described a cofibrantly generated model structure on C via the definition of its weak equivalences and its fibrations. In the p…
Let $ A $ be a finite connected graded cocommutative Hopf algebra over a field $ k $. There is an endofunctor $ \mathsf{tw} $ on the stable module category $ \mathrm{StMod}_A $ of $ A $ which twists the grading by $ 1 $. We show the categorical entropy of $ \mathsf{tw} $ is zero. We provide a lower bound for the categorical polynomial entropy of $ \mathsf{tw} $ in terms of the Krull dimension of the cohomology of $ A $, and an upper bound in terms of the existence of finite resolutions of $ A $-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of…
Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation
本文通过单纯对象、带乘法的余操作子和 Feynman 范畴，将数论、数学物理与代数拓扑中三个不同的霍普夫代数统一于一般范畴论框架，为新的构造与重新解释奠定基础。
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