$\mathcal{E}_\infty$ ring spectra and elements of Hopf invariant $1$

On a notion of homotopy Segal $ E_\infty $-Hopf cooperad

We define a notion of homotopy Segal cooperad in the category of $ E_\infty $-algebras. This model of Segal cooperad that we define in the paper, which we call homotopy Segal $ E_\infty $-Hopf cooperad, covers examples given by the cochain complex of topological operads and provides a framework for the study of the homotopy of such objects. In a first step, we consider a category of Segal $ E_\infty $-Hopf cooperads, which consists of collections of $ E_\infty $-algebras indexed by trees and equipped with coproduct operators, corresponding to tree morphisms, together with facet operators, corr…

The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra

The mod 2 Steenrod algebra $\mathcal{A}_2$ can be defined as the quotient of the mod 2 Leibniz--Hopf algebra $\mathcal{F}_2$ by the Adem relations. Dually, the mod 2 dual Steenrod algebra $\mathcal{A}_2^*$ can be thought of as a sub-Hopf algebra of the mod 2 dual Leibniz--Hopf algebra $\mathcal{F}_2^*$. We study $\mathcal{A}_2^*$ and $\mathcal{F}_2^*$ from this viewpoint and give generalisations of some classical results in the literature.

有限 Adams Hopf 代数胚的稳定 Picard 群及其在 $\mathbb{R}$-动机 Steenrod 子代数 $\mathcal{A}(1)^{\mathbb{R}}$ 上的应用

The stable Picard group of finite Adams Hopf algebroids with an application to the $\mathbb{R}$-motivic Steenrod subalgebra $\mathcal{A}(1)^{\mathbb{R}}$

本文研究有限Adams Hopf代数胚的稳定余模范畴刚性，揭示其Picard群，并通过基变换建立约化过程，计算R-动机模2 Steenrod子代数A(1)^R的Picard群，证明其同构于Z^4。

A_{\infty}-method in Lusternik-Schnirelmann category

To clarify the method behind the paper "Ganea's conjecture on Lusternik-Schnirelman category" by the author, a generalisation of Berstein-Hilton Hopf invariants is defined as `higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of "Ganea's ..." for all the cases except the case when …

一个导出的 Milnor-Moore 定理

A derived Milnor-Moore theorem

该文通过 Koszul 对偶构造包络 Hopf 代数函子，证明其在有理范畴全忠实，并在色同伦论中将其单位与 Goodwillie 完备化等同，推广了 Milnor-Moore 定理。

The stable Picard group of Hopf algebras via descent, and an application

Let $A$ be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over $A$. The starting point is the observation that the stable category of $A$-modules can be reconstructed, as an $\infty$-category, as the totalization of a cosimplicial $\infty$-category whose layers are related to the stable categories of modules over the quasi-elementary sub-Hopf-algebras of $A$. This leads to a spectral sequence computing the Picard group which…

Hodge theory for elliptic curves and the Hopf element $ν$

We show that the vector bundle on the moduli stack $M_\mathrm{ell}$ of elliptic curves associated to the $2$-cell complex $Cν$ is isomorphic to the de Rham cohomology sheaf $\mathrm{H}^1_\mathrm{dR}(\mathcal{E}/M_\mathrm{ell})$ of the universal elliptic curve $\mathcal{E}\to M_\mathrm{ell}$. We use this to calculate the homotopy groups of the $\mathbf{E}_{1}$-quotient $\mathrm{tmf} /\!\!/ ν$ of $\mathrm{tmf}$ by $ν$, called the spectrum of "topological quasimodular forms", by relating its Adams-Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitt…

The geometric Hopf invariant and surgery theory

The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$ up to homotopy. The explicit nature of the construction allows for a $π$-equivariant version of $h(F)$ in the case of a $π$-equivariant $F$, with $π$ a discrete group. In earlier joint work we applied the $π_1(N)$-equi…

如何构建一个 Hopf 代数

How to build a Hopf algebra

构造了一个从满足附加性条件的 $(\infty,3)$-范畴收缩到辫子幺半 $(\infty,1)$-范畴中 Hopf 代数的函子，为 Tannaka 重建提供了高阶范畴论解释。

Duoidal $\infty$-categories of operadic modules

In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a $\mathcal{P}\otimes\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^{\otimes}$ has a sufficient supply of colimits, then we show that the $\infty$-category ${\rm Mod}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in $\mathcal{C}^{\otimes}$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any $\mathcal{P}\otimes\mathcal{O}$-algebra object $A$.

Map monoidales and duoidal $\infty$-categories

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\i…

Hopf代数在$\mathcal{A}_g$、$\mathrm{GL}_n(\mathbb{Z})$和$\mathrm{SL}_n(\mathbb{Z})$的上同调中

Hopf algebras in the cohomology of $\mathcal{A}_g$, $\mathrm{GL}_n(\mathbb{Z})$, and $\mathrm{SL}_n(\mathbb{Z})$

本文在主极化阿贝尔簇模空间的权零紧支上同调上构造了双分次余交换Hopf代数结构，利用Waldhausen构造的滤过变体证明Quillen谱序列是Hopf代数谱序列，并推导出相关上同调群维数随参数呈指数级增长。

关于参数化∞-范畴的局部-整体原理

A local-global principle for parametrized $\infty$-categories

本文提出了一种关于参数化∞-范畴的局部-整体原理，通过局部数据和粘合数据描述了∞-范畴的结构，并推导了双∞-范畴CORR的分类。

Infinitesimal and $B\_\infty$-algebras, finite spaces, and quasi-symmetric functions

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop …

富预层∞-范畴的两种模型之间的等价性

An equivalence between two models of $\infty$-categories of enriched presheaves

本文构造了$\mathcal{V}$-富集函子的$\infty$-范畴在Hinich模型与作者构造模型之间的等价，建立了弱双张量背景下两种模型的一致性。

The stable hull of an exact $\infty$-category

We construct a left adjoint $\mathcal{H}^\text{st}\colon \mathbf{Ex}_{\infty} \rightarrow \mathbf{St}_{\infty}$ to the inclusion $\mathbf{St}_{\infty} \hookrightarrow \mathbf{Ex}_{\infty}$ of the $\infty$-category of stable $\infty$-categories into the $\infty$-category of exact $\infty$-categories, which we call the stable hull. For every exact $\infty$-category $\mathcal{E}$, the unit functor $\mathcal{E} \rightarrow \mathcal{H}^\text{st}(\mathcal{E})$ is fully faithful and preserves and reflects exact sequences. This provides an $\infty$-categorical variant of the Gabriel-Quillen embedding …