Operads of compatible structures and weighted partitions

空间上恒等函子的分划偏序复形与古德里导数

The partition poset complex and the Goodwillie derivatives of the identity in spaces

该研究通过上单纯形对象配对构造空间恒等函子导数的典范同伦协调 operad 结构，证明其与 Ching 描述的结构一致，还揭示谱上交换余代数的导出本原元为该 operad 上的代数。

划分上的一个prop结构

A prop structure on partitions

本文研究由Com算子的算畴悬垂自由生成的prop，给出特殊生成元及其复合与对称群作用的简单描述，并在划分上构造prop结构，用于计算自由群上简单函子间的扩张群。

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of t…

Coloring Operads for Algebraic Field Theory

In these proceedings we summarize previous work where we formalize a general concept of algebraic field theories using operads. After giving a gentle reminder of algebraic quantum field theory, operads and their algebras, we construct field theory operads, whose algebras are exactly algebraic field theories. Specifically, they satisfy a suitable version of the Einstein causality axiom. From this construction we get adjunctions between different types of field theories, including adjunctions related to local-to-global extensions and the time-slice axiom, and a quantization functor for linear fi…

自由对称余弦单纯操作范畴上的微分分次Hopf代数结构

Differential graded Hopf algebra structure on free symmetric cosimplicial operads

本文在自由对称连通乘法微分分次操作范畴上构造了新的链复形代数和双复形代数结构，并通过odot乘积和Alexander-Whitney同态类比构造了微分分次Hopf代数结构，推广了Malvenuto-Reutenauer的结果。

关于Λ序列的凯利单体结构和单位运算子的性质

On the Kelly monoidal structure of $Λ$-sequences and unital operads

本文研究了Λ序列的凯利单体结构，证明了其与对称序列右模的同构，并建立了普遍的正常弱单体结构，将单位运算子识别为单体。

Commutative ring objects in pro-categories and generalized Moore spectra

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have h…

排列多面体的胞腔对角线

Cellular diagonals of permutahedra

本文系统研究排列多面体的几何胞腔对角线，通过超平面排列对偶得到计数结果，刻画与operad结构相容的两条同构对角线的性质，并由此推出operahedra与multiplihedra上的两类拓扑胞腔operad结构及两类几何泛张量积。

BV代数与BV模之间的循环对偶性

Cyclic duality between BV algebras and BV modules

本文证明满足cosimplicial相容条件的循环operad其上同调具有BV代数结构，并给出BV模的循环对偶成为BV代数的条件，以Hopf代数理论中的同调群为例加以说明。

A chain level Batalin-Vilkovisky structure in string topology and decorated cacti

We show that a model of chain complex of the free loop space of a $C^\infty$-manifold, which is proposed in arxiv:1404.0153, admits an action of a certain dg operad. This is a chain level structure under the Chas-Sullivan BV structure on loop space homology. Our dg operad is a variant of the cacti operad, and we introduce combinatorial objects called "decorated cacti" to define it. We also define a chain level Gerstenhaber structure on Hochschild cochains of any differential graded algebra. Applied to the dga of differential forms, this structure is compatible with our chain level structure in…

关于 $\infty$-操作符理论的 dendroidal 空间的新模型结构

A new model structure on dendroidal spaces for the theory of $\infty$-operads

本文提出了一种新的 dendroidal 空间模型结构，用于 $\infty$-操作符理论，并证明其与 Cisinski-Moerdijk 模型结构的 Quillen 等价性。

盘状 $G$-转移系统的最大兼容性

Maximal compatibility of disklike $G$-transfer systems

本文研究盘状 $G$-转移系统与最大兼容转移系统的关系，通过显式公式确定兼容性，并证明兼容性在群商的膨胀映射下是函子性的。

On the deformation complex of homotopy affine actions

An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, where $V$ is a vector space and $V \rtimes {\rm End}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtim…

阿贝尔紧李群的全局转移系统

Global transfer systems of abelian compact Lie groups

本文通过将全局转移系统与全局 $N_\infty$-operad 的等价性应用于阿贝尔紧李群族，明确描述并完全分类了所有阿贝尔紧李群的全局转移系统。

Deformation theory with homotopy algebra structures on tensor products

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. We give …