富集的预李操作元与自由性定理

在操作符之间映射空间与双模的关系

Mapping spaces between operads in relation to bimodules

本文研究了增强操作符之间的映射空间，并通过方便的纤维序列将这些空间与操作符双模之间的空间联系起来，适用于仿射操作符、仿射在仿射模块中的操作符以及连通的dg操作符。

The Dwyer-Kan model structure for enriched coloured PROPs

Coloured PROPs are a generalisation of coloured operads. In this article, we prove the existence of a Dwyer-Kan model structure on the category of small coloured PROPs enriched in a (sufficiently nice) monoidal model category V. This model structure extends the known Dwyer-Kan model structures on the categories of (small) V-enriched categories and V-enriched coloured operads.

Enriched $\infty$-categories via non-symmetric $\infty$-operads

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…

富化的同伦相干结构

Enriched homotopy-coherent structures

本文基于“代数模式”框架，为Segal条件描述的同伦相干代数结构引入一般性富化概念，恢复了已知的富化∞-范畴等实例，并提出了富化模∞-代数操作等新实例。

Algebras of higher operads as enriched categories II

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimb…

Multitensors as monads on categories of enriched graphs

In this paper we unify previous developments on higher operads and multitensors into a single framework in which the interplay between multitensors on a category V, and monads on the category of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work in which the Gray tensor product of 2-categories and the Crans tensor product of Gray categories are exhibited as existing within our framework, and in an explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads.

循环 operad 的 Dwyer--Kan 同伦理论

Dwyer--Kan homotopy theory for cyclic operads

该研究提出对称幺半基范畴上着色循环 operad 的通用定义，借助其到着色 operad 的遗忘函子的左右伴随显式公式，将单纯集富集着色 operad 范畴的 Cisinski--Moerdijk 模型结构提升到对应循环 operad 范畴。

增广∞-范畴中内蕴算畴上的余积幂等代数

Coproduct idempotent algebras over internal operads in enriched $\infty$-categories

本文将内蕴算畴的作用解释为单子作用，证明在可展示性条件下余张量性质可传递到代数范畴，从而证明余积幂等代数在诱导张量作用下不变，并应用于稳定母题同伦范畴。

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。

Localization of algebras over coloured operads

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. I…

On measurings of algebras over operads and homology theories

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case …

伪对称多重函子的一个连贯性定理

A coherence theorem for pseudo symmetric multifunctors

本文证明伪对称 Cat-富集多重函子的连贯性定理，并由此推出 Mandell 逆 K-理论等函子在更换参数化 En-operad 的代价下保持 En-代数。

富化∞-范畴与带有弱作用的∞-范畴之间的等价性

An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action

该研究证明Lurie与Gepner-Haugseng两种意义下的$\u2118$-富化∞-范畴等价，建立了对应松弛幺半函子限制关系，并给出稳定∞-范畴嵌入与富化Yoneda嵌入两项应用。