论文标题
替代代数及其本地化的同喻理论
Homotopy theory of algebras of substitudes and their localisation
论文作者
论文摘要
我们研究了配备有（半）模型结构的替代代数类别（也已知等同于Getzler的常规模式），这些模型结构从基础预示的模型结构上提升。我们对预留模型结构是Cisinski风格的本地化时特别感兴趣的情况，就适当的Grothendieck基本当地人而言。例如，对于$ \ mathtt {w} = \ mathtt {w} _ {\ infty} $最小的基本原位者，这种本地化中的本地对象是本地恒定的预性，而替代的本地代数则是其基础预性的代数。 我们研究此本地化具有良好的属性。我们挑选了一类这样的替代品，我们称之为左右，并表明$ n $ operads，对称性和编织的替代品在此类中。作为一个应用程序，我们开发了一个高辫子作业的同义理论，并证明了其$ \ mathtt {w} _k $ - localisations的稳定定理。该定理尤其暗示着对更高类别的Baez-Dolan稳定假设的概括。
We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for $\mathtt{W}=\mathtt{W}_{\infty}$ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant. We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for $n$-operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their $\mathtt{W}_k$-localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan Stabilisation Hypothesis for higher categories.
