置换代数的同伦代数论及其局部化
Homotopy theory of algebras of substitudes and their localisation
摘要：
我们研究的范畴代数的替代品（也被称为是等价的定期模式的Getzler）配备了一个（半）模型结构解除模型结构的基础上的预层。我们特别感兴趣的情况下，当模型结构的presheaves是一个西辛斯基风格的本地化相对于一个适当的Grothendieck基本本地化。例如，对于最小基本定位子$\mathtt{W}=\mathtt{W}_{\infty}$，这种定位中的局部对象是局部常数预层，而置换的局部代数恰好是其基础预层是局部常数的代数。我们调查当这个本地化有很好的性能。我们挑出一类这样的替代品，我们称之为左本地化，并表明，替代$n$ -运算，对称，编织运算是在这个类。作为应用，我们发展了高阶辫子算子的同伦理论，并证明了它们的$\mathtt{W}_k$ -局部化的稳定性定理。这一定理意味着，特别是，一个推广的贝兹-多兰稳定假说的高阶范畴。
英文摘要：
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.
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