论文标题
曲面和弯曲代数的同义理论
Homotopy theory of curved operads and curved algebras
论文作者
论文摘要
弯曲的代数是赋予预定性的代数，这是-1的内态性，其正方形不一定为0。这使得对准同构的通常定义毫无意义，因此弯曲代数的同型研究不能像差异分析的代数相同的路径。 在本文中，我们建议通过弯曲的作战研究弯曲的代数。我们开发了适合这个新概念以及Koszul二元性理论的律师和柯比结构的理论。为了能够提供有意义的定义，我们在应用和完成的对象的上下文中工作，并在应用相关分级函数后变为差异。 这种设置带来了自己的困难，但是它允许我们定义一个组合模型类别结构，我们可以使用自由遗嘱的辅助功能将其转移到弯曲的曲面类别和代数类别。 我们解决了弯曲的联想代数的情况。我们恢复了弯曲的AOO-Elgebras的概念，并表明弯曲的关联代数和弯曲的AOO-Elgebras的同型类别是quillen等效的。
Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of curved algebras cannot follow the same path as differential graded algebras. In this article, we propose to study curved algebras by means of curved operads. We develop the theory of bar and cobar constructions adapted to this new notion as well as Koszul duality theory. To be able to provide meaningful definitions, we work in the context of objects which are filtered and complete and become differential graded after applying the associated graded functor. This setting brings its own difficulties but it nevertheless permits us to define a combinatorial model category structure that we can transfer to the category of curved operads and to the category of algebras over a curved operad using free-forgetful adjunctions. We address the case of curved associative algebras. We recover the notion of curved Aoo-algebras, and we show that the homotopy categories of curved associative algebras and of curved Aoo-algebras are Quillen equivalent.
