Title: Simplicial properadic homotopy Show Chinese title

Title: 单纯正则同伦

Abstract: In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with characterizations for the various types of infinity-morphisms, the most seminal one being the equivalence between infinity-quasi-isomorphisms and zig-zags of quasi-isomorphisms which plays a key role in the study the formality property. We establish a simplicial enrichment for the categories of gebras over some cofibrant properads together with their infinity-morphisms, whose homotopy category provides us with the localisation with respect to infinity-quasi-isomorphisms. These results extend to the properadic level known properties for operads, but the lack of the rectification procedure in this setting forces us to use different methods. Show Chinese abstract

Abstract: 本文中，我们解决了由 PROPERAD（PROPERAD 是一种代数结构，允许定义具有多个输入和输出的操作）上的同伦（双）代数的无穷 morphism 的同伦性质问题，即由多个输入和输出构成的操作组成的代数结构。我们首先通过提供各种类型无穷 morphism 的特征来丰富文献内容，其中最具开创性的结果是无穷准同构与准同构的 zigzag 等价关系，这一结果在形式性质的研究中起着关键作用。 我们建立了某些余纤维化 PROPERAD 上的代数及其无穷 morphism 的单纯性富集结构，其同伦范畴提供了关于无穷准同构的局部化。这些结果将运算的已知性质扩展到了 PROPERAD 水平，但由于该设置中缺乏归一化过程，迫使我们采用不同的方法。