prop（erad）s表示的变形理论
Deformation theory of representations of prop(erad)s
摘要：
本文研究了适形环和支撑环的态射的变形理论，从而推广到交换环的Quillen变形理论的非线性框架。相应的链复形具有李代数的同伦结构。它的Maurer-Cartan元素对应于变形结构，这使我们能够给出这些结果的几何解释。为此，我们赋予prop（erad）s范畴一个模型范畴结构。我们提供了一个完整的研究模型的道具（erad）。介绍了一种新的有效的方法，使极小模型显式，它扩展了Koszul对偶理论，相关的概念被称为同伦Koszul。作为推论，我们得到了prop（erad）上（al）代数的（co）同调论和同伦（al）代数的（co）同调论.它们的基本链复形在一般情况下具有标准李代数的同伦结构，而仅在Koszul情形下具有李代数的同伦结构.特别地，我们从结合双代数的适定性中明确了态射的变形复形。对这类适形的任意极小模型，证明了这类链复形的边界映射是Gerstenhaber和Schack定义的边界映射.作为推论，本文给出了Gerstenhaber-Schack双复形上与结合双代数的变形相关的李代数的同伦结构的存在性的完整证明.
英文摘要：
We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.
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