In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a series, we generalise the seminal notion of infini-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak--Fukaya--Latschev and we produce new explicit formulas.

中文翻译：

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Properadic同调微积分

在本文中，我们开始将控制同伦代数的性质的操作演算推广到控制同伦代数上的同伦代数的性质的properadic 演算。在该系列的第一篇文章中，我们概括了无穷同态的开创性概念和无处不在的同伦转移定理。作为一个应用，我们恢复了由 Cieliebak--Fukaya--Latschev 开发的对合李双代数的同伦性质，并产生了新的显式公式。