In this paper, we develop the \(A_\infty \)-analog of the Maurer-Cartan simplicial set associated to an \(L_\infty \)-algebra and show how we can use this to study the deformation theory of \(\infty \)-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of \(A_\infty \)-algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) \(A_\infty \)-algebras to simplicial sets, which sends a complete curved \(A_\infty \)-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of \(\infty \)-morphisms of algebras over non-symmetric operads.

在本文中，我们开发了与\(L_\infty \)代数相关联的 Maurer-Cartan 单纯集的\(A_\infty \)模拟，并展示了我们如何使用它来研究\( \infty \) -非对称操作数上的代数态射。更准确地说，我们首先回忆并证明\(A_\infty \) -代数的一些主要性质，如 Maurer-Cartan 方程和扭曲。我们在这里的主要创新之一是强调了 shuffle 产品的重要性。然后，我们定义一个从完全（弯曲）\(A_\infty \) -代数范畴到单纯集的函子，它发送一个完全弯曲的\(A_\infty \)-代数到相关的 Maurer-Cartan 元素的单纯集。这个函子的性质是它给出了一个 Kan 复形。在所有这些中，我们不需要对我们正在研究的领域进行任何假设。我们还表明，该函子可用于研究特征场大于或等于 0 的变形问题。作为此类变形问题的具体示例，我们研究了\(\infty \) -态射的变形理论非对称操作数上的代数。