We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger–Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer–Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block–Smith’s higher Riemann–Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces.

中文翻译：

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Maurer-Cartan模数和Riemann-Hilbert型定理

我们研究了dg代数和相关dg类别的Maurer-Cartan模空间，并表明，尽管它们不是准同构不变量，但它们是强同伦类型的不变量，这是以前没有研究过的自然概念。我们在几种不同的情况下证明了Schlessinger-Stasheff型定理，比较了Maurer-Cartan元素及其分类版本的同伦和规范等价的概念。作为应用，我们重新证明和归纳了Block-Smith的较高的Riemann-Hilbert对应关系，并开发了其用于简单复合物和拓扑空间的类似物。