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Maurer-Cartan moduli and theorems of Riemann-Hilbert type

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<mark>Journal publication date</mark>31/08/2021
<mark>Journal</mark>Applied Categorical Structures
Issue number4
Number of pages44
Pages (from-to)685-728
Publication StatusPublished
Early online date6/02/21
<mark>Original language</mark>English


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.