Maurer–Cartan Moduli and Theorems of Riemann–Hilbert Type

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...

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Bibliographic Details
Published in:Applied categorical structures 2021-08, Vol.29 (4), p.685-728
Main Authors: Chuang, Joseph, Holstein, Julian, Lazarev, Andrey
Format: Article
Language:English
Subjects:
Convex and Discrete Geometry
Geometry
Invariants
Isomorphism
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Theorems
Theory of Computation
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Summary: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.
ISSN:0927-2852
1572-9095
DOI:10.1007/s10485-021-09631-3