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Abstract

We develop an obstruction theory for the existence of gauge equivalences in complete differential graded Lie algebras. Specifically, this theory provides a characterization of homotopy equivalences between differential graded algebras governed by operads or properads, potentially colored in a groupoid. We apply this framework to establish new homotopy equivalence results in both algebraic topology and algebraic geometry, with a particular focus on the study of minimal models for highly connected varieties.