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Moduli stacks of algebraic structures and deformation theory

Abstract : We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial mod-uli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.
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Contributor : Sinan Yalin <>
Submitted on : Friday, February 16, 2018 - 11:41:42 AM
Last modification on : Monday, March 9, 2020 - 6:15:54 PM
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  • HAL Id : hal-01710705, version 1

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Sinan Yalin. Moduli stacks of algebraic structures and deformation theory. Journal of Noncommutative Geometry, European Mathematical Society, 2016, 10, pp.579-661. ⟨hal-01710705⟩

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