Moduli stacks of algebraic structures and deformation theory

TL;DR

This paper links the homotopy types of simplicial moduli spaces of algebraic structures to their deformation cohomology and shows that certain mapping spaces form affine stacks, providing a framework for deformation and obstruction theories.

Contribution

It establishes a connection between moduli space homotopy types and deformation cohomology, and proves that mapping spaces of algebraic structures form affine stacks under broad conditions.

Findings

Mapping spaces of algebras form affine stacks.

Cohomology of deformation complexes relates to tangent complexes.

Obstruction theory for deformations is developed.

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 moduli 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.