2022/05/26 by Alex Milham, Milham, Alex, Christopher L. Rogers +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2205.13099
Complete filtered A_∞-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the A_∞ analogs of several key theorems from the Maurer-Cartan theory for L_∞-algebras. In contrast with the L_∞ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the A_∞-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set N\bullet(A) to an A_∞-algebra A, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of N_\bullet(A) in terms of the cohomology algebra H(A), and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of A is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L_∞-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.