2023/03/21 by Rina Anno, Anno, Rina, Sergey Arkhipov +3
Mathematics · #14F08 #18F20 #18G35 #18G70 (primary) #18G80 #18M05 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2303.11826
openalex publication_date 2023/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define A∞-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong homotopy unitality and bimodule homotopy unitality to work in this level of generality. For a strong homotopy unital A∞-algebra we construct Free-Forgetful homotopy adjunction, its Kleisli category, and its derived category of modules. Analogous constructions for A∞-coalgebras require bicomodule homotopy counitality. We define homotopy adjunction for A∞-algebra and A∞-coalgebra and show such pair to be derived module-comodule equivalent. As an application, we obtain the notions of an A∞-monad and of an enhanced exact monad. We also show that for any adjoint triple (L,F,R) of functors between enhanced triangulated categories the adjunction monad RF and the adjunction comonad LF are derived module-comodule equivalent.