Absolute algebras, contramodules, and duality squares
Victor Roca i Lucio
TL;DR
Absolute algebras extend classical algebra by encoding infinite sums without topology as algebras over cooperads, enabling a homotopical duality framework. The authors construct a duality square connecting the standard bar-cobar theory with linear dualities, establish Quillen adjunctions for Sweedler and topological duals, and prove infinite-dimensional equivalences between dg $\mathcal{P}$-algebras and coalgebras under finiteness hypotheses. They develop the complete bar-cobar theory relative to curved twisting morphisms, compare absolute and classical structures via restriction functors and filtrations, and illustrate the theory with contramodules as well as absolute $\mathcal{A}_\infty$- and $\mathcal{L}_\infty$-algebras, including a universal enveloping construction. The results yield a robust homotopical framework for absolute Lie theory, with implications for formal moduli problems and derived geometry, and provide concrete instances such as absolute associative and Lie algebras that generalize classical enveloping constructions.
Abstract
Absolute algebras are a new type of algebraic structures, endowed with a meaningful notion of infinite sums of operations without supposing any underlying topology. Opposite to the usual definition of operadic calculus, they are defined as algebras over cooperads. The goal of this article is to develop this new theory. First, we relate the homotopy theory of absolute algebras to the homotopy theory of usual algebras via a duality square. It intertwines bar-cobar adjunctions with linear duality adjunctions. In particular, we show that linear duality functors between types of coalgebras and types of algebras are Quillen functors and that they induce equivalences between objects with finiteness conditions on their homology. We give general comparison results between absolute types of algebras and their classical counterparts. We work out examples of this theory such as absolute associative algebras and absolute Lie algebras, and show that it includes the theory of contramodules. Campos--Petersen--Robert-Nicoud--Wierstra showed that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic. As an application of our results, we generalize their theorem to the setting of absolute Lie algebras and absolute $\mathcal{L}_\infty$-algebras.