Absolute algebras, contramodules, and duality squares

TL;DR

This paper introduces absolute algebras, a new class of algebraic structures defined via cooperads, and explores their homotopy theory, dualities, and applications to Lie algebras and contramodules.

Contribution

It develops the theory of absolute algebras, relating them to classical algebras through duality squares and Quillen functors, and extends known results to this new framework.

Findings

Absolute algebras are defined via cooperads without topology assumptions.

Duality squares relate homotopy theories of absolute and classical algebras.

Generalization of a theorem on nilpotent Lie algebras to absolute Lie algebras.

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.