Curved homotopy coalgebras

TL;DR

This paper explores the structure and properties of homotopy coalgebras, focusing on cofree variants, morphisms, coderivations, and their relations via bar- and cobar-constructions, with applications to curved coalgebras and algebras.

Contribution

It provides a detailed categorical framework for homotopy coalgebras, including properties of cofree variants and the adjunction between bar- and cobar-constructions using twisting cochains.

Findings

Characterization of relatively cofree homotopy coalgebras

Descriptions of morphisms and coderivations between coalgebras

Establishment of an adjunction between cobar- and bar-constructions

Abstract

We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and coderivations between coalgebras of latter type. Cobar- and bar-constructions between counit-complemented curved coalgebras, unit-complemented curved algebras and curved homotopy coalgebras are described. Using twisting cochains an adjunction between cobar- and bar-constructions is derived under additional assumptions.