Homotopy Theory of linear coalgebras

TL;DR

This paper develops a homotopy theory framework for various types of coalgebras, including $ ext{E}_ ext{infinity}$, $ ext{A}_ ext{infinity}$, and $ ext{L}_ ext{infinity}$ coalgebras, by establishing a model category structure on complete curved algebras.

Contribution

It introduces a model category structure on complete curved algebras that is equivalent to the homotopy theory of general coalgebras, extending the theory to non-locally conilpotent cases.

Findings

Established a model category structure on complete curved algebras.

Proved the equivalence of homotopy theories between coalgebras and complete curved algebras.

Extended homotopy theory to include $ ext{E}_ ext{infinity}$, $ ext{A}_ ext{infinity}$, and $ ext{L}_ ext{infinity}$ coalgebras.

Abstract

We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras, $\mathcal L_\infty$-coalgebras etc. To do so, we define the category of complete curved algebras -- where the notion of quasi-isomorphims does not make sense -- and endow it with a model category structure, equivalent to that of the category of coalgebras.