# Goodwillie calculus in the category of algebras over a chain complex operad

## TL;DR

This paper develops a framework for Goodwillie calculus in categories derived from chain complexes, characterizing homogeneous functors and extending previous results to operad algebras over fields of characteristic zero.

## Contribution

It extends Goodwillie calculus to algebras over chain complex operads, providing new characterizations in characteristic zero fields.

## Key findings

- Characterization of homogeneous functors between chain complex categories
- Extension of Walter's results to operad algebra categories
- Applicability to categories of chain complexes and non-negatively graded complexes

## Abstract

The goal of this paper is to furnish a literature on Goodwillie calculus for functors defined between categories which derive from chain complexes over a ground field $\Bbbk.$ We characterize homogeneous functors $F: \mathcal{C} \longrightarrow \mathcal{D}$ where $\mathcal{C} ,\mathcal{D}= Ch$ (chain complexes), $Ch_+$(non-negatively graded chain complexes) or $\text{Alg}_\mathcal{O}$ (algebras over a chain complex operad $\mathcal{O}$). In the particular case when $\mathcal{D}= \text{Alg}_\mathcal{O},$ our characterization requires $\Bbbk$ to be of characteristics $0.$ We are then extending the results of Walter \cite{Walt06} who studied in characteristics $0$ the chain complex cases and when $\mathcal{O}$ is the Lie operad.