Rigidification of connective comodules

Abstract

Let $\mathbb{k}$ be a commutative ring with global dimension zero. We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of $\mathbb{k}$. That is, the $\infty$-category of homotopy coherent comodules is represented by a model category of strict comodules in non-negative chain complexes over $\mathbb{k}$. These comodules are over a coalgebra that is strictly coassociative and simply connected. The rigidification result allows us to derive the notion of cotensor product of comodules and endows the $\infty$-category of comodules with a symmetric monoidal structure via the two-sided cobar resolution.

Theorems & Definitions (27)

• Theorem 1.1: Theorem \ref{['main thm']}
• Theorem 1.2: Theorem \ref{['thm: derived cotensor product']}
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Theorem 3.1