The homotopy theory of coalgebras over a comonad

Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are specific instances of the following categories of comodules over a coring. For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category of right A-modules satisfying the conditions of our existence theorem with respect to the comonad given by tensoring over A with V and conclude that the category of V-comodules in the category of right A-modules admits a model category structure of the desired type. Finally, under extra conditions on R, A, and V, we describe fibrant replacements in this category of comodules in terms of a generalized cobar construction.Comment: 34 pages, minor corrections. To appear in the Proceedings of the London Mathematical Societ

A monoidal Dold-Kan correspondence for comodules

We provide examples of inductive fibrant replacements in fibrantly generated model categories constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond. We deduce that the rational $A$-theory of a simply connected space $X$ is equivalent to the $K$-theory of perfect chain complexes with a $C_*(X; \mathbb{Q})$-comodule structure.Comment: 31 pages. Added Theorem 1.2 on the rational $A$-theory. This paper contains some of the results in the original version of arXiv:2006.0939

Waldhausen K-theory of spaces via comodules

Let $X$ be a simplicial set. We construct a novel adjunction between the categories of retractive spaces over $X$ and of $X_{+}$-comodules, then apply recent work on left-induced model category structures (arXiv:1401.3651v2 [math.AT],arXiv:1509.08154 [math.AT]) to establish the existence of a left proper, simplicial model category structure on the category of $X_+$-comodules, with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory. We show moreover that this model category structure stabilizes, giving rise to a model category structure on the category of $\Sigma^\infty X_{+}$-comodule spectra. The Waldhausen $K$-theory of $X$, $A(X)$, is thus naturally weakly equivalent to the Waldhausen $K$-theory of the category of homotopically finite $\Sigma^\infty X_{+}$-comodule spectra, with weak equivalences given by twisted homology. For $X$ simply connected, we exhibit explicit, natural weak equivalences between the $K$-theory of this category and that of the category of homotopically finite $\Sigma^{\infty}(\Omega X)_+$-modules, a more familiar model for $A(X)$. For $X$ not necessarily simply connected, we have localized versions of these results. For $H$ a simplicial monoid, the category of $\Sigma^{\infty}H_{+}$-comodule algebras admits an induced model structure, providing a setting for defining homotopy coinvariants of the coaction of $\Sigma^{\infty}H_{+}$ on a $\Sigma^{\infty}H_{+}$-comodule algebra, which is essential for homotopic Hopf-Galois extensions of ring spectra as originally defined by Rognes in arXiv:math/0502183v2} and generalized in arXiv:0902.3393v2 [math.AT]. An algebraic analogue of this was only recently developed, and then only over a field (arXiv:1401.3651v2 [math.AT]).Comment: 48 pages, v3: some technical modifications, to appear in Advances in Mathematic