I Cue the Alphabet

A collection of words and phrases beginning with each of the letters

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

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

A uniqueness theorem for stable homotopy theory

In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra. One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres. In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of the stable homotopy groups of spheres. Another sufficient condition is the existence of a small generating object (corresponding to the sphere spectrum) for which a specific `unit map' from the infinite loop space QS^0 to the endomorphism space is a weak equivalence

Analysis of problems related to slingshot shock machine high-velocity shock testing

Slingshot device for high-velocity shock testing analyzed for impact delivery, shock pulse, and g-leve

Enriched model categories and an application to additive endomorphism spectra

We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in such a model category has a naturally associated endomorphism ring inside this spectra category. We establish the basic properties of this enrichment. We also develop some enriched model category theory. In particular, we have a notion of an adjoint pair of functors being a 'module' over another such pair. Such things are called "adjoint modules". We develop the general theory of these, and use them to prove a result about transporting enrichments over one symmetric monoidal model category to a Quillen equivalent one.Comment: Sections completely re-organized from previous version. Mathematical content all the sam