Related papers: Combinatorial parametrised spectra

We define a symmetric monoidal structure on the parametrised stable homotopy category over a base space with an action of an $E_\infty$ operad. We discuss products, orientations and push-forwards in parametrised cohomology theories…

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding…

(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy…

In this article we develop the cotangent complex and (co)homology theories for spectral categories. Along the way, we reproduce standard model structures on spectral categories. As applications, we show that the invariants to descend to…

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a…

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in…

We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (or more precisely, a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichments of model…

Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…

Let $\V$ be a symmetric monoidal model category and let $X$ be an object in $\V$. From this we can construct a new symmetric monoidal model category $Sp^{\Sigma}(\V,X)$ of symmetric spectra objects in $\V$ with respect to $X$, together with…

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we…

We give rigorous foundations for parametrized homotopy theory in this monograph. After preliminaries on point-set topology, base change functors, and proper actions of non-compact Lie groups, we develop the homotopy theory of equivariant…

Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of…

The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper…

In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…

We put a model structure on a full subcategory of based multicategories in which the weak equivalences are created by the K-theory functor of Elmendorf-Mandell, providing a model categorical lift of Thomason's theorem on the modeling of…

We develop a stable analogue to the theory of cosimplicial frames in model cagegories; this is used to enrich all homotopy categories of stable model categories over the usual stable homotopy category and to give a different description of…

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of…

The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model…

We extend Homotopy Type Theory with a novel modality that is simultaneously a monad and a comonad. Because this modality induces a non-trivial endomap on every type, it requires a more intricate judgemental structure than previous modal…