Admissibility and rectification of colored symmetric operads

Spectra and symmetric spectra in general model categories

We construct model category structures for monoids and modules in symmetric monoidal model categories which satisfy an extra axiom, the monoidal axiom, with applications to symmetric spectra and $\Gamma$-spaces.

In this chapter, we introduce the notions of 1-Segal and 2-Segal objects in a combinatorial model category C. If further C admits the structure of a left proper, tractable, symmetric monoidal model category, then we introduce model structures for 1-Segal and 2-Segal objects which arise as enriched Bousfield localizations of the injective model structure on CΔ. For \( {\mathbf C} = {\mathbb S}\), the model structure for 1-Segal objects in \({\mathbb S}\) recovers the Rezk model structure for 1-Segal spaces introduced in Rezk (Trans Am Math Soc 353(3):973–1007, 2001).

K-theory and the bridge from motives to noncommutative motives

In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.

Kan spectra provide a combinatorial model for the stable homotopy category. They were introduced by Dan Kan in 1963 under the name semisimplicial spectra. A Kan spectrum is similar to a pointed simplicial set, but it has simplices in negative degrees as well and all its simplices have infinitely many faces and degeneracies. In the first part of this thesis, we define a functor from the category of Gamma-spaces to the category of Kan spectra without passing through any other category of spectra. We show that the resulting Kan spectrum of a Gamma-space A agrees with the usual object associated to A in the stable homotopy category by comparing it to Bousfield-Friedlander's spectrum construction. In particular, applying our construction to the Gamma-space associated to a symmetric monoidal category provides a combinatorial model of its algebraic K-theory spectrum. For the Gamma-space associated to an abelian group, this yields via the stable Dold-Kan correspondence the unbounded chain complex with the abelian group concentrated in degree zero. The second part of this work concerns group spectra and twisting structures. Group spectra are the group objects in the category of Kan spectra. They provide an algebraic, combinatorial model for the stable homotopy category. We transfer Brown's model structure from the category of Kan spectra to a Quillen equivalent model structure on the category of group spectra. We then construct the analogues of Kan's loop group functor and its right adjoint Wbar together with corresponding classifying bundles, so that the category of Kan spectra becomes a twisted homotopical category in the sense of Farjoun and Hess.

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

For a completely distributive lattice V, a novel class of lattice-valued Scott open sets, referred to as Scott open V-sets, is introduced on the powerset. These sets are utilized to construct a monad over the category of sets, termed the Scott open V-set monad. It is demonstrated that the category of Eilenberg-Moore algebras for the Scott open V-set monad is isomorphic to that of algebraic V-modules, and the category of Kleisli monoids with respect to this monad is isomorphic not only to the category of algebraic V-closure spaces but also to that of lax algebras for the finite powerset monad.

We study the category of Reedy diagrams in a $\mm$-model category. Explicitly, we show that if K is a small category, V is a closed symmetric monoidal category and C is a closed V-module, then the diagram category V^K is a closed symmetric monoidal category and the diagram category C^K is a closed V^K-module. We then prove that if further K is a Reedy category, V is a monoidal model category and C is a V-model category, then with the Reedy model category structures, V^K is a monoidal model category and C^K$ is a $\mm^K-model category provided that either the unit 1 of V is cofibrant or V is cofibrantly generated.

We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

Analytification, localization and homotopy epimorphisms

Stone dualities from opfibrations

Our goal in this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the [math] –categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an [math] –combinatorial model category, which insure that its underlying [math] –category is [math] –presentable. In the second application we show that the topological realization of any Grothendieck topos coincides with the shape of the hypercompletion of the associated [math] –topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the [math] –category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by G Raptis.

This paper studies the category of posets Pos as a model for the homotopy theory of spaces. We prove that: (i) Pos admits a (cofibrantly generated and proper) model structure and the inclusion functor Pos → Cat into Thomason's model category is a right Quillen equivalence, and (ii) there is a proper class of different choices of cofibrations for a model structure on Pos or Cat where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the viewpoint of Alexandroff T0-spaces, and we apply a result of McCord to give a new proof of the classification theorems of Moerdijk and Weiss in the case of posets.

In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.