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In this paper, we consider the model structure on the category of cellular sets originally conjectured by Cisinski and Joyal to give a model for the homotopy theory of weak (\omega)-categories. We demonstrate first that any…

In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for…

This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of…

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe…

A manifold with fibered cusp metrics $X$ can be considered as a geometrical generalization of locally symmetric spaces of $\mathbb{Q}$-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find…

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…

This paper is a sequel to "Localization of $\frak{u}$-modules. I", hep-th/9411050. We are starting here the geometric study of the tensor category $\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of finite type)…

We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories…

Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not…

Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category of separable complex C*-algebras. Namely, they are in bijective…

Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models…

The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the…

Given a higher-rank graph $\Lambda$, we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $G_\Lambda$. We define an exact functor between the abelian category of right modules…

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $X\setminus Y_i$…

For a smooth affine algebraic group $G$, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on $G$, the stack $G/G_{ad}$ and the category of…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras. In the first part of this…

We axiomatize the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalized automorphic sheaf is proposed which includes sheaves of (meromorphic) sections…

This is a sequel to [Ca01]=math.AG/0110051. We define the bimeromorphic {\it category} of geometric orbifolds. These interpolate between (compact K\" ahler) manifolds and such manifolds with logarithmic structure. These geometric orbifolds…

Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the…