Related papers: A topos associated with a colored category

In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

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 extend the usual internal logic of a (pre)topos to a more general interpretation, called the stack semantics, which allows for "unbounded" quantifiers ranging over the class of objects of the topos. Using well-founded relations inside…

We generalise to a group homomorphism $\tau$ the $\chi$-graded categories of S\"{o}zer and Virelizier. These are categories in which both morphisms and objects have compatible degrees. We give a 'half-enriched' Yoneda lemma, a structure…

Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C…

A schemoid is a generalization of association schemes from the point of view of small categories. In this article, we discuss schemoid structures for two kinds of small categories; the canonical small category defined by a poset, and…

We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems…

A quick overview of category theory and topos theory including slice categories, monics, epics, isos, diagrams, cones, cocones, limits, colimits, products and coproducts, pushouts and pullbacks, equalizers and coequalizers, initial and…

Let k be an infinite perfect field. We provide a general criterion for a spectrum in the stable homotopy category over k to be effective, i.e. to be in the localizing subcategory generated by the suspension spectra of smooth schemes. As a…

Based on a Whitehead-type characterization of the sectional category we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of…

A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to…

In this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infinity-categories of quasi-coherent complexes. It can be viewed as a derived analogue of Tannaka…

With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the…

We investigate an algebraic problem related to the determination of the fundamental group of a class of spaces of configurations on surfaces. The configuration spaces are spaces of points grouped into colors. Whether two points are allowed…

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide…

Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full…

Let $H$ be a Hopf algebra over a field $k$, and $A$ an $H$-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom…

We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches…