197 research outputs found
On the tensor product of linear sites and Grothendieck categories
We define a tensor product of linear sites, and a resulting tensor product of
Grothendieck categories based upon their representations as categories of
linear sheaves. We show that our tensor product is a special case of the tensor
product of locally presentable linear categories, and that the tensor product
of locally coherent Grothendieck categories is locally coherent if and only if
the Deligne tensor product of their abelian categories of finitely presented
objects exists. We describe the tensor product of non-commutative projective
schemes in terms of Z-algebras, and show that for projective schemes our tensor
product corresponds to the usual product scheme.Comment: New sections 5.3 on the alpha-Deligne tensor product and 5.4 on
future prospect
Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory
Let X be a smooth projective variety. Using the idea of brane actions
discovered by To\"en, we construct a lax associative action of the operad of
stable curves of genus zero on the variety X seen as an object in
correspondences in derived stacks. This action encodes the Gromov-Witten theory
of X in purely geometrical terms and induces an action on the derived category
Qcoh(X) which allows us to recover the Quantum K-theory of Givental-Lee.Comment: final version, 64 pages, accepted for publication in Geometry &
Topolog
Chiral Koszul duality
We extend the theory of chiral and factorization algebras, developed for
curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties.
This extension entails the development of the homotopy theory of chiral and
factorization structures, in a sense analogous to Quillen's homotopy theory of
differential graded Lie algebras. We prove the equivalence of
higher-dimensional chiral and factorization algebras by embedding factorization
algebras into a larger category of chiral commutative coalgebras, then
realizing this interrelation as a chiral form of Koszul duality. We apply these
techniques to rederive some fundamental results of \cite{bd} on chiral
enveloping algebras of -Lie algebras
Indexed and Fibred Structures for Hoare Logic
Indexed and fibred categorical concepts are widely used in computer science as models of logical systems and type theories. Here we focus on Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed category and fibred category theory. The structural features of the language are presented in an indexed setting, while the logical features of deduction are modeled in the fibred one. Especially, Hoare triples arise naturally as special arrows in a fibred category over a syntactic category of programs, while deduction in the Hoare calculus can be characterized categorically by the heuristic deduction = generation of cartesian arrows + composition of arrows.publishedVersio
Noncommutative L-functions for varieties over finite fields
In this article we prove a Grothendieck trace formula for L-functions of not
necessarily commutative adic sheaves.Comment: 17 page
Towards Stone duality for topological theories
AbstractIn the context of categorical topology, more precisely that of T-categories (Hofmann, 2007 [8]), we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of “Stone duality”, and show that Cauchy completeness of a T-category is precisely its sobriety
The structure sheaf of the moduli of oriented -divisible groups
Using spectral algebraic geometry, we define a derived moduli stack of
oriented -divisible groups and study its structure sheaf. This stack is
consequently used to prove a theorem of Lurie which predicts the existence of a
certain sheaf of -rings
on a formally \'{e}tale site of
the classical moduli stack of -divisible groups. A variety of sections of
this sheaf are then shown to be equivalent to known -rings
in stable homotopy theory, and the natural symmetries on these sections also
recover well-studied actions and operations.Comment: 82 pages, comments always welcom
Topological Domain Theory
This thesis presents Topological Domain Theory as a powerful and flexible framework for denotational semantics. Topological Domain Theory models a wide range of type constructions and can interpret many computational features. Furthermore, it has close connections to established frameworks for denotational semantics, as well as to well-studied mathematical theories, such as topology and computable analysis.We begin by describing the categories of Topological Domain Theory, and their categorical structure. In particular, we recover the basic constructions of domain theory, such as products, function spaces, fixed points and recursive types, in the context of Topological Domain Theory.As a central contribution, we give a detailed account of how computational effects can be modelled in Topological Domain Theory. Following recent work of Plotkin and Power, who proposed to construct effect monads via free algebra functors, this is done by showing that free algebras for a large class of parametrised equational theories exist in Topological Domain Theory. These parametrised equational theories are expressive enough to generate most of the standard examples of effect monads. Moreover, the free algebras in Topological Domain Theory are obtained by an explicit inductive construction, using only basic topological and set-theoretical principles.We also give a comparison of Topological and Classical Domain Theory. The category of omega-continuous dcpos embeds into Topological Domain Theory, and we prove that this embedding preserves the basic domain-theoretic constructions in most cases. We show that the classical powerdomain constructions on omega-continuous dcpos, including the probabilistic powerdomain, can be recovered in Topological Domain Theory.Finally, we give a synthetic account of Topological Domain Theory. We show that Topological Domain Theory is a specific model of Synthetic Domain Theory in the realizability topos over Scott's graph model. We give internal characterisations of the categories of Topological Domain Theory in this realizability topos, and prove the corresponding categories to be internally complete and weakly small. This enables us to show that Topological Domain Theory can model the polymorphic lambda-calculus, and to obtain a richer collection of free algebras than those constructed earlier.In summary, this thesis shows that Topological Domain Theory supports a wide range of semantic constructions, including the standard domain-theoretic constructions, computational effects and polymorphism, all within a single setting
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