On the Largest Cartesian Closed Category of Stable Domains

AbstractLet SABC (resp., SABC˜) be the category of algebraic bounded complete domains with conditionally multiplicative mappings, that is, Scott-continuous mappings preserving meets of pairs of compatible elements (resp., stable mappings). Zhang showed that the category of dI-domains is the largest cartesian closed subcategory of ω-SABC and ω-SABC˜, with the exponential being the stable function space, where ω-SABC and ω-SABC˜ are full subcategories of SABC and SABC˜ respectively which contain countablly based algebraic bounded complete domains as objects. This paper shows that:i)The exponentials of any full subcategory of SABC or SABC˜ are exactly function spaces;ii)SDABC˜ the category of distributive algebraic bounded complete domains, is the largest cartesian closed subcategory of SABC˜; The compact elements of function spaces in the category SABC are also studied

On Linear Information Systems

Scott's information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a "set-theoretic" interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic

D-modules on Spaces of Rational Maps and on other Generic Data

Let X be an algebraic curve. We study the problem of parametrizing geometric data over X, which is only generically defined. E.g., parametrizing generically defined (aka rational) maps from X to a fixed target scheme Y. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of D-modules "on" B (K) \G (A) /G (O), and we combine results about this category coming from the different presentations.Comment: 55 page

Categories of embeddings

AbstractWe present a categorical generalisation of the notion of domains, which is closed under (suitable) exponentiation. The goal was originally to generalise Girard's model of polymorphism to Fω. If we specialise this notion in the poset case, we get new cartesian closed categories of domains

Algebraic Kasparov K-theory. I

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.Comment: This is the final revised versio

A General Framework for the Semantics of Type Theory

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory