In analogy with the varietal case, we give an abstract characterization of those categories occurring as regular epireflective subcategories of presheaf categories such that the inclusion functor preserves small sums. MSC 2000 : 18A40, 18F20. The aim of this short note is to add one step to the nice parallelism between presheaf categories and algebraic categories. Presheaf categories can be abs...

– We would like to solve the following problem: find a mathematical model formu­ lating I) quantum entanglement, II) particle-wave duality, III) universal objects (ur-sub-Planck objects): to be defined in terms of direct or inverse limits (defined by universal mapping proper­ ties) giving microcosm behaviors of space-time so as to give the smooth macrocosm space-time, and IV) the “curved” space...

– We would like to solve the following problem: find a mathematical model formu­ lating I) quantum entanglement, II) particle-wave duality, III) universal objects (ur-sub-Planck objects): to be defined in terms of direct or inverse limits (defined by universal mapping proper­ ties) giving microcosm behaviors of space-time so as to give the smooth macrocosm space-time, and IV) the “curved” space...

Presheaf models [7, 27, etc.] provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal an...

The cohomology theory known as Tmf, for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to re...

In the topos approach to quantum theory of Doering and Isham the Kochen–Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC∗–algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied withi...

We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) D in terms of the simplicial cohomologies of the dual complex ∆(D) with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of D and on their intersections. We use this formula to give a Hodge deco...

Abstract We provide a constructive version of the notion sheaf models univalent type theory. start by relativizing existing theory to presheaves over base category. Any Grothendieck topology category then gives rise family left-exact modalities, and we recover model localizing presheaf with respect this modalities. some examples.

The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that open-map bisimulations within a range of presheaf models are congruences for a general process language, in which CCS and related languages are easily encoded. The results ...

Recent work has shown that presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the π-calculus, a calculus for ‘mobile processes’ whose communication topology varies as channels are ...