In this paper, we propose a topological metaphor for computations: computing consists in moving through a path in a data space and making some elementary computations along this path. This idea underlies an experimental declarative programming language called mgs. mgs introduces the notion of topological collection: a set of values organized by a neighborhood relationship. The basic computation...

1. Introduction Mac Lane and Moerdijk, 1992, in their thorough introduction to topos theory, start their Prologue by saying – A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory. Indeed , a topos can be considered both as a " generalized space " an...

We introduce the operators modified limit and accumulation on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable metric space to a computable Banach space is internally computable. We motivate the need for internal concepts of computability by observing that the complexity of th...

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We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures-it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper L...

In this article we will propose a formal model for smart spaces. The key ingredients of this model are user, sensors and actuators. We show how we can derive a programming model based upon knowledge and software engineering. We describe how to interpret the world using an ontology and use this ontology to describe the intentional effect of available actuators. The model is loosely based on the ...