We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox [10], of the including topos....

Several papers and books by C. Isham, C.Isham-A. Doering, F. Van Oystaeyen, A. Mallios-I. Raptis, C. Mulvey, and Guts and Grinkevich, have been published on the methods of categories and sheaves to study quantum gravity. Needless to say, there are well-written treatises on quantum gravity whose methods are non-categorical and non-sheaf theoretic. This paper may be one of the first papers explai...

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...

It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of so-called hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of well-pointed topoi. Completeness theorems for higher-order logic result as corollaries. The main result of this paper is the following. Th...

This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal languag...

The purpose of this text is to equip the reader with an intuitive but precise understanding of elementary structures of category and topos theory. In order to achieve this goal, we provide a guided tour through category theory, leading to the definition of an elementary (Lawvere–Tierney) topos. Then we turn to the investigation of consequences of this definition. In particular, we analyse in de...

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where probabilities and probabilistic reasoning can be represented. In this way we obtain a mathematical definition of probabilistic sets. A valid and complete proof syst...

Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts’ to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize p...

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given `geometric' invariant as a property of its sites of de nition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to speci c invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equiva...

We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory)