A general path-lifting theorem, which fails in the context of topological spaces, is shown to hold for toposes, for locales (a slight generalization of topological spaces), and hence for complete separable metric spaces. This result generalizes the known fact that any connected locally connected topos (respectively complete separable metric space) is path-connected. In [MW] we proved that every...

However, the replete construction is of a very general nature and most of its properties are independent of the axioms for Σ and of E being a topos. We consider the replete construction in the more general setting of a cartesian closed B-fibration p: C → B, where B is a category with finite products. This encompasses the case of a cartesian closed category (i.e. p: C → 1) and that of a locally ...

In this paper I will give a summary of the description of State Vector reduction using Topos Theory put forward by Isham in [1]. I will first briefly explain some mathematical concepts regarding Categories and Topoi. For a more detailed and complete explanation of the subjects of Topos and Category Theory the readers should refer to Topos-Physics and reference therein. I would like to point out...

The pentalogy [60, 61, 62, 76, 63] is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure DTfcq. We...

In [3], the following semantical picture of modified realizability is sketched: Consider the topos Set, whose objects are functions between sets, and whose arrows are commutative squares. The logic of Set corresponds to “Kripke forcing over a 2-element linear order”. Over Set, one can define the standard realizability topos Eff→ (not to be confused with Eff!). The modified realizability topos, ...

The article studies the topos Sets of actions of an 8-element monoid T on sets. It is called the triadic topos as T is isomorphic to the monoid of affine transformations of the twelve tone system Z12, leaving a given major or minor triad invariant. The subobject classifier Ω of this topos and its Lawvere-TierneyTopologies j are calculated. We explore the characteristic functions of the restrict...

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a ‘mathematical universe’ with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how ...

We survey indications from different branches of Physics that the fine scale structure of spacetime is not adequately described by a man-ifold. Based on the hints we accumulate, we propose a new structure, which we call a quantum topos. In the process of constructing a quantum topos for quantum gravity, we propose a new, operational approach to the problem of the obervables in quantum gravity, ...

It is know in topos theory that axiom of choice implies that the topos is Boolean. In this paper we want to prove and generalize this result in the context of allegories. In particular, we will show that partial identities do have complements in distributive allegories with relational sums and total splittings assuming the axiom of choice. Furthermore, we will discuss possible modifications of ...