Boolean Topos Research Articles

Decidable objects and molecular toposes

We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particuar, we show that if $\mathcal{S}$ is a Boolean topos then, for every hyperconnected essential geometric morphism ${p : \mathcal{E} \rightarrow \mathcal{S}}$ such that the leftmost adjoint $p_!$ preserves finite products, $p$ is molecular and ${p^* : \mathcal{S} \rightarrow \mathcal{E}}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.

Measure theory over boolean toposes

AbstractIn this paper we develop a notion of measure theory over boolean toposes reminiscent of the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebras which take the form of a canonical$\mathbb{R}^{>0}$-principal bundle over any integrable locally separated boolean topos.

Structuring Co-constructive Logic for Proofs and Refutations

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following. It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such that classical logic (interpreted in a Boolean topos) is simulated where proofs and refutations are conclusive.

Categorical Formulation of Finite-Dimensional Quantum Algebras

We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.

Centralizer and faithful groupoid

We correlate existence and aspect of some properties of centralizers with the conceptual notion of Θ-faithful objects. We produce a large choice of examples from standard algebraic situations to topological models, going through less classical cases as the new notions of algebras introduced by Loday (Leibniz algebras, associative dialgebras and trialgebras), or the dual of any boolean topos.

The Michael completion of a topos spread

We continue the investigation of the extension into the topos realm of the concepts introduced by Fox (Cahiers Top. et Géometrie Diff. Catégoriques 36 (1995) 53) and Michael (Indag. Math. 25 (1963) 629) in connection with topological singular coverings. In particular, we construct an analog of the Michael completion of a spread and compare it with the analog of the Fox completion obtained earlier by the first two named authors (J. Appl. Algebra 113 (1996) 1). Two ingredients are present in our analysis of geometric morphisms ϕ : F→ E between toposes bounded over a base topos S . The first is the nature of the domain of ϕ, which need only be assumed to be a “definable dominance” over S , a condition that is trivially satisfied if S is a Boolean topos. The Heyting algebras arising from the object Ω S of truth values in the base topos play a special role in that they classify the definable monomorphisms in those toposes. The geometric morphisms F→ F′ over E which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of ϕ itself, which is assumed to be some kind of a spread. Applied to a spread, the (strongly pure, weakly entire) factorization obtained here gives what we call the “Michael completion” of the given spread. Whereas the Fox complete spreads over a topos E correspond to the S -valued Lawvere distributions on E (Acta Math. 111 (1964) 14) and relate to the distribution algebras (Adv. Math. 156 (2000) 133), the Michael complete spreads seem to correspond to some sort of “ S -additive measures” on E whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.

Linearization of graphic toposes via Coxeter groups

In an associative algebra over a field K of characteristic not 2, those idempotent elements a, for which the inner derivation [−, a] is also idempotent, form a monoid M satisfying the graphic identity aba= ab. In case K has three elements and M is such a graphic monoid, then the category of K-vector spaces in the topos of M-sets is a full exact subcategory of the vector spaces in the Boolean topos of G-sets, where G is a crystallographic Coxeter group which measures equality of levels in the category of M-sets.

Sheaf representation for topoi

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 and 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.

Categories, relations and dynamic programming

Dynamic programming is a strategy for solving optimisation problems. In this paper, we show how many problems that may be solved by dynamic programming are instances of the same abstract specification. This specification is phrased using the calculus of relations offered by topos theory. The main theorem underlying dynamic programming can then be proved by straightforward equational reasoning.The generic specification of dynamic programming makes use of higher-order operators on relations, akin to the fold operators found in functional programming languages. In the present context, a data type is modelled as an initial F-algebra, where F is an endofunctor on the topos under consideration. The mediating arrows from this initial F-algebra to other F-algebras are instances of fold – but only for total functions. For a regular category ε, it is possible to construct a category of relations Rel(ε). When a functor between regular categories is a so-called relator, it can be extended (in some canonical way) to a functor between the corresponding categories of relations. Applied to an endofunctor on a topos, this process of extending functors preserves initial algebras, and hence fold can be generalised from functions to relations.It is well-known that the use of dynamic programming is governed by the principle of optimality. Roughly, the principle of optimality says that an optimal solution is composed of optimal solutions to subproblems. In a first attempt, we formalise the principle of optimality as a distributivity condition. This distributivity condition is elegant, but difficult to check in practice. The difficulty arises because we consider minimum elements with respect to a preorder, and therefore minimum elements are not unique. Assuming that we are working in a Boolean topos, it can be proved that monotonicity implies distributivity, and this monotonicity condition is easy to verify in practice.

A topos with no geometric morphism to any Boolean one

In [1], Julian Cole asked whether every topos is definable over a Boolean one. Peter Johnstone partially answered this in [2] (corollary 3·7) by giving an example of a topos admitting no bounded morphism to any Boolean topos. Here we give a simple example where boundedness is unnecessary.

Finiteness and decidability: II

It has been known for some time ((6), p. 270; (4), theorem 9·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos ((1), example 7·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K-finite objects in , the Main Theorem of (1) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K-finite objects. The inclusionsdKf ⊆ Kf ⊆ sKf are clear.are clear.

Lawvere's basic theory of the category of categories

It is long known that Lawvere's theory in The category of categories as foundations of mathematics A[1] does not work, as indicated in Ishell's review [0]. Isbell there gives a counterexample that CDT—Category Description Theorem—[1, p. 15] is in fact not a theorem of BT (the Basic Theory of [1]) and suggests adding CDT to the axioms.Our starting point was the claim in [1] that “the basic theory needs no explicit axiom of infinity.” We define a model ℳ of BT in which all categories are finite. In particular, the “monoid of nonnegative integers N” coincides in ℳ with the terminal object 1. We study ℳ in some detail in order to establish the true status of various “theorems” or “metatheorems” of BT: The metatheorem of [1, p. 11] saying that the discrete categories form a category of sets, CDT, the theorem on p. 15, and the theorem on p. 16 of [1] are all nontheorems. The remaining results indicated in [1] concerning BT are provable. However, as the Predicative Functor Construction Schema—PFCS—are justified in [1] by using the “metatheorem” and CDT, we provide a proof of these two schemata by showing that the discrete categories of BT (or of convenient extensions of BT) form a two-valued Boolean topos.