Grothendieck Toposes Research Articles

Quotient toposes of discrete dynamical systems

Lawvere's open problem on quotient toposes has been solved for boolean Grothendieck toposes but not for non-boolean toposes. As a simple and non-trivial example of a non-boolean topos, this paper provides a complete classification of the quotient toposes of the topos of discrete dynamical systems, which, in this context, are sets equipped with an endofunction. This paper also offers an order-theoretic framework to address the open problem, particularly useful for locally connected toposes.Our result is deeply related to monoid epimorphisms. At the end of this paper, utilizing the theory of lax epimorphisms in the 2-category Cat, we explain how (non-surjective) monoid epimorphisms from N correspond to (non-periodic) behaviors in discrete dynamical systems.

Toposes of Topological Monoid Actions

We demonstrate that categories of continuous actions of topological monoids on discrete spaces are Grothendieck toposes. We exhibit properties of these toposes, giving a solution to the corresponding Morita-equivalence problem. We characterize these toposes in terms of their canonical points. We identify natural classes of representatives with good topological properties, 'powder monoids' and then 'complete monoids', for the Morita-equivalence classes of topological monoids. Finally, we show that the construction of these toposes can be made (2-)functorial by considering geometric morphisms induced by continuous semigroup homomorphisms.

Covariant Isotropy of Grothendieck Toposes and Extensive Categories

Covariant Isotropy of Grothendieck Toposes and Extensive Categories

On sheaf cohomology and natural expansions

In this survey paper, we present \v{C}ech and sheaf cohomologies -- themes that were presented by Koszul in University of S\~ao Paulo during his visit in the late 1950s -- we present expansions for categories of generalized sheaves (i.e, Grothendieck toposes), with examples of applications in other cohomology theories and other areas of mathematics, besides providing motivations and historical notes. We conclude explaining the difficulties in establishing a cohomology theory for elementary toposes, presenting alternative approaches by considering constructions over quantales, that provide structures similar to sheaves, and indicating researches related to logic: constructive (intuitionistic and linear) logic for toposes, sheaves over quantales, and homological algebra.

New Trends in Stochastic Geometry for Wireless Networks: A Tutorial and Survey

Next-generation wireless networks are expected to be highly heterogeneous, multilayered, with embedded intelligence at both the core and edge of the network. In such a context, system-level performance evaluation will be very important to formulate relevant insights into tradeoffs that govern such a complex system. Over the past decade, SG has emerged as a powerful analytical tool to evaluate the system-level performance of wireless networks and capture their tendency toward heterogeneity. However, with the imminent onset of this crucial new decade, where global commercialization of fifth generation (5G) is expected to emerge and essential research questions related to beyond 5G (B5G) are intended to be identified, we are wondering about the role that a powerful tool, such as SG, should play. In this article, we first aim to track and summarize the novel SG models and techniques developed during the last decade in the evaluation of wireless networks. Next, we will outline how SG has been used to capture the properties of emerging RANs for 5G/B5G and quantify the benefits of key enabling technologies. Finally, we will discuss new horizons that will breathe new life into the use of SG in the foreseeable future, for instance, using SG to evaluate performance metrics in the visionary paradigm of molecular communications. Also, we will review how SG is envisioned to cooperate with machine learning that is seen as a crucial component in the race toward ubiquitous wireless intelligence. Another important insight is Grothendieck's toposes, which is considered as a powerful mathematical concept that can help to solve long-standing problems formulated in SG.

Topologies for intermediate logics

We investigate the problem of characterizing the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras, and introduce natural analogues of the double negation and De Morgan topologies on an elementary topos for a wide class of intermediate logics.

A short proof of the localic groupoid representation of Grothendieck toposes

It is known that each Grothendieck topos is the category of G \mathbb {G} -equivariant sheaves for some localic groupoid G \mathbb {G} . A simple proof of this is given which relies on the recently observed fact that the pullback adjunction between locales induced by any geometric morphism satisfies Frobenius reciprocity.

Sheaves on involutive quantales: Grothendieck quantales

We show that sheaves on Grothendieck quantales, which are the quantales that correspond to Grothendieck toposes, may be defined in several equivalent ways. The main topic of this paper is the characterization of internal posets and suplattices in the topos of sheaves on a Grothendieck quantale Q, in terms of categories enriched in a suitable quantaloid and by means of Q-modules. We thereby extend the existing results concerning locales.

A General Method for Building Reflections

We establish a general method for generating reflections between categories. We then apply our technique to generate adjunctions starting from geometric morphisms between Grothendieck toposes; as particular cases, we recover various well-known Stone-type adjunctions and establish several new ones.

Totally distributive toposes

A locally small category E is totally distributive (as defined by Rosebrugh and Wood) if there exists a string of adjoint functors t⊣c⊣y, where y:E→Ê is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by Kelly and Lawvere and by Kennett, Riehl, Roy, and Zaks.

Universal models and definability

AbstractWe establish some general results on universal models in Topos Theory and show that the investigation of such models can shed light on problems of definability in Logic as well as on De Morgan's law and the law of excluded middle for Grothendieck toposes.

A characterization theorem for geometric logic

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.

Forcing for IZF in Sheaf Toposes

Abstract In [Category-theoretic models for intuitionistic set theory, 1985] D. Scott showed how the interpretation of intuitionistic set theory IZF in presheaf toposes can be reformulated in a more concrete fashion à la forcing as known to set theorists. In this note we show how this can be adapted to the more general case of Grothendieck toposes dealt with abstractly in [Fourman, J. Pure Appl. Algebra 19: 91–101, 1980, Hayashi, On set theories in toposes, Springer, 1981].

Sheaf toposes for realizability

We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over different partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott.

Cocomplete toposes whose exact completions are toposes

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos. We also show that for any topological space X , the exact completion of Sh ( X ) is a topos if and only if X is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.

Saturated models of intuitionistic theories

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with.

Topical categories of domains

This paper shows how it is possible to express many techniques of categorical domain theory in the general context of topical categories (where ‘topical’ means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as ‘generalized topological spaces’ to be treated in a transparently spatial way, and also shows the constructivity of the arguments. The theory of strongly algebraic (SFP) domains is given as a case study in which the topical category is Cartesian closed.

The locally connected coclosure of a Grothendieck topos

Locally connected Grothendieck toposes are shown to be coreflective in Grothendieck toposes. We refer to this coreflection as locally connected coclosure. The locally connected coclosure of a localic topos is localic. A topos and its locally connected coclosure are seen to have equivalent categories of Lawvere distributions. A class of locales is produced whose locally connected coclosures are trivial. We show how the locally connected coclosure can be described in terms of the locally connected coclosure of a localic cover.

Definability and descent

The present note offers a short argument for the descent theorems of Zawadowski [10] (originally [9]) and Makkai [6], which were conjectured by Pitts after the descent theorem of Joyal and Tierney [3] for open geometric morphisms of (Grothendieck) toposes. The original proofs, which involve variants of Makkai's [5] duality for first order logic, are rather involved and there has been considerable interest in locating simpler proofs. Viewed categorically, the descent theorems establish a bicategorical exactness property (conservative morphisms are effective descent) for pretop (the 2-category of small pretoposes, pretopos functors, and natural transformations), for exact (exact categories, exact functors, and natural transformations), and for bpretop* (Boolean pretoposes, pretopos functors, and natural isomorphisms). Viewed logically, they fragment into a familiar Beth/Tarski-type definability theorem and a covering theorem for certain functors on PCΔ-categories (groupoids, in the Boolean case); the latter (as the former) is a arithmetical statement about the syntax of first order logic ([6, §3])).The argument here involves special models and, independently, a continuity lemma of Makkai. The use of special models is axiomatic in that only a few properties (listed below) are needed. The continuity lemma, 9.1 of [6], is established via forcing and can be read independently of the rest of that paper. Because of its interest to both the model theorist and the category theorist, the argument is first given as straight model theory and afterwards it is briefly indicated how the descent theorems follow.

Regular and exact completions

The regular and exact completions of categories with weak limits are proved to exist and to be determined by an appropriate universal property. Several examples are discussed, and in particular the class of examples given by categories monadic over a power of Set: any such a category is in fact the exact completion of the full subcategory of free algebras. Applications to Grothendieck toposes and geometric morphisms, and to epireflective hulls are also discussed.