Marta Bunge

The symmetric topos

Abstract We show that the 2-category of toposes and inverse images of geometric morphisms is 2-monadic over the 2-category of locally presentable categories and cocontinuous functors between them.

Spreads and the symmetric topos II

Abstract We introduce a new intrinsic notion of spread for toposes and geometric morphisms, and use it to give a “topological” characterization of Lawvere distributions on a topos. In the process, we relate spreads to zero-dimensional locales, and establish two new pure/spread factorizations for geometric morphisms. Our results are then applied to the study of the symmetric topos as a generalized lower power locale. In particular, we show that the symmetric topos is part of a Kock-Zoberlein 2-monad on toposes, give a new construction of bicomma squares in which the “lower” leg is an essential geometric morphism, characterize local connectedness in terms of the symmetric topos, and relate the symmetric and bagdomain constructions via the classifier of probability distributions.

Exponentiability and single universes

Abstract In this paper, we first consider known universes for pairs of opposite notions such as those of discrete fibrations/discrete opfibrations and of open/closed locale inclusions, and then extrapolate these in order to introduce new single universes for open/closed inclusions of subcategories and for functions/distributions on a topos. A key factor that these notions have in common is exponentiability in the ambient category. Along the way, we (1) prove that, for a factorization linearly ordered small category B , the category of discrete Giraud-Conduche fibrations over B is a (model generated) topos, (2) characterize locally closed inclusions in the category Cat of small categories, and (3) investigate “generalized coverings” in topos theory, as one of several possible single universes for local homeomorphisms and complete spreads over a topos.

Constructive theory of the lower power locale

This paper considers two main aspects of the lower power locale PL(X): first, its relation to the symmetric topos construction of Bunge and Carboni; and second, its points, which, it is shown, are equivalent to the weakly closed sublocales of X with open domain. This is done as part of a more general discussion of arbitrary weakly closed sublocales, including a new characterization using suplattice homomorphisms from 0{X) to Sub{\), and a new proof of a theorem of Jibladze relating them to fi-nuclei.

Unique factorisation lifting functors and categories of linearly-controlled processes

We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearly-controlled ones. To this end, we introduce and study a notion of path-linearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearly-controlled processes (viz., processes controlled by path-linearisable categories) are sheaf models.

On a bicomma object condition for KZ-doctrines

Abstract We study Kock–Zoberlein doctrines that satisfy a certain bicomma object condition. Such KZ-doctrines we call admissible. Our investigation is mainly motivated by the example of the symmetric monad on toposes. For an admissible KZ-doctrine, we characterize its algebras in terms of cocompleteness, and we describe its Kleisi 2-category by means of its bifibrations. We obtain in terms of bifibrations a “comprehensive” factorization of 1-cells (and 2-cells). Then we investigate admissibility when the KZ-doctrine is stable under change of base, thus obtaining a characterization of the algebras as linear objects, and the classification of discrete fibrations. Known facts about the symmetric monad are revisited, such as the Waelbroeck theorems. We obtain new results for complete spreads in topos theory. Finally, we apply the theory to the similar examples of the lower power locale and the bagdomain constructions. There is in domain theory an example of a different kind.

On the construction of the Grothendieck fundamental group of a topos by paths

Abstract The purpose of this paper is to compare the construction of the Grothendieck fundamental group of a topos using locally constant sheaves, with the construction using paths given by Moerdijk and Wraith. Our discussion focuses on the Grothendieck fundamental group in the general case of an unpointed (possibly pointless) topos, as constructed by Bunge. Corresponding results for topoi with a chosen base-point are then easily derived. The main result states that the basic comparison map from the paths fundamental group to the (unpointed version of the) Grothendieck fundamental group is an equivalence, under assumptions of the “locally paths simply connected” sort, as for topological spaces.

van Kampen theorems for toposes

Abstract In this paper we introduce the notion of an extensive 2-category, to be thought of as a “2-category of generalized spaces”. We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K op → CAT , which we think of as specifying the “coverings” of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Brown and Janelidze. The local properties required in this theorem are stated in terms of morphisms of effective descent for the pseudofunctor C . We specialize the general van Kampen theorem to the 2-category Top S of toposes bounded over an elementary topos S , and to its full sub 2-category LTop S determined by the locally connected toposes, after showing both of these 2-categories to be extensive. We then consider three particular notions of coverings on toposes corresponding, respectively, to local homeomorphisms, covering projections, and unramified morphisms; in each case we deduce a suitable version of a van Kampen theorem in terms of coverings and, under further hypotheses, also one in terms of fundamental groupoids. An application is also given to knot groupoids and branched coverings. Along the way we are led to investigate locally constant objects in a topos bounded over an arbitrary base topos S and to establish some new facts about them.

Cosheaves and distributions on toposes

Distributions on a Grothendieck topos were introduced by Lawvere [12] (cf. also [13]) as a generalization of the classical notion (cf. [20]) of real-valued distributions on a topological space. The cosheaves approach to distributions which is implicit in work of Pitts [19] is used here first, in order to answer affirmatively a question posed in [12] concerning the existence of the “symmetric topos” and next, in order to prove a structure theorem for categories of distributions on Grothendieck toposes that is similar in spirit to the Joyal-Tierney [11] structure theorem for Grothendieck toposes.

Toposes in Logic and Logic in Toposes

The purpose of this paper is to justify the claim that Topos theory and Logic (the latter interpreted in a wide enough sense to include Model theory and Set theory) may interact to the advantage of both fields. Once the necessity of utilizing toposes (other than the topos of Sets) becomes apparent, workers in Topos theory try to make this task as easy as possible by employing a variety of methods which, in the last instance, find their justification in metatheorems from Logic. Some concrete instances of this assertion will be given in the form of simple proofs that certain theorems of Algebra hold in any (Grothendieck) topos, in order to illustrate the various techniques that are used. In the other direction, Topos theory can also be a useful tool in Logic. Examples of this are independence proofs in (classical as well as intuitionistic) Set theory, as well as transfer methods in the presence of a sheaf representation theorem, the latter applied, in particular, to model theoretic properties of certain theories.