Peter T. Johnstone

Connected limits, familial representability and Artin glueing

We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.

On the structure of categories of coalgebras

Abstract Consideration of categories of transition systems and related constructions leads to the study of categories of F -coalgebras, where F is an endofunctor of the category of sets, or of some more general ‘set-like’ category. It is fairly well known that if E is a topos and F : E → E preserves pullbacks and generates a cofree comonad, then the category of F -coalgebras is a topos. Unfortunately, in most of the examples of interest in computer science, the endofunctor F does not preserve pullbacks, though it comes close to doing so. In this paper we investigate what can be said about the category of coalgebras under various weakenings of the hypothesis that F preserves pullbacks. It turns out that almost all the elementary properties of a topos, except for effectiveness of equivalence relations, are still inherited by the category of coalgebras; and the latter can be recovered by embedding the category in its effective completion. However, we also show that, in the particular cases of greatest interest, the category of coalgebras is not itself a topos.

Open maps of toposes

We present a concept of “openness” for geometric morphisms between toposes, which extends the usual notion of “open map of spaces”, and investigate some of its properties. Our two main results are that open maps are precisely those whose inverse image functors preserve full first-order logic, and that open maps and open surjections are stable under bounded pullback in Top. These two results extend earlier work of C. J. Mikkelsen [11] and M. Coste [3] respectively; they are also closely related to recent work of A. Joyal and M. Tierney [9].

The gleason cover of a topos, II

In [13], it was observed that the topos Shv(X) of sheaves on a topological space X satisfies De Morgan’s law iff X is extremally disconnected, and it was claimed that most of the occurrences of extremally disconnected spaces in general topology and functional analysis could in fact be explained as appeals to De Morgan’s law in the internal logic of Shv(X). In this and a subsequent paper [17], we set out to justify this claim in the case of a particular topological construction involving extremally disconnected spaces: the Gleason cover. In [7], A.IM. Gleason proved the following two results:

Rings, fields, and spectra

Let T be a finitary algebraic theory [15] (or equivalently a variety of universal algebras [l]). It is well known that if d is any category with finite products, we may define the notion of “T-model in 8” by interpreting each m-ary operation of T as a morphism A” -+ A (A being the underlying object of the T-model), and each equation of T as a commutative diagram. However, if we wish to impose additional (nonequational) first-order axioms on a T-model, we must demand that d have certain additional structure: for example, that of a regular category [22] or of a logical category [25]. Throughout this paper, we shall assume that d is a topos (in the sense of Lawvere and Tierney, see [9] or [27], though readers unfamiliar with this notion of topos may substitute that of “Grothendieck topos” [5] without serious damage). We shall also require that d have a natural number object; it is by now well known that this assumption implies the existence of a free T-model functor for any finitely presented algebraic theory T [IO, 171. The axioms of a topos are certainly sufficient to permit the interpretation of arbitrary first-order formulas (and indeed of higher-order formulas) (cf. [18, 191). However, we shall find it convenient to distinguish certain formulas called geometric formulas, which are built up from the atomic formulas by the use of the connectives A, V and 3 (but not G-, 1, or V). Ageometric sequent is a formula of the form (v 3 #)>, where y and 9 are geometric formulas; and a geometric theory is a pair T = (L, A) consisting of a languagei L and a set A of geometric sequents of L, called axioms of T. (By convention, we also include the vacuous sentences true and false as geometric formulas; this enables us to replace any geometric formula y by the sequent (true 3 v), and its negation by the sequent (p’ + false),

Classifying toposes for first-order theories

Abstract By a classifying topos for a first-order theory T , we mean a topos ∄ such that, for any topos F models of T in F correspond exactly to open geometric morphisms F → E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic.

Fibrations and partial products in a 2-category

We introduce a new intrinsic definition of fibrations in a 2-category, and show how it may be used (in conjunction with a suitable limit-colimit commutation condition) to define a 2-categorical version of the notion of partial product. We use these notions to show that partial products exist for all fibrations in the 2-category of (small) categories, and to identify the fibrations in the 2-category of toposes and geometric morphisms.

Quasitoposes, quasiadhesive categories and artin glueing

Adhesive categories are a class of categories in which pushouts along monos are well-behaved with respect to pullbacks. Recently it has been shown that any topos is adhesive. Many examples of interest to computer scientists are not adhesive, a fact which motivated the introduction of quasiadhesive categories. We show that several of these examples arise via a glueing construction which yields quasitoposes. We show that, surprisingly, not all such quasitoposes are quasiadhesive and characterise precisely those which are by giving a succinct necessary and sufficient condition on the lattice of subobjects.

An axiomatics for categories of transition systems as coalgebras

We consider a finitely branching transition system as a coalgebra for an endofunctor on the category Set of small sets. A map in that category is a functional bisimulation. So, we study the structure of the category of finitely branching transition systems and functional bisimulations by proving general results about the category H-Coalg of H-coalgebras for an endofunctor H on Set. We give conditions under which H-Coalg is complete, cocomplete, symmetric monoidal closed, regular, and has a subobject classifier.

Corrigenda for ‘Connected limits, familial representability and Artin glueing’

Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main results of the paper, concerning when a category obtained by Artin glueing is a topos, we feel it is appropriate to publish this correction in the hope that it may prevent future readers of Carboni and Johnstone (1995) from being misled. We are grateful to Tom Leinster and to Marek Zawadowski for drawing our attention to the two errors.