We define a point-free construction of real exponentiation and logarithms, i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto \log_b(y)$, and we develop familiar algebraic rules for them. The point-free approach is constructive, and defines the points of a space as models of a geometric theory, rather than as elements of a set - in particular, this allows geometric constructions to be applied to points living in toposes other than Set. Our geometric development includes new lifting and gluing techniques in point-free topology, which highlight how properties of $\mathbb{Q}$ determine properties of real exponentiation. This work is motivated by our broader research programme of developing a version of adelic geometry via topos theory. In particular, we wish to construct the classifying topos of places of $\mathbb{Q}$, which will provide a geometric perspective into the subtle relationship between $\mathbb{R}$ and $\mathbb{Q}_p$, a question of longstanding number-theoretic interest.

This paper introduces an endofunctor VT on the category of frames that is parametrised by an endofunctor T on the category Set that satisfies certain constraints. This generalises Johnstone's construction of the Vietoris powerlocale in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T-powerlocale VT out of a frame is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss's coalgebraic cover modality. We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality and the combination of zero-dimensionality and compactness.

Abstract The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure. Overt sublocales and weakly closed sublocales are described, and related via a new notion of “rest closed” sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with “lower powerpoints”, arising from the impredicative theory of the lower powerlocale. Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with “upper powerpoints”, arising from the impredicative theory of the upper powerlocale.

Abstract. The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is known that these can be expressed by presheaves and the present paper applies this idea to configuration. We interpret the category theory informally as follows. Suppose ? is a category whose objects are interpreted as specifications, and for which each morphismu:X→Yis interpreted as contravariant ‘instance reduction’, reducing instances of specificationYto instances ofX. Then a presheafP: Set?oprepresents a collection of instances that is closed under reduction. We develop an algebraic account of presheaves in which we present configurations by generators (for components) and relations (for shared reducts), and we outline a proposed configuration language based on the techniques. Oriat uses diagrams to express colimits of specifications, and we show that Oriat's category Diag(?) of finite diagrams is equivalent to the category of finitely presented presheaves over ?.

Certain ‘Finite Structure Conditions’ on a geometric theory are shown to be sufficient for its classifying topos to be a presheaf topos. The conditions assert that every homomorphism from a finite structure of the theory to a model factors via a finite model, and they hold in cases where the finitely presentable models are all finite. The conditions are shown to hold for the theory of strongly algebraic (or SFP) information systems and some variants, as well as for some other theories already known to be classified by presheaf toposes. The work adheres to geometric constructivism throughout, and in consequence provides ‘topical’ categories of domains (internal in the category of toposes and geometric morphisms) with an analogue of Plotkin's double characterization of strongly algebraic domains, by sets of minimal upper bounds and by sequences of finite posets.

Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications.