Susan B. Niefield

Spatial sublocales and essential primes

Abstract In this paper, we characterize those locales L such that every sublocale is spatial. The notion of essential prime is introduced and it is shown that every sublocale of L is spatial iff every element of L is the meet of its essential primes. This is related to work of Simmons characterizing when the assembly NL of nuclei on L is spatial. If L is the open subsets of a space X , these conditions correspond to X being weakly scattered. Finally, the main theorem is applied to the case X = spec R , where R is a commutative ring with identity.

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.

Strong de Morgan's law and the spectrum of a commutative ring

The logical principle (A G-B) V (B +A) = true, known as the Strong de Morgan’s law, is not in general valid in intuitionistic logic. P. T. Johnstone (in [6]) showed that this principle holds in the topos sh(X), of set-valued sheaves on a topological space X, and hence also in the locale O(X) of open subsets of X, if and only if every closed subspace of X is extremally disconnected. We investigate this property for X = Spec R, the spectrum of a commutative ring R with identity, and obtain ideal theoretic conditions characterizing those R whose spectra satisfy the Strong de Morgan’s law. These ideal theoretic properties are closely related to ones which characterize Dedekind domains, however, they involve the consideration of radical ideals. Section one develops the notion of a closed poset, which is a closed category whose underlying category is a partially ordered set. The main examples of closed posets that we consider are locales and ideals of a commutative ring R. We carry the analogy between the two examples further by establishing an identification between the locale B(Spec R) and the locale RIdl(R) of radical ideals of R. Section two presents the Strong de Morgan’s law and Johnstone’s results about the de Morgan laws for sh(X) and B(X). Using the analogy developed in section one, we define algebraic de Morgan’s laws for rings. This leads directly to our main theorem, which gives equivalent ideal theoretic conditions characterizing rings R, such that Spec R satisfies the Strong de Morgan’s law. If Spec R is Noetherian, we obtain several additional equivalences.

Sheaves of integral domains on stone spaces

It is well known in the theory of ring representations on Stone spaces, that if R is a commutative ring with 1, it is representable as a sheaf of fields (local rings) on a Stone space iff R is a von Neumann regular (exchange) ring. These results make use of the Pierce representation of R. The question of necessary and sufficient conditions on R to guarantee representability as a sheaf of integral domains is answered in this article. The appropriate condition on R is that of being an ‘almost weak Baer’ ring, where this means that Ann(a) is generated by its idempotent elements for all a e R. Two examples from rings of continuous functions distinguish this property from several closely related ring theoretic conditions.

Ideals of closed categories

A complete closed poset can be viewed as a commutative monoid in the closed category SI of complete lattices and sup-preserving maps. A lax adjunction between closed posets and the 2-category CSI of symmetric, monoidal closed categories over sup-lattices is described. This makes use of categories of ‘modules’ over a closed poset. If V is a suitably complete and cocomplete symmetric monoidal closed category, it is shown that the subobjects of the unit for ⊗ in V form a closed poset. The functoriality of this ideal construction is investigated; it is functorial in two different ways depending upon the type of morphism we consider for our closed categories. It alternately provides a right lax or right colax adjoint to the inclusion of closed posets into the 2-category of closed categories under consideration.

Locally Compact Path Spaces

Abstract It is shown that the space X[0,1], of continuous maps [0,1]→X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T1-space X, it follows that X[0,1] is locally compact if and only if X is locally compact and totally path-disconnected.

An algebraic approach to chaos

The aim of this paper is to study chaotic actions on objects other than metric spaces, e.g. locales and commutative rings. To do so, point-free versions of “topologically transitivity” and the “density of periodic points” are obtained for actions on a locale, and then generalized to a category which includes the desired objects of study.

Implementing finite structures in mathematica via a skeletal topos of finite sets

To implement finite structures in a symbolic computation program such as Mathematica, we consider a skeletal topos N which is equivalent to the category Setf of finite sets. Objects of N are nonnegative integers, and morphisms f : n → m are lists (f1,..., fn) of integers such that 1 ≤ fi ≤ m, for all i. A full and faithful functor from N to Setf is obtained by identifying n with the set [n] = {1,..., n} and identifying (f1,..., fn) with the function i ↦ fi. A topos structure on N (appropriate for Mathematica) is obtained by transporting the topos structure of Setf along a suitable pseudo-inverse C of the functor from N to Setf described above. The code for the Mathematica implementation included below is also available as a Mathematica Notebook.