Abstracts

Smooth and proper morphisms: a dictionary between algebra and topology

Jonathan Weinberger

I'll explain how the notion of smooth and proper morphisms w.r.t. to a Grothendieck fibration implements the idea of morphisms with respect to which one can take sums and products, resp. This establishes a dictionary between the category of categories, locales, and toposes. This is joint work with Mathieu Anel (https://arxiv.org/abs/2402.00331) and the perspective is informed by Mathieu Anel and André Joyal's Topo-logie (http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf).

On prime morphisms of algebraic frames

Oghenetega Ighedo

Abstract

A ring homomorphism φ : A → B is said to be prime at a prime ideal P of A if for a ∈ A and b ∈ B, φ(a)b ∈ PB implies that either a ∈ A or b ∈ PB. In other words, if the induced ring homomorphism φ : A/P → B/PB is torsion-free. Also, φ is called a prime morphism if it is prime at every prime ideal of its domain.

We investigate this notion in the category FIPFrm, of algebraic frames with the finite intersection property, whose maps are coherent maps. We give a condition for when a FIPFrm-morphism is prime at every prime element of L, where L is an object of FIPFrm. A justification that the condition cannot be relaxed is given in the form of an example. Compositions of prime morphisms will also be discussed.

We also introduce prime morphisms in the category CRFrm, of completely regular frames with frame homomorphisms. This we do with the help of the functor ZId: CRFrm → FIPFrm, which sends L ∈ CRFrm to ZId(RL), the lattice of z-ideals of RL, where RL is the ring of all continuous real- valued functions on L, and sends h ∈ CRFrm to the coherent map Rh in FIPFrm given by composition. We say that an ideal of a commutative ring with identity is a z-ideal in case it has the property that if two elements belong to the same set of maximal ideals, and one of them is in the ideal, then the other is in the ideal. 

Title: A crash course in Bornologies

Speaker: Gerald Beer CSULA

Abstract: By a bornology on a nonempty set  X, we mean a family of subsets that contains the singletons, that is stable under finite unions, and that is stable under taking subsets.  The prototype for a bornology is the so-called metric bornology: the family of metrically bounded subsets of a metric space.  Bornologies help us to understand large structure.  We enumerate some basic bornologies and give a few applications.  We give an old result of S.-T. Hu characterizing the bornologies on a metrizable space that are metric bornologies with respect to some compatible metric, and give a fairly recent result of J. Cabello-Sanchez characterizing those metric spaces (X,d) for which UC(X,R) is a ring.  We introduce the notion of bornological convergence of a sequence or net of closed subsets, of which Attouch-Wets convergence is the prototype, and give two applications to functional analysis.

Title: Some special classes of localic maps 

Speaker: Ana Belen Avilez Garcia 

Abstract: In this talk I would like to share my interest in some classes of localic maps (right adjoints of frame homomorphisms). A lot about localic maps is unknown since, in practice, point-free topologists tend to use frame homomorphisms. The motivation for studying these maps came from classical topology where some special classes of continuous maps can be characterized if they preserve certain order relations. In the point-free context we wanted to answer the question: when do localic maps preserve the rather below or completely rather below relation?

In order to give a general picture of the localic framework, I will first recall some notation I use for sublocales and a few basic results and notions about localic maps. Then I will introduce some classes of localic maps, some of which have interesting characterizations. Finally, I will present a characterization of those maps that preserve the completely rather below relation. 

Title: Frame Relations

Speaker: Andrew Moshier

Abstract: Though frames are ordered structures, frame theory rarely considers the category of frames to be order enriched (where comparison of morphisms is the important feature). This is likely because any separation as strong as T1 forces morphisms to be incomparable. Here I will begin a development of an enlarged category of frames and frame relations in which morphisms can meaningfully be compared, and in which many classical frame theoretic notions are manifested by placing conditions on these frame relations. With time (perhaps in another talk), this leads to a construction of the assembly of a frame, and a quite efficient proof that the assembly is ultraparacompact.