arXiv subjects

João Paulo Santos

Publications and source records attributed to João Paulo Santos.

5 recordsLinked to original sources

arXiv2025

Scott locales

We prove some facts about locales $L$ equipped with the Scott topology $Ω(L)$, in particular studying a canonical frame homomorphism $ϕ:Ω(L)\to L$ which is motivated by an application to cognitive science. Such a topological locale $L$ is called a Scott locale if the inclusion of primes $p:Σ(L)\to L$ is continuous. We prove that the spectrum $Σ(L)$ of a Scott locale $L$ is necessarily $T_1$, and that preregular locales (a generalization of regular locales) are Scott locales. If $L$ is the topology of a topological space $X$ we find a (necessarily unique) continuous map $f:X\to L$ such that $f^{-1}=ϕ$ and compare it with the points-to-primes map $p:X\to L$, showing that $f=p$ if and only if $X$ is preregular, and that a sober space $X$ is Hausdorff if and only if $X$ is $T_1$ and $f(X)\subseteqΣ(L)$.

math.CT↗

arXiv2018

The codual quotient vector bundle

Given a quotient vector bundle $\mathcal A$ over $X$ with kernel map $κ: X\to\mathrm{Max}\,A$ we study the codual bundle with fiber at each point $x\in X$ isomorphic to the dual of $κ(x)$. Applying the adjunction between quotient vector bundles and linearized locales to the codual bundle leads to a new adjunction between the category of all quotient vector bundles and a category whose objects are locales $L$ equipped with a contravariant morphism $L\to\mathrm{Max}\, A$.

math.CT↗

arXiv2017

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space $A$, a quotient vector bundle for $A$ is a quotient of a trivial vector bundle $π_2:A\times X\to X$ by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space $X$. Hyperspaces consisting of linear subspaces of $A$, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If $A$ is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.

math.FA↗

arXiv2016

Linear structures on locales

We define a notion of morphism for quotient vector bundles that yields both a category $\textit{QVBun}$ and a contravariant global sections functor $C:\textit{QVBun}^{\textrm{op}}\to\textit{Vect}$ whose restriction to trivial vector bundles with fiber $F$ coincides with the contravariant functor $\textit{Top}^{\textrm{op}}\to\textit{Vect}$ of $F$-valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological spaces and locales: (i) a linearized topological space is a spectral vector bundle, by which is meant a mildly restricted type of quotient vector bundle; (ii) a linearized locale is a locale $\triangle$ equipped with both a topological vector space $A$ and a $\triangle$-valued support map for the elements of $A$ satisfying a continuity condition relative to the spectrum of $\triangle$ and the lower Vietoris topology on $\operatorname{Sub} A$; (iii) we obtain an adjunction between the full subcategory of spectral vector bundles $\textit{QVBun}_Σ$ and the category of linearized locales $\textit{LinLoc}$, which restricts to an equivalence of categories between sober spectral vector bundles and spatial linearized locales. The spectral vector bundles are classified by a finer topology on $\operatorname{Sub} A$, called the open support topology, but there is no notion of universal spectral vector bundle for an arbitrary topological vector space $A$.

math.CT↗

arXiv2013

Rank-stable limit of completed moduli spaces of instantons

Nakajima introduced a resolution of singularities of the Donaldson-Uhlenbeck completion of the moduli space of based instantons over $S^4$. For $k\leq 4$, we extend this result to $\mathbb P^2$ and compute, in the rank-stable limit, the homotopy type of these spaces by showing that, in this limit, these constructions yield universal bundles.

math.AT↗