arXiv subjects

Willian Ribeiro

Publications and source records attributed to Willian Ribeiro.

3 recordsLinked to original sources

arXiv2019

Compactly generated spaces and quasi-spaces in topology

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated spaces}$ and $\textit{quasi-spaces}$ in this setting. Moreover, for a class $\mathcal{C}$ of objects we generalize the notion of $\textit{$\mathcal{C}$-generated spaces}$, from which we derive, for instance, a general concept of $\textit{Alexandroff spaces}$. Furthermore, as done for $\mathsf{Top}$, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

math.CT↗

arXiv2019

Cartesian closed exact completions in topology

Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over $\mathsf{Set}$, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of $(\mathbb{T},V)$-categories and show that, under suitable conditions, every $(\mathbb{T},V)$-injective category is exponentiable in $(\mathbb{T},V)\text{-}\mathsf{Cat}$.

math.CT↗

arXiv2018

On generalized equilogical spaces

In this paper we carry the construction of equilogical spaces into an arbitrary category $\mathsf{X}$ topological over $\mathsf{Set}$, introducing the category $\mathsf{X}$-$\mathsf{Equ}$ of equilogical objects. Similar to what is done for the category $\mathsf{Top}$ of topological spaces and continuous functions, we study some features of the new category as (co)completeness and regular (co-)well-poweredness, as well as the fact that, under some conditions, it is a quasitopos. We achieve these various properties of the category $\mathsf{X}$-$\mathsf{Equ}$ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion $\mathsf{X}_{_{\rm ex}}$, and as the regular completion $\mathsf{X}_{_{\rm reg}}$. We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the $(\mathbb{T},\mathsf{V})$-$\mathsf{Cat}$ setting.

math.CT↗