arXiv subjects

David Holgate

Publications and source records attributed to David Holgate.

3 recordsLinked to original sources

arXiv2023

Topogenous structures and related families of morphisms

In a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous structures induced by pointed and co-pointed endofunctors. We also lift the topogenous structures along an $\mathcal{M}$-fibration. This permits one to obtain the lifting of interior and neighbourhood operators along an $\mathcal{M}$-fibration and includes the lifting of closure operators found in the literature. A number of examples presented at the end of the paper demonstrates our results.

math.CT↗

arXiv2023

Quasi-uniform structures and functors

We study a number of categorical quasi-uniform structures induced by functors. We depart from a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, then define the continuity of a $\mathcal{C}$-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) on $\mathcal{C}$ and use it to describe the quasi-uniformities induced by pointed and copointed endofunctors of $\mathcal{C}$. In particular, we demonstrate that every quasi-uniformity on a reflective subcategory of $\mathcal{C}$ can be lifted to a coarsest quasi-uniformity on $\mathcal{C}$ for which every reflection morphism is continuous. Thinking of categories supplied with quasi-uniformities as large ``spaces'', we generalize the continuity of $\mathcal{C}$-morphisms (with respect to a quasi-uniformity) to functors. We prove that for an $\mathcal{M}$-fibration or a functor that has a right adjoint, we can obtain a concrete construction of the coarsest quasi-uniformity for which the functor is $continuous$. The results proved are shown to yield those obtained for categorical closure operators. Various examples considered at the end of the paper illustrate our results.

math.CT↗

arXiv2023

Topogenous structures on faithful and amnestic functors

Departing from a suitable categorical concept of topogenous orders defined relative to the bifibration of subobjects, this note introduces and studies topogenous orders on faithful and amnestic functors. Amongst other things, it is shown that this approach captures the formal closure operators and leads to the introduction of formal interior operators. Turning to special morphisms relative to the orders introduced, we show that a morphism is strict relative to an order if the order preserves codomains of its cocartesian liftings while a morphism is final if the order reflects domains of its cartesian liftings. Key examples in topology and algebra that demonstrate our results are included.

math.CT↗