arXiv subjects

José Siqueira

Publications and source records attributed to José Siqueira.

3 recordsLinked to original sources

arXiv2026

Double-functorial representation of regular hyperdoctrines

It is well-known that pseudofunctors from bicategories of spans are equivalent to Beck-Chevalley bifibrations, and therefore capture the relationships underlying the adjunctions suitable as semantics for existential quantification. This was further expanded upon by Dawson, Paré and Pronk in the context of double categories. By viewing hyperdoctrines from a double-categorical lens, this paper shows that we can also characterise the Frobënius property: (generalised) regular hyperdoctrines correspond to certain lax symmetric monoidal pseudo double functors from spans to quintets whose monoidal laxators provide companion commuter cells (in the sense of Paré). This facilitates the study of the compositionality of regular hyperdoctrines and hints at a new notion of regular double hyperdoctrine. As an application, we discuss how we can recover a form of graphical regular logic suitable for modelling specifications of systems (e.g., port-plugging systems) that compose operadically.

math.CT↗

arXiv2023

Nonstandard proof methods in toposes

We determine sufficient structure for an elementary topos to emulate E. Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of $G$-sets and Boolean étendues.

math.CT↗

arXiv2022

Artin glueings of toposes as adjoint split extensions

Artin glueings of frames correspond to adjoint split extensions in the category of frames and finite-meet-preserving maps. We extend these ideas to the setting of toposes and show that Artin glueings of toposes correspond to a 2-categorical notion of adjoint split extensions in the 2-category of toposes, finite-limit-preserving functors and natural transformations. A notion of morphism between these split extensions is introduced, which allows the category Ext(H,N) to be constructed. We show that Ext(H,N) is contravariantly equivalent to Hom(H,N), and moreover, that this can be extended to a 2-natural contravariant equivalence between the Hom 2-functor and a naturally defined Ext 2-functor.

math.CT↗