On Geometric Implications

TL;DR

This paper explores the concept of geometric implications in topology, defining a stable family of implications that are invariant under inverse images, and characterizes their algebraic and categorical properties.

Contribution

It introduces a new notion of geometric implications using categorical fibrations, identifying the greatest geometric categories and providing representation theorems.

Findings

No non-trivial geometric category over the full category of spaces.

Characterization of geometric categories over subcategories of spaces.

Representation theorem for implications via Yoneda and Kripke-style models.

Abstract

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in [1]. Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category S, provided that S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication.