  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.
