Connection between Frobenius-defined “open” elements and openness of the range map

Determine the precise connection between Cockett–Guo–Hofstra’s characterization of restriction categories admitting a range category structure via “open” elements defined by the Frobenius condition (as in Proposition 2.13 of Cockett–Guo–Hofstra) and the conditions established in this paper whereby a preBoolean range semigroup yields a topological category whose range map r is open (Proposition 8.10) and compact slices form a Boolean range semigroup (Proposition 7.9(1)). In particular, clarify how the Frobenius-based notion of openness for elements in a restriction semigroup relates to, or is equivalent to, the openness of the range map r in the associated étale or ample topological category, thereby characterizing precisely when a restriction semigroup admits a compatible coEhresmann (cosupport) operation.

Background

Restriction categories that admit a range category structure are characterized in Cockett–Guo–Hofstra by requiring that all elements are “open,” a notion defined via the Frobenius condition and inspired by open maps in locale theory. This yields that restriction semigroups admitting a compatible coEhresmann (cosupport) operation correspond to those whose elements are open in that sense.

In contrast, this paper develops a topological perspective: starting from a preBoolean restriction or range semigroup with local units, one constructs a category of germs and shows that the openness of the topological range map r (and, in the strongly ample case, properties of compact slices) encodes the cosupport and yields Boolean range semigroups. The author notes that these approaches are parallel but that their precise relationship has not been established.

Clarifying the equivalence, implications, or exact correspondence between these two frameworks would unify the algebraic Frobenius-based criterion with the topological openness of r, providing a comprehensive characterization of when a restriction semigroup admits a compatible cosupport operation.

References

In [12, Proposition 2.13] restriction categories which admit the structure of a range category are characterized, which implies that restriction semigroups which admit a compatible coEhresmann structure are precisely those all whose elements are open in the sense of [12]. This notion is defined by means of the Frobenius condition and resembles the definition of an open locale map. Our Proposition 8.10 and Proposition 7.9(1) are somewhat parallel to this, but involve openness of the range map r of a topological category. The precise connection between the two approaches needs further investigation.

— Relating ample and biample topological categories with Boolean restriction and range semigroups  (2410.22181 - Kudryavtseva, 2024) in Remark 3.10, Section 3.2