# Seminar

Monday, 23 January 2023, Aula F (Ca’ Vignal 1).

15:00

Tatsuji Kawai (Japan Advanced Institute of Science and Technology)

*Predicative theory of stably locally compact locales*

We give a predicative presentation of stably locally compact locales, the class of locales which includes locally compact regular locales (e.g., localic reals) as its subclass. In our setting, a stably locally compact locale is presented as a quasi-proximity lattice, a quasi-bounded distributive lattice (distributive lattice without top) together with a certain idempotent relation on it. Using this structure, we construct a coreflection from the category of locally compact regular locales and cobounded maps to that of stably locally compact locales and perfect maps. The construction of this coreflection generalizes Dedekind’s construction of real numbers as pairs of a lower and an upper cut.

16:30

Hajime Ishihara (Japan Advanced Institute of Science and Technology)

*Reflexive combinatory algebras*

We introduce the notion of reflexivity for combinatory algebras. Reflexivity can be thought of as an equational counterpart of the Meyer-Scott axiom of combinatory models, which indeed allows us to characterise an equationally definable counterpart of combinatory models. This new structure, called strongly reflexive combinatory algebra, admits a finite axiomatisation with seven closed equations, and the structure is shown to be exactly the retract of combinatory models. Lambda algebras can be characterised as strongly reflexive combinatory algebras which are stable.

This is a joint work with Marlou M. Gijzen and Tatsuji Kawai.