basic constructions:
strong axioms
further
The concept of meros (plural meroi or meroses) is a relational analogue of that of a topos; where the morphisms in a topos are like functions, the morphisms in a meros are like relations. Like Set is the canonical base topos, so Rel is the archetypical meros. More generally, like topoi provide a context for generalized constructive set theory, so meroi provide a context for generalized relational set theory.
Where the boolean truth values of set theory are identified/generalized to elements of the subobject classifier of a topos, a meros instead represents Boolean values as degrees of inclusion between relations: between any two objects, there is a false relation (a zero morphism) and a true relation.
Following Kawahara 1995, we define an auxiliary venue for meroi first. An I-category is a dagger category where every hom-set has lattice structure. Specifically,
An I-category is a dagger category where, for all objects , there is a partial order on the morphisms , a least relation and greatest relation such that
and the partial order commutes with the dagger structure:
I-categories can have a zero object like typical dagger categories, but also a terminal object.
When it exists, a terminal object in an I-category (Def. ) is an object such that and for all objects , .
We next highlight specific morphisms within I-categories.
A morphism in an I-category (Def. ) is called a partial function, or univalent, when .
A morphism in an I-category (Def. ) is rational when there exist functions and (in the sense of Def. ) such that and .
For all there exists a quotient such that .
We also need the Dedekind formula, which relates any three morphisms between three objects.
We are now ready to define meroi. Instead of a lattice, we will now expect a complete Heyting algebra for each hom-set.
A meros is an I-category (Def. ) where:
The partial order is not just a lattice, but a complete Heyting algebra.
The Dedekind formula (1) holds whenever possible.
There is a terminal object (in the sense of Def. ).
For all we have .
Rel is the classic example of a meros, analogous to how Set is the archetypical topos.
Kawahara 1995 uses also the example of .
The notion of meroi was introduced in:
Last revised on October 9, 2021 at 11:32:40. See the history of this page for a list of all contributions to it.