natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
In set theory, a setoid is usually described as a set with a pseudo-equivalence relation. However, in homotopy type theory or more generally in intensional type theory, there are multiple different ways to define the concept of a “collection with a pseudo-equivalence relation”. Presetoids arise from the most general possible notion of “collection with a pseudo-equivalence relation” in homotopy type theory , in the same way that the notion of precategory and pregroupoid arise when defining categories and groupoids.
A pseudo-equivalence relation on a type is a family of sets of morphisms for each element and , which comes with the following additional structure
for each element a family of morphisms
for each object and a family of functions
for each object , , and , a family of functions
A presetoid consists of a type of objects and a pseudo-equivalence relation .
The structure-preserving maps between two presetoids are called dagger-functors. A dagger-functor between two presetoids and consists of
a function
a family of functions
and identifications indicating that the identity morphism, dagger, and the composition operation are preserved:
An dagger functor between two presetoids is faithful if every morphism function is an injection. A dagger functor is embedding-on-objects if the object function is an embedding, and equivalent-on-objects if the object function is a equivalence of types.
A subpresetoid of a presetoid is a presetoid with an faithful embedding-on-objects dagger functor . A subpresetoid of is a subpregroupoid if the pseudo-equivalence relation of additionally come with:
for every object , , , and and morphism , , and , an identification
for every object , and morphism , an identification
for every object , and morphism , an identification
for every object , and morphism , an identification
for every object , and morphism , an identification
A subpregroupoid of a presetoid is a maximal subpregroupoid if the dagger functor is equivalent-on-objects, and additionally if, for every other subgroupoid of with a faithful embedding-on-objects dagger functor , the type of faithful embedding-on-objects dagger functors with an identification is contractible.
The core pregroupoid of a setoid is the maximal subpregroupoid of . Since the core is a pregroupoid, every morphism in the core is an isomorphism, and the type of morphisms between two objects and in the core of is denoted as .
There are two ways to show that two objects of a presetoid are identified with each other: by way of the identity type, and by way of the type of isomorphisms in the core pregroupoid of a presetoid. Univalent setoids are precisely the presetoids which the two notions of identifying objects with each other coincide.
For each object and of a presetoid , there is a function
from the identity type to the type of isomorphisms in the core pregroupoid of . A presetoid is a univalent setoid if for each object and the function
is an equivalence of types.
Since every type of morphisms in the core of the presetoid is a h-set, this means that every univalent setoid is an h-groupoid.
Last revised on September 22, 2022 at 11:47:18. See the history of this page for a list of all contributions to it.