Category theoretic structure of setoids

Constructing Categories and Setoids of Setoids in Type Theory

In this paper we consider the problem of building rich categories of setoids, in standard intensional Martin-Löf type theory (MLTT), and in particular how to handle the problem of equality on objects in this context. Any (proof-irrelevant) family F of setoids over a setoid A gives rise to a category C(A,F ) of setoids with objects A. We may regard the family F as a setoid of setoids, and a cruc...

Constructions of categories of setoids from proof-irrelevant families

When formalizing mathematics in constructive type theories, or more practically in proof assistants such as Coq or Agda, one is often using setoids (types with explicit equivalence relations). In this note we consider two categories of setoids with equality on objects and show, within intensional Martin-Löf type theory, that they are isomorphic. Both categories are constructed from a fixed proo...

Setoids and universes

Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disjoint sums. It is shown that, given the minimal type universe, all epimorphisms are surjections, and disjoin...

Remarks on the relation between families of setoids and identity in type theory

can be turned into a family represented by a function. This can be contrasted to MartinLöf type theory [10], and other theories of dependent types, where a family of types is a basic mathematical object. Following the tradition in constructive mathematics (see [2]) a set is commonly understood in type theory as a setoid, that is, a type together with an equivalence relation. However the notion ...

Fuzzy sets form A meta-system-theoretic point of view