Abstract
AbstractIn the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e., objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.
Highlights
Following a tradition initiated by Bishop [4], the constructive notion of set is taken to be a collection of elements together with an equivalence relation on it, seen as the equality of the set
In Martin-Lof type theory this is realised with the notion of setoid, which consists of a type together with a type-theoretic equivalence relation on it [29]
A category-theoretic counterpart is provided by the exact completion construction Cex, which freely adds quotients of equivalence relations to a category C with finite limits [6, 8]
Summary
The same situation arises for any model of the Constructive Elementary Theory of the Category of Sets (CETCS), a first order theory introduced by the second author in [26] in order to formalise properties of the category of sets in the informal set theory used by Bishop. Any model E of CETCS is the exact completion of its projective objects, which form a quasi-cartesian category P. As shown by Carboni and Vitale [8], any category with weak finite limits C can be regarded as a projective cover of an exact category, known as the exact completion of C This construction consists in freely adding quotients of pseudo-equivalence relations and we describe it in the case of a quasi-cartesian category C. The following result proves that, in every quasi-cartesian category, extensionality of presubobjects is equivalent to a categorical choice principle. Let us assume (∀x, x′, x′′ ∈ X )(x ∼r x′ ∧ x′ ∼r x′′ =⇒ x ∼r x′′)
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