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Exact completion and constructive theories of sets
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Loef 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.

Keywords [en]
setoids, exact completion, local cartesian closure, constructive set theory, categorical logic
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-151876OAI: oai:DiVA.org:su-151876DiVA, id: diva2:1176023
Available from: 2018-01-19 Created: 2018-01-19 Last updated: 2018-11-21Bibliographically approved
In thesis
1. Exact completion and type-theoretic structures
Open this publication in new window or tab >>Exact completion and type-theoretic structures
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers and is a contribution to the study of representations of extensional properties in intensional type theories using, mainly, the language and tools from category theory. Our main focus is on exact completions of categories with weak finite limits as a category-theoretic description of the setoid construction in Martin-Löf's intensional type theory.

Paper I, which is joint work with Erik Palmgren, provides sufficient conditions for such an exact completion to produce a model of the system CETCS (Constructive Elementary Theory of the Category of Sets), a finite axiomatisation of the theory of well-pointed locally cartesian closed pretoposes with a natural numbers object and enough projectives. In particular, we use a condition inspired by Aczel's set-theoretic Fullness Axiom to obtain the local cartesian closure of an exact completion. As an application, we obtain a simple  uniform proof that the category of setoids is a model of CETCS.

Paper II was prompted by the discovery of an overlooked issue in the characterisationof local cartesian closure for exact completions due to Carboni and Rosolini. In this paper we clarify the problem, show that their characterisation is still valid when the base category has finite limits, and provide a complete solution in the general case of a category with weak finite limits.

In paper III we generalise the approach used in paper I to obtain the local cartesian closure of an exact completion to arbitrary categories with finite limits. We then show how this condition inspired by the Fullness Axiom naturally arises in several homotopy categories and apply this result to obtain the local cartesian closure of the exact completion of the homotopy category of spaces, thus answering a question left open by Marino Gran and Enrico Vitale.

Finally, in paper IV we abandon the pure category-theoretic approach and instead present a type-theoretic construction, formalised in Coq, of W-types in the category of setoids from dependent W-types in the underlying intensional theory. In particular, contrary to previous approaches, this construction does not require the assumption of Uniqueness of Identity Proofs nor recursion into a type universe.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019. p. 23
Keywords
exact completion, type theory, setoid, weak limits, cartesian closure, inductive types
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-162275 (URN)978-91-7797-526-7 (ISBN)978-91-7797-527-4 (ISBN)
Public defence
2019-01-18, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2018-12-19 Created: 2018-11-21 Last updated: 2018-12-07Bibliographically approved

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