uu.seUppsala University Publications
Change search
ReferencesLink to record
Permanent link

Direct link
Setoids and universes
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
2010 (English)In: Mathematical Structures in Computer Science, ISSN 0960-1295, E-ISSN 1469-8072, Vol. 20, no 4, 563-576 p.Article in journal (Refereed) Published
Abstract [en]

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 universes that are as small as possible (and thus, the type theory is as weak as possible). In particular, we will consider epimorphisms and disjoint sums. We show that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably non-derivable.

Place, publisher, year, edition, pages
2010. Vol. 20, no 4, 563-576 p.
National Category
URN: urn:nbn:se:uu:diva-135600DOI: 10.1017/S0960129510000071ISI: 000280672800003OAI: oai:DiVA.org:uu-135600DiVA: diva2:375252
Available from: 2010-12-07 Created: 2010-12-07 Last updated: 2011-12-15Bibliographically approved
In thesis
1. On Constructive Sets and Partial Structures
Open this publication in new window or tab >>On Constructive Sets and Partial Structures
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The first three papers in this thesis study the formalisation of a set in type theory as a data type with an equivalence relation – an object usually known as a setoid. The corresponding formalisation of a locally small category is called an E-category.

In Paper I, we show that type theory without universes is insufficient for proving that some expected properties hold of the E-category of setoids, but that a minimal universe is sufficient.

In Paper II, we show that although the collection of all E-categories does not form a category, we can introduce a type-theoretic version of bicategories, and the E-categories form such an E-bicategory.

In Paper III, we consider the setoids inside a type-theoretic universe. The axiom of unique substitutions is proposed and used to show that these form a small category (that is, a category witha setoid of objects and a single setoid of all arrows). We demonstrate that this construction can not be carried out without adding some new axiom to type theory. We also show that the axiom of unique substitutions is strictly weaker than the axiom of unique identity proofs.

In Paper IV, we investigate partial equivalence relations, also known as partial setoids, in Heyting arithmetic in all finite types, and adapt the result that the extensional axiom of choice is equivalent to the combination of the intensional axiom of choice, classical logic, and an extensionality axiom.

In Paper V, we investigate PHL, a logic of partial terms, and prove a cut elimination theorem for it and for a related calculus.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2011. 33 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 74
National Category
Algebra and Logic
Research subject
Mathematical Logic
urn:nbn:se:uu:diva-160605 (URN)978-91-506-2245-4 (ISBN)
Public defence
2011-12-13, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Available from: 2011-11-22 Created: 2011-10-27 Last updated: 2011-12-15Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full text
By organisation
Algebra, Geometry and Logic
In the same journal
Mathematical Structures in Computer Science

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 190 hits
ReferencesLink to record
Permanent link

Direct link