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Constructing a small category of setoids

Published online by Cambridge University Press:  13 September 2011

OLOV WILANDER
OLOV WILANDER*
Affiliation:
Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden Email: wilander@math.uu.se
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Abstract

Consider the first-order theory of a category.d It has a sort of objects, and a sort of arrows (so we may think of it as a small category). We show that, assuming the principle of unique substitutions, the setoids inside a type theoretic universe provide a model for this first-order theory. We also show that the principle of unique substitutions is not derivable in type theory, but that it is strictly weaker than the principle of unique identity proofs.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 1 , February 2012 , pp. 103 - 121
Copyright
Copyright © Cambridge University Press 2011

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References

Aczel, P. (1995) Galois: a theory development project. Unpublished manuscript, available for download at http://www.cs.man.ac.uk/~petera/papers.html.Google Scholar
Awodey, S., Garner, R., Martin-Löf, P. and Voevodsky, V. (organisers) (2011) Mini-workshop: the homotopy interpretation of constructive type theory. Oberwolfach Report No. 11/2011.CrossRefGoogle Scholar
Bishop, E. (1967) Foundations of Constructive Analysis, McGraw-Hill.Google Scholar
Bishop, E. and Bridges, D. (1985) Constructive Analysis, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag 279.CrossRefGoogle Scholar
Dybjer, P. and Gaspes, V. (1994) Implementing a category of sets in ALF. Unpublished manuscript, available for download at http://www.cse.chalmers.se/~peterd/papers/categorytypetheory.html.Google Scholar
Fridlender, D. (2002) A proof-irrelevant model of Martin-Löf's logical framework. Mathematical Structures in Computer Science 12 (6)771–795.Google Scholar
Hedberg, M. (1998) A coherence theorem for Martin-Löf's type theory. Journal of Functional Programming 8 (4)413–436.Google Scholar
Hofmann, M. (1997) Extensional Constructs in Intensional Type Theory, CPHC/BCS Distinguished Dissertations, Springer-Verlag.CrossRefGoogle Scholar
Hofmann, M. and Streicher, T. (1998) The groupoid interpretation of type theory. In: Twenty-five Years of Constructive Type Theory (Venice, 1995), Oxford Logic Guides 36 83–111.Google Scholar
Johnstone, P. T. (2002) Sketches of an Elephant: A Topos Theory Compendium, Volume 1, Oxford Logic Guides 43.CrossRefGoogle Scholar
Lawvere, F. W. (1964) An elementary theory of the category of sets. Proceedings of the National academy of Sciences of the United States of America 52 1506–1511.Google Scholar
Lawvere, F. W. (2005) An elementary theory of the category of sets (long version). Reprints in Theory and Applications of Categories 11 1–35. (Reprinted and expanded from Lawvere (1964), with comments by the author and C. McLarty.)Google Scholar
Mac Lane, S. (1998) Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer-Verlag.Google Scholar
Nordström, B., Petersson, K. and Smith, J. M. (1990) Programming in Martin-Löf's Type Theory, An Introduction, International Series of Monographs on Computer Science 7, The Clarendon Press.Google Scholar
Nordström, B., Petersson, K. and Smith, J. M. (2000) Martin-Löf's type theory. In: Handbook of logic in computer science 5, Oxford University Press 1–37.Google Scholar
Palmgren, E. (2009a) Constructivist and structuralist foundations: Bishop's and Lawvere's theories of sets. Technical Report 4, Fall 2009, Institut Mittag-Leffler, Sweden.Google Scholar
Palmgren, E. (2009b) Remarks on the relation between families of setoids and identity in type theory. Technical Report 36, Fall 2009, Institut Mittag-Leffler, Sweden.Google Scholar
Smith, J. M. (1988) The independence of Peano's fourth axiom from Martin-Löf's type theory. Journal of Symbolic Logic 53 (3)840–845.CrossRefGoogle Scholar
Streicher, T. (1993) Semantical investigations into intensional type theory, Habilitationsschrift, LMU München.Google Scholar
Voevodsky, V. (2010) Univalent Foundations Project. Unpublished manuscript, available for download at http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html.Google Scholar