Algebraic Set Theory

Algebraic set theory uses the methods of category theory to study elementary set theory. The purpose of this website is to link together current research in algebraic set theory and make it easily available. It is hoped that this will encourage and facilitate further development of the subject.

Why do you call it "algebraic set theory"?

Researchers in Algebraic Set Theory

Steve Awodey (Carnegie Mellon University)
Benno van den Berg (Darmstadt University of Technology)
Carsten Butz (IT University, Copenhagen)
Henrik Forssell (University of Oslo)
Nicola Gambino (University of Palermo)
F. William Lawvere (University at Buffalo, SUNY)
Peter LeFanu Lumsdaine (Carnegie Mellon University)
Ieke Moerdijk (Utrecht University)
Erik Palmgren (Uppsala University)
Ivar Rummelhoff (University of Oslo)
Dana Scott (Carnegie Mellon University)
Alex Simpson (University of Edinburgh)
Thomas Streicher (Darmstadt University of Technology)
Michael A. Warren (University of Ottawa)

Bibliography

The following is a brief survey of the current literature on algebraic set theory. The bibliography is ordered chronologically by year of publication and then alphabetically by the author's surname (by the first author's surname in the case of works with multiple authors). Draft and preprint versions of papers are listed as they become available. Upon publication preprints are removed and papers are listed under year of publication. In most cases preprints are still available on individual author homepages or on the arXiv.
The ordering of papers below does not, in all cases, do full justice to the historical development of the subject due, among other things, to the delay between completion of a paper and its subsequent publication. Forthcoming annotations will provide additional details regarding the historical development of the research listed below. If there is anything you feel that we have left out, then please feel free to contact us.

1991

André Joyal and Ieke Moerdijk.
A categorical theory of cumulative hierarchies of sets.
Comptes Rendus Mathématiques de l'Académie des Science,
13:55-58, 1991.

1994

André Joyal and Ieke Moerdijk.
A completeness theorem for open maps.
Annals of Pure and Applied Logic, 70(1):51-86, 1994.
[Abstract and download (if available)]

1995

André Joyal and Ieke Moerdijk.
Algebraic Set Theory.
Cambridge University Press, 1995.

1999

Alex Simpson.
Elementary axioms for categories of classes.
In Proceedings of the 14th Annual Symposium on Logic in Computer Science, pages 77-85, 1999.
[Abstract and download (if available)]

2000

Ieke Moerdijk and Erik Palmgren.
Wellfounded trees in categories.
Annals of Pure and Applied Logic, 104(1-3):189-218, 2000.
[Abstract and download (if available)]

2002

Nicola Gambino.
Sheaf interpretations for generalised predicative intuitionistic systems.
PhD thesis, University of Manchester, 2002.
Ieke Moerdijk and Erik Palmgren.
Type theories, toposes and constructive set theory:
Predicative aspects of ast.
Annals of Pure and Applied Logic, 114(1-3):155-201, 2002.
[Abstract and download (if available)]
Alex Simpson.
Computational adequacy for recursive types in models of intuitionistic set theory (conference version).
Seventeenth Annual IEEE Symposium on Logic in Computer Science, pp. 287-298, 2002.

2003

Carsten Butz.
Bernays-Gödel type theory.
Journal of Pure and Applied Algebra, 178(1):1-23, 2003.
[Abstract and download (if available)]

2004

Henrik Forssell.
Categorical Models of Intuitionistic Theories of Sets and Classes.
Master's thesis, Carnegie Mellon University, 2004.
[PS] [PDF]
Alex Simpson.
Computational adequacy for recursive types in models of intuitionistic set theory (journal version).
Annals of Pure and Applied Logic, 130(1-3):207-275, 2004.
[Abstract and download (if available)]
Michael A. Warren.
Predicative Categories of Classes.
Master's thesis, Carnegie Mellon University, 2004.
[PS] [PDF]

2005

Steve Awodey.
Notes on algebraic set theory.
Notes for lectures given at the Summer School on Topos Theory,
Haute-Bodeux, Belgium. May 29 to June 5, 2005.
Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005.
[PDF]
Steve Awodey and Henrik Forssell.
Algebraic models of intuitionistic theories of sets and classes.
Theory and Applications of Categories, 15(5):147-163, 2005.
[Abstract and download (if available)]
Steve Awodey and Michael A. Warren.
Predicative algebraic set theory.
Theory and Applications of Categories, 15(1):1-39, 2005.
[Abstract and download (if available)]
Benno van den Berg.
Sheaves for predicative toposes.
To appear, draft of 2005.
[PS] [PDF]
Nicola Gambino.
Presheaf models of constructive set theories.
In From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics,
edited by L. Crosilla and P. Schuster,
Oxford University Press, 2005.
Claire Kouwenhoven-Gentil and Jaap van Oosten.
Algebraic set theory and the effective topos.
Journal of Symbolic Logic, 70(3):879-890, 2005.
[Abstract and download (if available)]
Alex Simpson.
Constructive set theories and their category-theoretic models.
In From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics,
edited by L. Crosilla and P. Schuster,
Oxford University Press, 2005.
Thomas Streicher.
Universes in toposes.
In From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics,
edited by L. Crosilla and P. Schuster,
Oxford University Press, 2005.

2006

Steve Awodey, Henrik Forssell and Michael A. Warren.
Algebraic models of sets and classes in categories of ideals.
Unpublished note, 2006.
[PS] [PDF]
Benno van den Berg.
Predicative topos theory and models for constructive set theory.
PhD thesis, Utrecht University, 2006.

2007

Steve Awodey, Carsten Butz, Alex Simpson and Thomas Streicher.
Relating first-order set theories and elementary toposes (communication).
Bulletin of Symbolic Logic, 13(3):340-358, 2007.
[Abstract and download (if available)]
Steve Awodey, Carsten Butz, Alex Simpson and Thomas Streicher.
Relating first-order set theories, toposes and categories of classes.
In preparation, 2007.
[PDF]
Benno van den Berg and Federico De Marchi.
Models of non-well-founded sets via an indexed final coalgebra theorem.
Journal of Symbolic Logic, 72(3):767-791, 2007.
[Abstract and download (if available)]
Nicola Gambino.
The associated sheaf functor theorem in algebraic set theory.
To appear, 2007.
Michael A. Warren.
Coalgebras in a category of classes.
Annals of Pure and Applied Logic, 146(1):60-71, 2007.
[Abstract and download (if available)]

2008

Steve Awodey.
A brief introduction to algebraic set theory.
The Bulletin of Symbolic Logic, 14(3):281-298, 2008.
[PS]
Benno van den Berg and Ieke Moerdijk.
Aspects of predicative algebraic set theory I: Exact completion.
Annals of Pure and Applied Logic, 156(1):123-159, 2008.
[Abstract and download (if available)]
Benno van den Berg and Ieke Moerdijk.
Aspects of predicative algebraic set theory II: Realizability.
To appear in Theoretical Computer Science, 2008.
[PDF]

Benno van den Berg and Ieke Moerdijk.
W-types in sheaves.
Submitted, 2008.
[PDF] [PS]

2009

Steve Awodey, Nicola Gambino, Peter L. Lumsdaine and Michael A. Warren.
Lawvere-Tierney sheaves in algebraic set theory.
Journal of Symbolic Logic, 74(3):862-890, 2009.
[Abstract and download (if available)]
Benno van den Berg and Ieke Moerdijk.
Aspects of predicative algebraic set theory III: Sheaves.
Submitted, 2009.
[PDF]
Benno van den Berg and Ieke Moerdijk.
A unified approach to Algebraic Set Theory.
Logic Colloquium 2006, Lecture Notes in Logic, 18-37, 2009.
[arXiv]

2013

Steve Awodey, Carsten Butz, Alex Simpson, and Thomas Streicher.
Relating first-order set theories, toposes and categories of classes.
Annals of Pure and applied Logic,, 165(2):428–502, 2013.
[PDF] [journal]

Additional Resources and References

For further relevant resources and references to work outside of the Joyal and Moerdijk framework see our links page.

This page is an archive and is no longer being maintained.