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Invited Talk

**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

Constructs that involve taking a quotient are commonplace in mathematics. Here I will consider how they are treated within dependent type theory. The notion of quotient type has its origins in the Nuprl theorem-proving system [R. L. Constable et al., 1986] for extensional type theory. Later Hofmann formulated a version for intensional type theory in his thesis [M. Hofmann, 1995]. This depends on having a pre-existing notion of intensional identity type. Hofmann used Martin-Löf’s notion, the indexed family inductively generated from proofs of reflexivity [B. Nordström et al., 1990]. The recent homotopical view of identity in terms of path types [The Univalent Foundations Program, 2013] gives a more liberal perspective and has brought with it the notion of higher inductive type (HIT) [P. L. Lumsdaine and M. Shulman, 2019], subsuming both inductive and quotient types.
Inductively defined indexed families of types (in all their various forms) are perhaps the most useful concept that dependent type theory has contributed to the practice of computer assistance for formalizing mathematical proofs. However, it is often the case that a particular application of such types needs not only to inductively generate a collection of objects, but also to make identifications between the objects. In classical mathematics one can first generate and then identify, using the Axiom of Choice to lift infinitary constructions to the quotient. HITs can allow one to avoid such non-constructive uses of choice by inter-twining generation and identification. Perhaps more important than the constructive/non-constructive issue is that simultaneously declaring how to generate and how to identify can be a very natural way of defining some construct from the user’s point of view. This is why HITs promise to be so useful, once we have robust and convenient mechanisms in theorem-proving systems for defining HITs and defining functions out of HITs. Although some HITs have been axiomatized in various systems, at the moment the only system I know of with built in support for defining quite general forms of HIT and using them is the implementation of cubical type theory [C. Cohen et al., 2018] within recent versions of the Agda system [A. Vezzosi et al., 2019].
The higher dimensional aspect of identity in cubical type theory is fascinating; nevertheless, the simpler one-dimensional version of identity, in which one has uniqueness of identity proofs (UIP), is adequate for many applications. Although some regard UIP as a bug of early versions of Agda with it’s original form of dependent pattern matching [T. Coquand, 1992], it is by choice a feature of the Lean prover [J. Avigad et al., 2020]. Altenkirch and Kaposi [T. Altenkirch and A. Kaposi, 2016] have termed the one-dimensional version of HITs quotient inductive types (QITs) and they promise to be very useful even in the setting of type theory with UIP.
In this talk I survey some of these developments, including a recent reduction of QITs to quotients [M. P. Fiore et al., 2020], and the prospects for better support in theorem-provers for quotient constructions.

Andrew M. Pitts. Quotients in Dependent Type Theory (Invited Talk). In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pitts:LIPIcs.FSCD.2020.3, author = {Pitts, Andrew M.}, title = {{Quotients in Dependent Type Theory}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {3:1--3:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.3}, URN = {urn:nbn:de:0030-drops-123256}, doi = {10.4230/LIPIcs.FSCD.2020.3}, annote = {Keywords: dependent type theory, quotient, quotient inductive type, theorem-proving systems} }

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**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

This paper investigates Voevodsky's univalence axiom in intensional Martin-Löf type theory. In particular, it looks at how univalence can be derived from simpler axioms. We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms. We argue that these axioms are easier to verify in certain potential models of univalent type theory, particularly those models based on cubical sets. Finally we show how this decomposition is relevant to an open problem in type theory.

Ian Orton and Andrew M. Pitts. Decomposing the Univalence Axiom. In 23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{orton_et_al:LIPIcs.TYPES.2017.6, author = {Orton, Ian and Pitts, Andrew M.}, title = {{Decomposing the Univalence Axiom}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, pages = {6:1--6:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.6}, URN = {urn:nbn:de:0030-drops-100546}, doi = {10.4230/LIPIcs.TYPES.2017.6}, annote = {Keywords: dependent type theory, homotopy type theory, univalent type theory, univalence, cubical type theory, cubical sets} }

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**Published in:** LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.

Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters. Internal Universes in Models of Homotopy Type Theory. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{licata_et_al:LIPIcs.FSCD.2018.22, author = {Licata, Daniel R. and Orton, Ian and Pitts, Andrew M. and Spitters, Bas}, title = {{Internal Universes in Models of Homotopy Type Theory}}, booktitle = {3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-077-4}, ISSN = {1868-8969}, year = {2018}, volume = {108}, editor = {Kirchner, H\'{e}l\`{e}ne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.22}, URN = {urn:nbn:de:0030-drops-91929}, doi = {10.4230/LIPIcs.FSCD.2018.22}, annote = {Keywords: cubical sets, dependent type theory, homotopy type theory, internal language, modalities, univalent foundations, universes} }

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**Published in:** LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.

Ian Orton and Andrew M. Pitts. Models of Type Theory Based on Moore Paths. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{orton_et_al:LIPIcs.FSCD.2017.28, author = {Orton, Ian and Pitts, Andrew M.}, title = {{Models of Type Theory Based on Moore Paths}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {28:1--28:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-047-7}, ISSN = {1868-8969}, year = {2017}, volume = {84}, editor = {Miller, Dale}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.28}, URN = {urn:nbn:de:0030-drops-77149}, doi = {10.4230/LIPIcs.FSCD.2017.28}, annote = {Keywords: dependent type theory, homotopy theory, Moore path, topos} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

The homotopical approach to intensional type theory views proofs of equality as paths. We explore what is required of an interval-like object I in a topos to give a model of type theory in which elements of identity types are functions with domain I. Cohen, Coquand, Huber and Mörtberg give such a model using a particular category of presheaves. We investigate the extent to which their model construction can be expressed in the internal type theory of any topos and identify a collection of quite weak axioms for this purpose. This clarifies the definition and properties of the notion of uniform Kan filling that lies at the heart of their constructive interpretation of Voevodsky's univalence axiom. Furthermore, since our axioms can be satisfied in a number of different ways, we show that there is a range of topos-theoretic models of homotopy type theory in this style.

Ian Orton and Andrew M. Pitts. Axioms for Modelling Cubical Type Theory in a Topos. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{orton_et_al:LIPIcs.CSL.2016.24, author = {Orton, Ian and Pitts, Andrew M.}, title = {{Axioms for Modelling Cubical Type Theory in a Topos}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {24:1--24:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.24}, URN = {urn:nbn:de:0030-drops-65647}, doi = {10.4230/LIPIcs.CSL.2016.24}, annote = {Keywords: models of dependent type theory, homotopy type theory, cubical sets, cubical type theory, topos, univalence} }

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**Published in:** LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)

The cubical sets model of Homotopy Type Theory introduced by Bezem, Coquand and Huber uses a particular category of presheaves. We show that this presheaf category is equivalent to a category of sets equipped with an action of a monoid of name substitutions for which a finite support property holds. That category is in turn isomorphic to a category of nominal sets equipped with operations for substituting constants 0 and 1 for names. This formulation of cubical sets brings out the potentially useful connection that exists between the homotopical notion of path and the nominal sets notion of name abstraction. The formulation in terms of actions of monoids of name substitutions also encompasses a variant category of cubical sets with diagonals, equivalent to presheaves on Grothendieck's "smallest test category." We show that this category has the pleasant property that path objects given by name abstraction are exponentials with respect to an interval object.

Andrew M. Pitts. Nominal Presentation of Cubical Sets Models of Type Theory. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 202-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{pitts:LIPIcs.TYPES.2014.202, author = {Pitts, Andrew M.}, title = {{Nominal Presentation of Cubical Sets Models of Type Theory}}, booktitle = {20th International Conference on Types for Proofs and Programs (TYPES 2014)}, pages = {202--220}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-88-0}, ISSN = {1868-8969}, year = {2015}, volume = {39}, editor = {Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.202}, URN = {urn:nbn:de:0030-drops-54980}, doi = {10.4230/LIPIcs.TYPES.2014.202}, annote = {Keywords: models of dependent type theory, homotopy type theory, cubical sets, nominal sets, monoids} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 10 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 13422 "Nominal Computation Theory". The underlying theme of the seminar was nominal sets (also known as sets with atoms or Fraenkel-Mostowski sets) and they role and applications in three distinct research areas: automata over infinite alphabets, program semantics using nominal sets and nominal calculi of concurrent processes.

Mikolaj Bojanczyk, Bartek Klin, Alexander Kurz, and Andrew M. Pitts. Nominal Computation Theory (Dagstuhl Seminar 13422). In Dagstuhl Reports, Volume 3, Issue 10, pp. 58-71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{bojanczyk_et_al:DagRep.3.10.58, author = {Bojanczyk, Mikolaj and Klin, Bartek and Kurz, Alexander and Pitts, Andrew M.}, title = {{Nominal Computation Theory (Dagstuhl Seminar 13422)}}, pages = {58--71}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {3}, number = {10}, editor = {Bojanczyk, Mikolaj and Klin, Bartek and Kurz, Alexander and Pitts, Andrew M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.3.10.58}, URN = {urn:nbn:de:0030-drops-44285}, doi = {10.4230/DagRep.3.10.58}, annote = {Keywords: nominal sets, Fraenkel-Mostowski sets} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

The operational semantics of programming constructs involving locally
scoped names typically makes use of stateful "dynamic allocation": a
set of currently-used names forms part of the state and upon entering
a scope the set is augmented by a new name bound to the scoped
identifier. More abstractly, one can see this as a transformation of
local scopes by expanding them outward to an implicit top-level. By
contrast, in a neglected paper from 1994, Odersky gave a stateless
lambda calculus with locally scoped names whose dynamics contracts
scopes inward. The properties of "Odersky-style" local names are quite
different from dynamically allocated ones and it has not been clear,
until now, what is the expressive power of Odersky's notion. We show
that in fact it provides a direct semantics of locally scoped names
from which the more familiar dynamic allocation semantics can be
obtained by continuation-passing style (CPS) translation. More
precisely, we show that there is a CPS translation of typed lambda
calculus with dynamically allocated names (the Pitts-Stark
nu-calculus) into Odersky's lambda-nu-calculus which is
computationally adequate with respect to observational equivalence in
the two calculi.

Steffen Lösch and Andrew M. Pitts. Relating Two Semantics of Locally Scoped Names. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 396-411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{losch_et_al:LIPIcs.CSL.2011.396, author = {L\"{o}sch, Steffen and Pitts, Andrew M.}, title = {{Relating Two Semantics of Locally Scoped Names}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {396--411}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.396}, URN = {urn:nbn:de:0030-drops-32454}, doi = {10.4230/LIPIcs.CSL.2011.396}, annote = {Keywords: local names, continuations, typed lambda-calculus, observational equivalence} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10351, Modelling, Controlling and Reasoning About State (2010)

The purpose of this note is to illustrate the use of step-indexing combined with biorthogonality to construct syntactical logical relations. It walks through the details of a syntactically simple, yet non-trivial example: a proof of the "CIU Theorem'' for contextual equivalence in the untyped call-by-value $lambda$-calculus with recursively defined functions.

Andrew M. Pitts. Step-Indexed Biorthogonality: a Tutorial Example. In Modelling, Controlling and Reasoning About State. Dagstuhl Seminar Proceedings, Volume 10351, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{pitts:DagSemProc.10351.6, author = {Pitts, Andrew M.}, title = {{Step-Indexed Biorthogonality: a Tutorial Example}}, booktitle = {Modelling, Controlling and Reasoning About State}, pages = {1--10}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10351}, editor = {Amal Ahmed and Nick Benton and Lars Birkedal and Martin Hofmann}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10351.6}, URN = {urn:nbn:de:0030-drops-28067}, doi = {10.4230/DagSemProc.10351.6}, annote = {Keywords: Biorthogonality, logical relations, operational semantics, step-indexing} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

John W. Gray, Andrew M. Pitts, and Kurt Sieber. Interactions between Category Theory and Computer Science (Dagstuhl Seminar 9329). Dagstuhl Seminar Report 69, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1993)

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@TechReport{gray_et_al:DagSemRep.69, author = {Gray, John W. and Pitts, Andrew M. and Sieber, Kurt}, title = {{Interactions between Category Theory and Computer Science (Dagstuhl Seminar 9329)}}, pages = {1--28}, ISSN = {1619-0203}, year = {1993}, type = {Dagstuhl Seminar Report}, number = {69}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemRep.69}, URN = {urn:nbn:de:0030-drops-149575}, doi = {10.4230/DagSemRep.69}, }

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