homotopy type theory - references

A roughly taxonomised listing of some of the papers on [[homotopy type theory]]. Titles link to more details, bibdata, etc.  Currently very incomplete; please add!

[A good place to start](https://arxiv.org/search/?searchtype=all&query=%22homotopy+type+theory%22&abstracts=show&size=200&order=-announced_date_first)

# Contents
* table of contents
{:toc}

##Surveys##

* _Type theory and homotopy_ [[Steve Awodey]], 2010. (To appear.) [PDF](http://www.andrew.cmu.edu/user/awodey/preprints/TTH.pdf)
* _Homotopy type theory and Voevodsky's univalent foundations._ [[Álvaro Pelayo]] and [[Michael A. Warren]], 2012. (Bulletin of the AMS, forthcoming) [arXiv](http://arxiv.org/abs/1210.5658)
* _Voevodsky's Univalence Axiom in homotopy type theory._ [[Steve Awodey]], [[Álvaro Pelayo]], and [[Michael A. Warren]], October 2013, Notices of the American Mathematical Society 60(08), pp.1164-1167. [arXiv](http://arxiv.org/abs/1302.4731)
* _Homotopy Type Theory: A synthetic approach to higher equalities_.  [[Michael Shulman]].  To appear in _Categories for the working philosopher_; [arXiv](http://arxiv.org/abs/1601.05035)
* _Univalent Foundations and the UniMath library._ [[Anthony Bordg]], 2017. [PDF](https://drive.google.com/open?id=0B63Vr5tVwiAjdEJhSW03YXBNMVE)
* _Homotopy type theory: the logic of space_.  [[Michael Shulman]].  To appear in _New Spaces in Mathematics and Physics_:  [arxiv](https://arxiv.org/abs/1703.03007)
* _An introduction to univalent foundations for mathematicians_.  [[Dan Grayson]], [arxiv](https://arxiv.org/abs/1711.01477)
* _A self-contained, brief and complete formulation of Voevodsky's Univalence Axiom_.  Martín Escardó, [web](http://www.cs.bham.ac.uk/~mhe/agda-new/UnivalenceFromScratch.html), [arxiv](https://arxiv.org/abs/1803.02294)
* _A proposition is the (homotopy) type of its proofs_. [[Steve Awodey]].  [arxiv](https://arxiv.org/abs/1701.02024), 2017
* _Introduction to Univalent Foundations of Mathematics with Agda_.  Martín Escardó,[web](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/), [arxiv](https://arxiv.org/abs/1911.00580), 2019
* _[[Introduction to Homotopy Type Theory]].  Egbert Rijke, to appear 2021
* _[[HoTT Book|Homotopy Type Theory: Univalent Foundations of Mathematics]]_, The Univalent Foundations Program, Institute for Advanced Study, 2013. [homotopytypetheory.org/book](http://homotopytypetheory.org/book)

See also [Philosophy](#Philosophy), below.

##Synthetic homotopy theory##

* _Calculating the fundamental group of the circle in homotopy type theory_. [[Dan Licata]] and [[Michael Shulman]], LICS 2013, available [here](http://www.cs.cmu.edu/~drl/pubs.html) and on [arXiv](http://arxiv.org/abs/1301.3443)
* _$\pi_n(S^n)$ in Homotopy Type Theory_, [[Dan Licata]] and [[Guillaume Brunerie]], Invited Paper, CPP 2013, [PDF](http://dlicata.web.wesleyan.edu/pubs/lb13cpp/lb13cpp.pdf)
* _Homotopy limits in type theory_. [[Jeremy Avigad]], [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], Math. Structures Comput. Sci. 25 (2015), no. 5, 1040?1070. [arXiv](http://arxiv.org/abs/1304.0680)
* _[[Eilenberg-MacLane Spaces in Homotopy Type Theory]]_. [[Dan Licata]] and [[Eric Finster]], LICS 2014, [PDF](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) and [code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda)
* _[[A Cubical Approach to Synthetic Homotopy Theory]]_.  [[Dan Licata]] and [[Guillaume Brunerie]], LICS 2015, [PDF](http://dlicata.web.wesleyan.edu/pubs/lb15cubicalsynth/lb15cubicalsynth.pdf)
* _Synthetic Cohomology in Homotopy Type Theory_, [[Evan Cavallo]], [PDF](http://www.cs.cmu.edu/~rwh/theses/cavallo.pdf)
* _[[A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory]]_, LICS 2016  [[Kuen-Bang Hou (Favonia)]], [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], [arXiv](http://arxiv.org/abs/1605.03227)
* _The Seifert-van Kampen Theorem in Homotopy Type Theory_,  [[Kuen-Bang Hou]] and [[Michael Shulman]], [PDF]( http://www.cs.cmu.edu/~kuenbanh/files/vankampen.pdf)
* _[[On the homotopy groups of spheres in homotopy type theory]]_, [[Guillaume Brunerie]], Ph.D. Thesis, 2016, [arxiv](https://arxiv.org/abs/1606.05916)
* _[[The real projective spaces in homotopy type theory]]_, [[Ulrik Buchholtz]] and [[Egbert Rijke]], LICS 2017, [arxiv](https://arxiv.org/abs/1704.05770)
* _Covering Spaces in Homotopy Type Theory_, [[Kuen-Bang Hou (Favonia)]], [doi](https://doi.org/10.1007/978-3-319-21284-5_15)
* _[[Higher-Dimensional Types in the Mechanization of Homotopy Theory]], [[Kuen-Bang Hou (Favonia)]], Ph. D. Thesis 2017, [pdf](https://favonia.org/files/thesis.pdf)
* _[[Higher Groups in Homotopy Type Theory]]_, [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], [arXiv:1802.04315](https://arxiv.org/abs/1802.04315)
* _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_, [[Floris van Doorn]], Ph.D. Thesis 2018 [arXiv:1808.10690](https://arxiv.org/abs/1808.10690)
* _[[Localization in Homotopy Type Theory]]_, [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], [arXiv:1807.04155](https://arxiv.org/abs/1807.04155)
* _[[Cellular Cohomology in Homotopy Type Theory]]_, [[Ulrik Buchholtz]], [[Kuen-Bang Hou (Favonia)]], [arXiv:1802.02191](https://arxiv.org/abs/1802.02191)
* _[[The join construction]]_.  [[Egbert Rijke]], [arXiv:1701.07538](https://arxiv.org/abs/1701.07538)
* _Nilpotent Types and Fracture Squares in Homotopy Type Theory_. [[Luis Scoccola]], [arXiv:1903.03245](https://arxiv.org/abs/1903.03245)

## Higher category theory

* _[[Univalent categories and the Rezk completion]]_. [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], Math. Structures Comput. Sci. 25 (2015), no. 5, 1010?1039. [arXiv:1303.0584](http://arxiv.org/abs/1303.0584) (on _[[nLab:internal categories in HoTT]]_)
* _A type theory for synthetic $\infty$-categories_. [[Emily Riehl]], [[Michael Shulman]]. [arxiv](https://arxiv.org/abs/1705.07442), 2017
* _Univalent Higher Categories via Complete Semi-Segal Types_.  [[Paolo Capriotti]], [[Nicolai Kraus]], [arxiv](https://arxiv.org/abs/1707.03693), 2017
* _Bicategories in Univalent Foundations_, [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], [arXiv:1903.01152](https://arxiv.org/abs/1903.01152)

##Homotopical ideas and truncations in type theory##

* _Generalizations of Hedberg?s Theorem_. [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], and [[Thorsten Altenkirch]]. TLCA 2013, [pdf](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf)
* _Notions of anonymous existence in Martin-Lof type theory_.  [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], and [[Thorsten Altenkirch]]. [pdf](http://www.cs.nott.ac.uk/~txa/publ/jhedberg.pdf)
* _Idempotents in intensional type theory_.  [[Michael Shulman]], [arXiv](http://arxiv.org/abs/1507.03634)
* _Functions out of Higher Truncations_.  [[Paolo Capriotti]], [[Nicolai Kraus]], and [[Andrea Vezzosi]]. CSL 2015 [arxiv](http://arxiv.org/abs/1507.01150)
* _Truncation levels in homotopy type theory_. [[Nicolai Kraus]], PhD Thesis: University of Nottingham, 2015. [pdf](http://www.cs.nott.ac.uk/~psznk/docs/thesis_nicolai.pdf)
* _Parametricity, automorphisms of the universe, and excluded middle_.  [[Auke Bart Booij]], [[Martín Hötzel Escardó]], [[Peter LeFanu Lumsdaine]], [[Michael Shulman]].  [arxiv](https://arxiv.org/abs/1701.05617)

##General models##

* [_The groupoid interpretation of type theory._](http://homotopytypetheory.org/references/hofmann-streicher-the-groupoid-interpretation-of-type-theory/) [[Thomas Streicher]] and [[Martin Hofmann]], in Sambin (ed.) et al., Twenty-five years of constructive type theory. Proceedings of a congress, Venice, Italy, October 19?21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. 36, 83-111 (1998). [PostScript](http://www.mathematik.tu-darmstadt.de/~streicher/venedig.ps.gz)
* [_Homotopy theoretic models of identity types._](http://homotopytypetheory.org/references/awodey-warren-homotopy-theoretic-models-of-identity-types/) [[Steve Awodey]] and [[Michael Warren]], Mathematical Proceedings of the Cambridge Philosophical Society, 2009. [PDF](http://www.andrew.cmu.edu/user/awodey/preprints/homotopy.pdf)
* _Homotopy theoretic aspects of constructive type theory._ [[Michael Warren]], Ph.D. thesis: Carnegie Mellon University, 2008. [PDF](http://mawarren.net/papers/phd.pdf)
* _Two-dimensional models of type theory_, [[Richard Garner]], Mathematical Structures in Computer Science 19 (2009), no. 4, pages 687--736. [RG's website](http://web.science.mq.edu.au/~rgarner/Papers/2-LCCC.pdf)
* _Topological and simplicial models of identity types._ [[Richard Garner]] and [[Benno van den Berg]], to appear in ACM Transactions on Computational Logic (TOCL). [PDF](http://tocl.acm.org/accepted/467garner.pdf)
* _The strict ∞-groupoid interpretation of type theory_ [[Michael Warren]], in Models, Logics and Higher-Dimensional Categories: A Tribute to the Work of Mihály Makkai, AMS/CRM, 2011. [PDF](http://mawarren.net/papers/crmp1295.pdf)
* _Homotopy-Theoretic Models of Type Theory._ [[Peter Arndt]] and [[Chris Kapulkin]]. In Typed Lambda Calculi and Applications, volume 6690 of Lecture Notes in Computer Science, pages 45?60. [arXiv](http://arxiv.org/abs/1208.5683)
* _Combinatorial realizability models of type theory_, [[Pieter Hofstra]] and [[Michael Warren]], 2013, Annals of Pure and Applied Logic 164(10), pp. 957-988. [arXiv](http://arxiv.org/abs/1205.5527)
* _Natural models of homotopy type theory_, [[Steve Awodey]], 2015. [arXiv](http://arxiv.org/abs/1406.3219)
* _Subsystems and regular quotients of C-systems_, [[Vladimir Voevodsky]], 2014. [arXiv](http://arxiv.org/abs/1406.7413)
* _C-system of a module over a monad on sets_, [[Vladimir Voevodsky]], 2014. [arXiv](http://arxiv.org/abs/1407.3394)
* _A C-system defined by a universe category_, [[Vladimir Voevodsky]], 2014. [arXiv](http://arxiv.org/abs/1409.7925)
* _B-systems_, [[Vladimir Voevodsky]], 2014. [arXiv](http://arxiv.org/abs/1410.5389)
* _The local universes model: an overlooked coherence construction for dependent type theories_, [[Peter LeFanu Lumsdaine]], [[Michael Warren]], to appear in ACM Transactions on Computational Logic, 2014. [arXiv](http://arxiv.org/abs/1411.1736)
* _Products of families of types in the C-systems defined by a universe category_, [[Vladimir Voevodsky]], 2015. [arXiv](http://arxiv.org/abs/1503.07072)
* _Martin-Lof identity types in the C-systems defined by a universe category_, [[Vladimir Voevodsky]], 2015. [arXiv](http://arxiv.org/abs/1505.06446)
* _The Frobenius Condition, Right Properness, and Uniform Fibrations_, [[Nicola Gambino]], [[Christian Sattler]].  [arXiv](http://arxiv.org/abs/1510.00669)
* _Fibred fibration categories_, [[Taichi Uemura]], 2016, [arXiv](http://arxiv.org/abs/1602.08206)
* _A homotopy-theoretic model of function extensionality in the effective topos_, [[Daniil Frumin]], [[Benno van den Berg]], [arxiv](https://arxiv.org/abs/1701.08369)
* _Polynomial pseudomonads and dependent type theory_, [[Steve Awodey]], [[Clive Newstead]], 2018, [arxiv](https://arxiv.org/abs/1802.00997)
* _Towards a Topological Model of Homotopy Type Theory_, [[Paige North]], [doi](https://doi.org/10.1007/978-3-319-21284-5_16)
* _The Equivalence Extension Property and Model Structures_, [[Christian Sattler]], [arxiv](https://arxiv.org/abs/1704.06911)
* _Univalent polymorphism_, [[Benno van den Berg]], [arxiv](https://arxiv.org/abs/1803.10113)
* _All (∞,1)-toposes have strict univalent universes_, [[Mike Shulman]], [arXiv:1904.07004](https://arxiv.org/abs/1904.07004)

##Univalence##

* _The equivalence axiom and univalent models of type theory_ (talk at CMU, February 4th, 2010), [[Vladimir Voevodsky]]. [arXiv](http://arxiv.org/abs/1402.5556)
* _Univalence in simplicial sets._ [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]]. [arXiv](http://arxiv.org/abs/1203.2553)
* _Univalence for inverse diagrams and homotopy canonicity._ [[Michael Shulman]]. [arXiv](http://arxiv.org/abs/1203.3253)
* _Fiber bundles and univalence._  Lecture by [[Ieke Moerdijk]] at the Lorentz Center, Leiden, December 2011.  [Lecture notes by Chris Kapulkin](http://www-home.math.uwo.ca/~kkapulki/notes/fiber_bundles_univalence.pdf)
* _A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory._  [[Thomas Streicher]], 2011. [PDF](http://www.mathematik.tu-darmstadt.de/~streicher/sstt.pdf)
* _Univalence and Function Extensionality._  Lecture by [[Nicola Gambino]] at Oberwohlfach, February 2011.  [Lecture notes by Chris Kapulkin and Peter Lumsdaine](http://www.pitt.edu/~krk56/UA_implies_FE.pdf)
* _The Simplicial Model of Univalent Foundations._ [[Chris Kapulkin]] and [[Peter LeFanu Lumsdaine]] and [[Vladimir Voevodsky]], 2012. [arXiv](http://arxiv.org/abs/1211.2851)
* _The univalence axiom for elegant Reedy presheaves_. [[Michael Shulman]], [arXiv](http://arxiv.org/abs/1307.6248)
* _Univalent universes for elegant models of homotopy types_.  [[Denis-Charles Cisinski]], [arXiv](http://arxiv.org/abs/1406.0058)
* _A univalent universe in finite order arithmetic_.  [[Colin McLarty]], [arXiv](http://arxiv.org/abs/1412.6714)
* _Constructive Simplicial Homotopy_. Wouter Pieter Stekelenburg, [arXiv](http://arxiv.org/abs/1604.04746)
* _Higher Homotopies in a Hierarchy of Univalent Universes_. [[Nicolai Kraus]] and [[Christian Sattler]], [arXiv](http://arxiv.org/abs/1311.4002), [DOI](http://dl.acm.org/citation.cfm?doid=2737801.2729979)
* _Univalence for inverse EI diagrams_.  [[Michael Shulman]], [arXiv](http://arxiv.org/abs/1508.02410)
* _Univalent completion_.  [[Benno van den Berg]], [[Ieke Moerdijk]], [arXiv](http://arxiv.org/abs/1508.04021)
* _On lifting univalence to the equivariant setting_. [[Anthony Bordg]], [arXiv](http://arxiv.org/abs/1512.04083)
* _Univalence in locally cartesian closed $\infty$-categories_.  [[David Gepner]] and [[Joachim Kock]]. Forum Math. 29 (2017), no. 3, 617--652

##Inductive, coinductive, and higher-inductive types##

* _Inductive Types in Homotopy Type Theory._ [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]]. To appear in LICS 2012.  [arXiv](http://arxiv.org/abs/1201.3898)
* _W-types in homotopy type theory_.  [[Benno van den Berg]] and [[Ieke Moerdijk]], [arXiv](http://arxiv.org/abs/1307.2765)
* _Homotopy-initial algebras in type theory_ [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]]. [arXiv](http://arxiv.org/abs/1504.05531), [Coq code](https://github.com/kristinas/hinitiality)
* _The General Universal Property of the Propositional Truncation_.  [[Nicolai Kraus]], [arXiv](http://arxiv.org/abs/1411.2682)
* _Non-wellfounded trees in Homotopy Type Theory_. [[Benedikt Ahrens]], [[Paolo Capriotti]], [[Régis Spadotti]]. TLCA 2015, [doi:10.4230/LIPIcs.TLCA.2015.17](http://dx.doi.org/10.4230/LIPIcs.TLCA.2015.17), [arXiv](https://arxiv.org/abs/1504.02949)
* _Constructing the Propositional Truncation using Non-recursive HITs_. [[Floris van Doorn]], [arXiV](http://arxiv.org/abs/1512.02274)
* _Constructions with non-recursive higher inductive types_. [[Nicolai Kraus]], LiCS 2016, [pdf](http://www.cs.nott.ac.uk/~psznk/docs/pseudotruncations.pdf)
* _Semantics of higher inductive types_.  [[Michael Shulman]] and [[Peter LeFanu Lumsdaine]], [arXiv](https://arxiv.org/abs/1705.07088)
* _A Descent Property for the Univalent Foundations_, [[Egbert Rijke]], [doi](https://doi.org/10.1007/978-3-319-21284-5_18)
* _Impredicative Encodings of (Higher) Inductive Types_. [[Steve Awodey]], [[Jonas Frey]], and [[Sam Speight]].  [arxiv](https://arxiv.org/abs/1802.02820), 2018
* _W-Types with Reductions and the Small Object Argument_, [[Andrew Swan]], [arxiv](https://arxiv.org/abs/1802.07588)
* _Bisimulation as path type for guarded recursive types_, [[Rasmus Ejlers Møgelberg]], [[Niccolò Veltri]], [arxiv](https://arxiv.org/abs/1810.13261)
* _Signatures and Induction Principles for Higher Inductive-Inductive Types_, [[Ambrus Kaposi]], [[András Kovács]] [arXiv:1902.00297](https://arxiv.org/abs/1902.00297)

##Formalizations##

* _An experimental library of formalized Mathematics based on the univalent foundations_, [[Vladimir Voevodsky]], Math. Structures Comput. Sci. 25 (2015), no. 5, pp 1278-1294, 2015. [arXiv](http://arxiv.org/abs/1401.0053) [journal](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9679046)
* _A preliminary univalent formalization of the p-adic numbers._ Álvaro Pelayo, [[Vladimir Voevodsky]], [[Michael Warren]], 2012. [arXiv](http://arxiv.org/abs/1302.1207)
* _[[Univalent categories and the Rezk completion]]_. [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], Math. Structures Comput. Sci. 25 (2015), no. 5, 1010?1039. [arXiv:1303.0584](http://arxiv.org/abs/1303.0584) (on _[[nLab:internal categories in HoTT]]_)
* _The HoTT Library: A formalization of homotopy type theory in Coq_, [[Andrej Bauer]], [[Jason Gross]], [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], [[Matthieu Sozeau]], [[Bas Spitters]], 2016 [arxiv](https://arxiv.org/abs/1610.04591)

##Applications to computing##

* _Homotopical patch theory_.  [[Carlo Angiuli]], [[Ed Morehouse]], [[Dan Licata]], [[Robert Harper]], [PDF](http://dlicata.web.wesleyan.edu/pubs/amlh14patch/amlh14patch.pdf)
* _Guarded Cubical Type Theory: Path Equality for Guarded Recursion_, [[Lars Birkedal]], [[Ale? Bizjak]], [[Ranald Clouston]], [[Hans Bugge Grathwohl]], [[Bas Spitters]], [[Andrea Vezzosi]], ([arXiv](https://arxiv.org/abs/1606.05223))

##Cubical models and cubical type theory##

* _A Cubical Approach to Synthetic Homotopy Theory_.  [[Dan Licata]] and [[Guillaume Brunerie]], LICS 2015, [PDF](http://dlicata.web.wesleyan.edu/pubs/lb15cubicalsynth/lb15cubicalsynth.pdf)
* _A syntax for cubical type theory_.  [[Thorsten Altenkirch]] and [[Ambrus Kaposi]], [PDF](http://www.cs.nott.ac.uk/~txa/publ/ctt.pdf)
* _Implementation of Univalence in Cubical Sets_, [github](https://github.com/simhu/cubical)
* _A Note on the Uniform Kan Condition in Nominal Cubical Sets_, [[Robert Harper]] and [[Kuen-Bang Hou]].  [arXiv](http://arxiv.org/abs/1501.05691)
* _The Frobenius Condition, Right Properness, and Uniform Fibrations_, [[Nicola Gambino]], [[Christian Sattler]].  (Note: this is a duplicate of an entry in the section "General Models" above; accident?)  [arXiv](http://arxiv.org/abs/1510.00669)
* _Cubical Type Theory:  a constructive interpretation of the univalence axiom_, [[Cyril Cohen]], [[Thierry Coquand]], [[Simon Huber]], and [[Anders Mortberg]], [arxiv](https://arxiv.org/abs/1611.02108) and [github implementation](https://github.com/mortberg/cubicaltt)
* _Canonicity for cubical type theory_, [[Simon Huber]], [arxiv](https://arxiv.org/abs/1607.04156).
* _Nominal Presentation of Cubical Sets Models of Type Theory]]_, [[Andrew M. Pitts]], [pdf](http://drops.dagstuhl.de/opus/volltexte/2015/5498/pdf/12.pdf)
* _Axioms for Modelling Cubical Type Theory in a Topos_, [[Ian Orton]] and [[Andrew M. Pitts]], [pdf](http://drops.dagstuhl.de/opus/volltexte/2016/6564/pdf/LIPIcs-CSL-2016-24.pdf) [Agda code](http://www.cl.cam.ac.uk/~rio22/agda/cubical-topos/root.html)
* _Computational Higher Type Theory I: Abstract Cubical Realizability_, [[Carlo Angiuli]], [[Robert Harper]], [[Todd Wilson]], [arxiv](https://arxiv.org/abs/1604.08873), 2016
* _Computational Higher Type Theory II: Dependent Cubical Realizability_, [[Carlo Angiuli]], [[Robert Harper]], [arxiv](https://arxiv.org/abs/1606.09638), 2016
* _Computational Higher Type Theory III: Univalent Universes and Exact Equality_, [[Carlo Angiuli]], [[Kuen-Bang Hou]], [[Robert Harper]], [arxiv](https://arxiv.org/abs/1712.01800), 2017
* _Computational Higher Type Theory IV: Inductive Types_, [[Evan Cavallo]], [[Robert Harper]], [arxiv](https://arxiv.org/abs/1801.01568), 2018
* _The univalence axiom in cubical sets_. [[Marc Bezem]], [[Thierry Coquand]], [[Simon Huber]].  [arxiv](https://arxiv.org/abs/1710.10941), 2017
* _A Cubical Model of Homotopy Type Theory_. [[Steve Awodey]].  [arxiv](https://arxiv.org/abs/1607.06413), 2016
* _Cartesian Cubical Computational Type Theory_, [[Carlo Angiuli]], [[Favonia]], [[Robert Harper]].  [pdf](https://www.cs.cmu.edu/~cangiuli/papers/ccctt.pdf)
* _On Higher Inductive Types in Cubical Type Theory, [[Thierry Coquand]], [[Simon Huber]], [[Andes Mortberg]], [arxiv](https://arxiv.org/abs/1802.01170), 2018
* _Homotopy canonicity for cubical type theory_, [[Thierry Coquand]], [[Simon Huber]], [[Christian Sattler]], [arXiv](https://arxiv.org/abs/1902.06572), 2019

##Syntax of type theory##

* [_The identity type weak factorisation system._](http://homotopytypetheory.org/references/gambino-garner-the-identity-type-weak-factorisation-system/) [[Nicola Gambino]] and [[Richard Garner]], Theoretical Computer Science 409 (2008), no. 3, pages 94?109. [RG?s website](http://www.comp.mq.edu.au/~rgarner/Idtype/Idtype.html)
* _Types are weak ∞-groupoids._ [[Richard Garner]] and [[Benno van den Berg]], to appear. [RG?s website](http://www.comp.mq.edu.au/~rgarner/Typesom/Typesom.html)
* _Weak ∞-Categories from Intensional Type Theory._ [[Peter LeFanu Lumsdaine]], TLCA 2009, Brasília, Logical Methods in Computer Science, Vol. 6, issue 23, paper 24. [PDF](http://www.mathstat.dal.ca/~p.l.lumsdaine/research/Lumsdaine-Weak-omega-cats-from-ITT-LMCS.pdf)
* _Higher Categories from Type Theories._ [[Peter LeFanu Lumsdaine]], PhD Thesis: Carnegie Mellon University, 2010. [PDF](http://www.mathstat.dal.ca/~p.l.lumsdaine/research/Lumsdaine-2010-Thesis.pdf)
* [_A coherence theorem for Martin-Löf?s type theory.]]_](http://homotopytypetheory.org/references/hedberg/) [[Michael Hedberg]], Journal of Functional Programming 8 (4): 413?436, July 1998.
* _Model Structures from Higher Inductive Types._  [[Peter LeFanu Lumsdaine]].  Unpublished note, Dec. 2011. [PDF](http://www.mathstat.dal.ca/~p.l.lumsdaine/research/Lumsdaine-Model-strux-from-HITs.pdf)
* _A category-theoretic version of the identity type weak factorization system_.  [[Jacopo Emmenegger]], [arXiv](http://arxiv.org/abs/1412.0153)
* _Locally cartesian closed quasicategories from type theory_. [[Chris Kapulkin]], [arXiv](http://arxiv.org/abs/1507.02648).
* _Note on the construction of globular weak omega-groupoids from types, topological spaces etc_.  [[John Bourke]], [arXiv](http://arxiv.org/abs/1602.07962)
* _The Homotopy Theory of Type Theories_. [[Chris Kapulkin]] and [[Peter LeFanu Lumsdaine]], [arXiv](https://arxiv.org/abs/1610.00037).
* _Polynomial Pseudomonads and Dependent Type Theory_. [[Steve Awodey]] and [[Clive Newstead]].  [arxiv](https://arxiv.org/abs/1802.00997), 2018

##Strict equality types##

* [[Homotopy Type System]], [[Vladimir Voevodsky]]
* _Extending Homotopy Type Theory with Strict Equality_,  [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Nicolai Kraus]], CSL 2016, [arXiv](http://arxiv.org/abs/1604.03799)
* _Two-Level Type Theory and Applications_, [[Danil Annenkov]], [[Paolo Capriotti]], [[Nicolai Kraus]], [arxiv](https://arxiv.org/abs/1705.03307), 2017

##Directed type theory##

* _2-Dimensional Directed Dependent Type Theory._ [[Dan Licata]] and [[Robert Harper]]. MFPS 2011. See also Chapters 7 and 8 of [Dan?s thesis](http://www.cs.cmu.edu/~drl/pubs/thesis/thesis.pdf). [PDF](http://www.cs.cmu.edu/~drl/pubs/lh102dtt/lh102dtt.pdf)
* _A type theory for synthetic $\infty$-categories_. [[Emily Riehl]], [[Michael Shulman]]. [arxiv](https://arxiv.org/abs/1705.07442)
* _Synthetic fibered (∞,1)-category theory_.  Ulrik Buchholtz, Jonathan Weinberger.  [arxiv](https://arxiv.org/abs/2105.01724), 2021.
* _Limits and colimits of synthetic ∞-categories_.  César Bardomiano Martínez.  [arxiv](https://arxiv.org/abs/2202.12386), 2022.
* _A Synthetic Perspective on (∞,1)-Category Theory: Fibrational and Semantic Aspects_.  Jonathan Weinberger.  PhD Thesis; [arxiv](https://arxiv.org/abs/2202.13132), 2022.
* _Two-sided cartesian fibrations of synthetic (∞,1)-categories_.  Jonathan Weinberger, [arxiv](https://arxiv.org/abs/2204.00938).

##Cohesion and modalities##

* _Quantum gauge field theory in cohesive homotopy type theory_.  [[Urs Schreiber]] and [[Michael Shulman]].  [arxiv](https://arxiv.org/abs/1408.0054)
* _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_.  [[Michael Shulman]].  To appear in MSCS; [arxiv](http://arxiv.org/abs/1509.07584)
* _[[Modalities in homotopy type theory]]_.  [[Egbert Rijke]], [[Michael Shulman]], [[Bas Spitters]].  [arxiv](https://arxiv.org/abs/1706.07526)

##Theories and models##

* _Homotopy Model Theory I: Syntax and Semantics_, [[Dimitris Tsementzis]], [arXiv](http://arxiv.org/abs/1603.03092)

##Computational interpretation##

* _Canonicity for 2-Dimensional Type Theory._ [[Dan Licata]] and [[Robert Harper]]. POPL 2012. [PDF](http://www.cs.cmu.edu/~drl/pubs/lh112tt/lh122tt-final.pdf)

##Philosophy##
 {#Philosophy}

* _Structuralism, Invariance, and Univalence_.  [[Steve Awodey]].   Philosophia Mathematica (2014) 22 (1): 1-11. [online](http://philmat.oxfordjournals.org/content/22/1/1.abstract)
* _Identity in Homotopy Type Theory, Part I: The Justification of Path Induction_.     [[James Ladyman]] and [[Stuart Presnell]].   Philosophia Mathematica (2015), [online](http://philmat.oxfordjournals.org/content/early/2015/06/01/philmat.nkv014.abstract#fn-2)
* _Homotopy Type Theory: A synthetic approach to higher equalities_.  [[Michael Shulman]].  To appear in _Categories for the working philosopher_; [arXiv](http://arxiv.org/abs/1601.05035)
* _Univalent Foundations as Structuralist Foundations_.  [[Dimitris Tsementzis]].  Forthcoming in  _Synthese_;
[Pitt-PhilSci] (http://philsci-archive.pitt.edu/12070/)
* _Homotopy type theory: the logic of space_.  [[Michael Shulman]].  To appear in _New Spaces in Mathematics and Physics_:  [arxiv](https://arxiv.org/abs/1703.03007)

##Other##

* _Martin-Löf Complexes._ [[Steve Awodey]], [[Pieter Hofstra]] and [[Michael Warren]], 2013, Annals of Pure and Applied Logic, 164(10), pp. 928-956. [PDF](http://mathstat.dal.ca/~mwarren/Papers/mc.pdf), [arXiv](http://arxiv.org/abs/0906.4521)
* _Space-Valued Diagrams, Type-Theoretically (Extended Abstract)_.  [[Nicolai Kraus]] and [[Christian Sattler]]. [arXiv](https://arxiv.org/abs/1704.04543)
* _Homotopy and Type Theory (Project description)_. [[Steve Awodey]], 2009. Unpublished grant proposal. [[proposal2009.pdf:file]].

[[!redirects references]]

category: reference, navigation


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

An introductory textbook on [[homotopy type theory]] and its use as a [[univalent foundations for mathematics]]:

* [[UF-IAS-2012|Univalent Foundations Project]]

  _Homotopy Type Theory -- Univalent Foundations of Mathematics_

  2013

  (**[web](http://homotopytypetheory.org/book/)**, **[pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)**)

**About the book** (from the cover)

> [[homotopy type theory|Homotopy type theory]] is a new branch of [[mathematics]] that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between [[homotopy theory]] and [[type theory]]. It touches on topics as seemingly distant as the [[homotopy groups of spheres]], the [[algorithms]] for [[type checking]], and the definition of [[weak ∞-groupoids]]. Homotopy type theory offers a new "[[univalence|univalent]]" [[foundation of mathematics]], in which a central role is played by [[Vladimir Voevodsky|Voevodsky]]'s [[univalence axiom]] and [[higher inductive types]]. The present book is intended as a first systematic exposition of the basics of [[univalent foundations]], and a collection of examples of this new style of [[reasoning]] — but without requiring the reader to know or learn any [[formal logic]], or to use any [[computer proof assistant]]. We believe that [[univalent foundations]] will eventually become a viable alternative to [[set theory]] as the "implicit foundation" for the unformalized mathematics done by most mathematicians.

* A more detailed description of the contents here:

  [http://golem.ph.utexas.edu/category/2013/06/the_hott_book.html#more](http://golem.ph.utexas.edu/category/2013/06/the_hott_book.html#more)

* And a description of the "sociology" of the writing of the book is here:

  [[Andrej Bauer]], _The HoTT book_ ([web](http://audrey.fmf.uni-lj.si/hott.html), [alternative](http://math.andrej.com/2013/06/20/the-hott-book/))

* It can be cited as:

Homotopy Type Theory: Univalent Foundations of Mathematics,
The Univalent Foundations Program, Institute for Advanced Study, 2013.

category: reference

[[!redirects HoTT Book]]
[[!redirects The HoTT Book]]
[[!redirects HoTT book]]
[[!redirects The HoTT book]]

[[!redirects the HoTT book]]

[[!redirects Homotopy Type Theory – Univalent Foundations of Mathematics]]


\tableofcontents

## Idea

The **Homotopy Type Theory Electronic Seminar Talks**, abbreviated as HoTTEST, is a series of talks on various topics related to [[homotopy type theory]] and [[dependent type theory]] more generally, run by [[Carlo Angiuli]], [[Dan Christensen]], [[Chris Kapulkin]], and [[Emily Riehl]].

## List of Talks

### Spring 2018

* [[Peter LeFanu Lumsdaine]], *Inverse diagram models of type theory*, 15 February 2018 ([video](https://www.youtube.com/watch?v=Td4UcFI76DM), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Lumsdaine-2018-02-15-HoTTEST.pdf))

* [[Emily Riehl]], *The synthetic theory of infinity-categories vs the synthetic theory of infinity-categories*, 1 March 2018, ([video](https://www.youtube.com/watch?v=ge-9m1SsEmc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Riehl-2018-03-01-HoTTEST.pdf))

  > on [[synthetic (infinity,1)-category theory]] in [[simplicial type theory]] and [[infinity-cosmos]] theory

* [[Carlo Angiuli]], *Computational semantics of Cartesian cubical type theory*, 15 March 2018 ([video](https://www.youtube.com/watch?v=3iKwq9iRHWk), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Angiuli-2018-03-15-HoTTEST.pdf))

* [[Ulrik Buchholtz]], *From Higher Groups to Homotopy Surfaces*, 29 March 2018 ([video](https://www.youtube.com/watch?v=0UlLwf8ttUw), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Buchholtz-2018-03-29-HoTTEST.pdf))

* [[Michael Shulman]], *Type 2-theories*, 12 April 2018 ([video](https://www.youtube.com/watch?v=hEhFeatKchQ), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Shulman-2018-04-12-HoTTEST.pdf))

* [[Martín Hötzel Escardó]], *Constructive mathematics in univalent type theory*, 26 April 2018 ([video](https://www.youtube.com/watch?v=rM484kI7OAA), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Escardo-2018-04-26-HoTTEST.pdf))

* [[Thorsten Altenkirch]], *Towards higher models and syntax of type theory*, 10 May 2018 ([video](https://www.youtube.com/watch?v=UgNYKxZKzsk), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Altenkirch-2018-05-10-HoTTEST.pdf))

* [[Thierry Coquand]], *A survey of constructive models of univalence*, 24 May 2018 ([video](https://www.youtube.com/watch?v=fv_dcmcQec4&), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Coquand-2018-05-24-HoTTEST.pdf))

### Autumn 2018

* [[Andy Pitts]], *Axiomatizing Cubical Sets Models of Univalent Foundations*, 20 September 2018 ([video](https://youtu.be/sy1-toqYRrk), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Pitts-2018-09-20-HoTTEST.pdf))

* [[Dimitris Tsementzis]], *First-Order Logic with Isomorphism*, 27 September 2018 ([video](https://www.youtube.com/watch?v=9fK9eEMYjII))

* [[Kuen-Bang Hou]], *Towards efficient cubical type theory*, 11 October 2018 ([video](https://www.youtube.com/watch?v=9pc4QkMtrnQ), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Favonia-2018-10-11-HoTTEST.pdf))

* [[Nicolai Kraus]], *Some connections between open problems*, 25 October 2018 ([video](https://www.youtube.com/watch?v=l8S0Ll4PhoI), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kraus-2018-10-25-HoTTEST.pdf))

* [[Guillaume Brunerie]], *Computer-generated proofs for the monoidal structure of the smash product*, 8 November 2018 ([video](https://youtu.be/JEUvWyd1mTk), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Brunerie-2018-11-08-HoTTEST.pdf))

* [[Floris van Doorn]], *Towards Spectral Sequences for Homology*, 22 November 2018 ([video](https://www.youtube.com/watch?v=Q3zaqeKhUKg), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/van-Doorn-2018-11-22-HoTTEST.pdf))

* [[Eric Finster]], *Towards Higher Universal Algebra in Type Theory*, 6 December 2018 ([video](https://www.youtube.com/watch?v=hlCVHVtAlqQ), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Finster-2018-12-6-HoTTEST.pdf))

### Spring 2019

* [[Nicola Gambino]], *From algebraic weak factorisation systems to models of type theory*, 31 January 2019 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Gambino-2019-01-31-HoTTEST.pdf))

* [[Nima Rasekh]], *Algebraic Topology in an Elementary Higher Topos*, 7 February 2019 ([video](https://www.youtube.com/watch?v=6fe7GjpSj6k), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Rasekh-2019-02-07-HoTTEST.pdf))

* [[Simon Huber]], *Homotopy canonicity for cubical type theory*, 21 February 2019 ([video](https://www.youtube.com/watch?v=Q7-I8Lh4-kg), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Huber-2019-02-21-HoTTEST.pdf))

* [[Evan Cavallo]], *Internal Parametricity and Cubical Type Theory*, 7 March 2019 ([video](https://www.youtube.com/watch?v=0C06XYL2Oro), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cavallo-2019-03-07-HoTTEST.pdf))

* [[Dan Licata]], *A Fibrational Framework for Substructural and Modal Dependent Type Theories*, 21 March 2019 ([video](https://youtu.be/-DP0wY2FBs4), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Licata-2019-03-21-HoTTEST.pdf))

* [[Joachim Kock]], *Infinity-operads as polynomial monads*, 4 April 2019 ([video](https://www.youtube.com/watch?v=Ljfz83Q41vs), [lecture notes](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kock-2019-04-04-HoTTEST.pdf))

* [[Paolo Capriotti]], *Polynomial monads as opetopic types*, 18 April 2019 ([video](https://www.youtube.com/watch?v=8P5bWQhKcG4), [lecture notes](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Capriotti-2019-04-18-HoTTEST.pdf))

* [[Mathieu Anel]], *Descent v. Univalence*, 2 May 2019 ([video](https://www.youtube.com/watch?v=86s0BCB3HGo), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Anel-2019-05-2-HoTTEST.pdf))

### Autumn 2019

* [[Andrej Bauer]], *General type theories*, 9 October 2019 ([video](https://www.youtube.com/watch?v=mdVmF9FB0LM), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Bauer-2019-10-09-HoTTEST.pdf))

* [[Anders Mörtberg]], *Unifying Cubical Models of Homotopy Type Theory*, 23 October 2019 ([video](https://www.youtube.com/watch?v=KxQYT4SrENg), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Mortberg-2019-10-23-HoTTEST.pdf))

* [[Andrew Swan]], *Choice, Collection and Covering in Cubical Sets*, 6 November 2019 ([video](https://www.youtube.com/watch?v=r9KbEOzyr1g), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Swan-2019-11-06-HoTTEST.pdf))

* [[Benno van den Berg]], *Uniform Kan fibrations in simplicial sets*, 20 November 2019 ([video](https://www.youtube.com/watch?v=5Mgau0Uh8IM), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Berg-2019-11-20-HoTTEST.pdf))

* [[Richard Garner]], *Polynomial comonads and comodules*, 11 December 2019 ([video](https://www.youtube.com/watch?v=tW6HYnqn6eI), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Garner-2019-12-11-HoTTEST.pdf))

### Spring 2020

* [[Simon Henry]], *The language of a model category*, 23 January 2020 ([video](https://www.youtube.com/watch?v=7_X0qbSXlfk), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Henry-2020-01-23-HoTTEST.pdf))

* [[Niels van der Weide]], *Constructing 1-Truncated Finitary Higher Inductive Types as Groupoid Quotients*, 6 February 2020 ([video](https://www.youtube.com/watch?v=sQhqQTMcMKw), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weide-2020-02-06-HoTTEST.pdf))

* [[Karol Szumiło]], *The Constructive Kan-Quillen Model Structure*, 20 February 2020 ([video](https://www.youtube.com/watch?v=xjghkPwLFTo), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Szumilo-2020-02-20-HoTTEST.pdf))

* [[Jon Sterling]], *Objective Metatheory of Dependent Type Theories*, 19 March 2020 ([video](https://www.youtube.com/watch?v=cyBn-lkvXTc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Sterling-2020-03-19-HoTTEST.pdf))

* [[Denis-Charles Cisinski]], *Univalence of the universal coCartesian fibration*, 2 April 2020 ([video](https://www.youtube.com/watch?v=OnMUka9bLAw), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cisinski-2020-04-02-HoTTEST.pdf))

* [[Matthew Weaver]], *A constructive model of directed univalence in bicubical sets*, 16 April 2020 ([video](https://www.youtube.com/watch?v=kkfNjqSx4Nw), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weaver-2020-04-16-HoTTEST.pdf))

### Autumn 2020

* [[Guillaume Brunerie]], [[Peter LeFanu Lumsdaine]], *Initiality for Martin-Löf type theory*, 10 September 2020 ([video](https://www.youtube.com/watch?v=1ogUFFUfU_M), [slides-GB](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Brunerie-2020-09-10-HoTTEST.pdf), [slides-PLL](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Lumsdaine-2020-09-10-HoTTEST.pdf))

* [[Jakob von Raumer]], *Coherence via Well-Foundedness*, 24 September 2020 ([video](https://www.youtube.com/watch?v=5iAQ-JVdlmc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/von-Raumer-2020-09-24-HoTTEST.pdf))

* [[Yuki Maehara]], *A cubical model for weak omega-categories*, 8 October 2020 ([video](https://www.youtube.com/watch?v=AIBTh7MIbew), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Maehara-2020-10-08-HoTTEST.pdf))

* [[Ambrus Kaposi]], *Quotient inductive-inductive types and higher friends*, 22 October 2020 ([video](https://www.youtube.com/watch?v=a9_KjX1WM84), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kaposi-2020-10-22-HoTTEST.pdf))

* [[Nima Rasekh]], *Filter Products and Elementary Models of Homotopy Type Theory*, 5 November 2020 ([video](https://www.youtube.com/watch?v=Yk6yEv02PbY), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Rasekh-2020-11-05-HoTTEST.pdf))

* [[Pierre Cagne]], *On the symmetries of the spheres in univalent foundations*, 19 November 2020 ([video](https://www.youtube.com/watch?v=u8wuoq_MQz8), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cagne-2020-11-19-HoTTEST.pdf))

* [[Jamie Vicary]], *A type theory for strictly unital infinity-categories*, 3 December 2020 ([video](https://www.youtube.com/watch?v=TSCggv_YE7M), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Vicary-2020-12-03-HoTTEST.pdf))

### Spring 2021

* [[Thierry Coquand]], *Sheaf models and constructive mathematics*, 28 January 2021 ([video](https://www.youtube.com/watch?v=_H-CD4B8pLQ), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Coquand-2021-01-28-HoTTEST.pdf))

* [[Norihiro Yamada]], *Game semantics of homotopy type theory*, 11 February 2021 ([video](https://www.youtube.com/watch?v=uY5LubwJ1Xo), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Yamada-2021-02-11-HoTTEST.pdf))

* [[Carlo Angiuli]], *Internalizing Representation Independence with Univalence*, 25 February 2021 ([video](https://www.youtube.com/watch?v=e3wGUuAxfG8), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Angiuli-2021-02-25-HoTTEST.pdf))

* [[Andrea Vezzosi]], *Cubical Agda and its Extensions*, 11 March 2021 ([video](https://www.youtube.com/watch?v=9RFt1Q2pHE8), [Agda files](https://github.com/Saizan/hottest-talk))

* [[Håkon Robbestad Gylterud]], *Defining and relating theories*, 25 March 2021 ([video](https://www.youtube.com/watch?v=8aTDEn2D2pA), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Gylterud-2021-03-25-HoTTEST.pdf))

* [[Egbert Rijke]], *A higher encode decode method*, 8 April 2021 ([video](https://www.youtube.com/watch?v=KvHGds-AL8M), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Rijke-2021-04-08-HoTTEST.pdf))

* [[Ulrik Buchholtz]], *(Co)cartesian families in simplicial type theory*, 22 April 2021 ([video](https://www.youtube.com/watch?v=TOGx2F-MLi0), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Buchholtz-2021-04-22-HoTTEST.pdf))

### Autumn 2021

* [[Rafaël Bocquet]], *Coherence of definitional equality in type theory*, 23 September 2021 ([video](https://www.youtube.com/watch?v=WKCxBygnYqE), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Bocquet-2021-09-23-HoTTEST.pdf))

* [[Matthieu Sozeau]], *The MetaCoq Project*, 7 October 2021 ([video](https://www.youtube.com/watch?v=azlhMVH1f98), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Sozeau-2021-10-07-HoTTEST.pdf))

* [[Hugo Moeneclaey]], *Parametricity and cubes*, 21 October 2021 ([video](https://www.youtube.com/watch?v=79Ik6KkoZ-w), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Moeneclaey-2021-10-21-HoTTEST.pdf))

* [[Kristina Sojakova]], *Syllepsis in Homotopy Type Theory*, 4 November 2021 ([video](https://www.youtube.com/watch?v=k07gFYzYvng), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Sojakova-2021-11-04-HoTTEST.pdf))

* [[Raffael Stenzel]], *Higher sites and their higher categorical logic*, 18 November 2021 ([video](https://www.youtube.com/watch?v=hUJlK8XgZxQ), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Stenzel-2021-11-18-HoTTEST.pdf))

* [[Taichi Uemura]], *Infinity-type theories and internal language conjectures*, 2 December 2021 ([video](https://www.youtube.com/watch?v=DY42DgCxQls), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Uemura-2021-12-02-HoTTEST.pdf))

### Spring 2022

* [[Jonas Frey]], *Characterizing clan-algebraic categories*, 3 February 2022 ([video](https://www.youtube.com/watch?v=0SAQSx0vPmU), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Frey-2022-02-03-HoTTEST.pdf))

* [[Marcelo Fiore]], *Mathematical and Computational Metatheory of Second-Order Algebraic Theories*, 17 February 2022 ([video](https://www.youtube.com/watch?v=nP2J9SJ9DVM), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Fiore-2022-02-17-HoTTEST.pdf))

* [[Fredrik Nordvall Forsberg]], *Different Notions of Ordinals in Homotopy Type Theory*, 3 March 2022 ([video](https://www.youtube.com/watch?v=8_3tLrrfrLY), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Forsberg-2022-03-03-HoTTEST.pdf))

* [[Chaitanya Leena Subramaniam]], *Dependently typed algebraic theories and their homotopy algebras*, 17 March 2022 ([video](https://www.youtube.com/watch?v=NAT02hbwOUs), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Subramaniam-2022-03-17-HoTTEST.pdf))

* [[Assia Mahboubi]], *Continuity in dependent type theory*, 31 March 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Mahboubi-2022-03-31-HoTTEST.pdf))

### Autumn 2022

* [[Amélia Liao]], *Univalent Category Theory*, 6 October 2022, ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Liao-2022-10-06-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=asY6dXkR2yg))

  > on [[category theory]] in [[univalent type theory|univalent]] [[cubical type theory]] in the [[1lab]]

* [[Chris Grossack]], *Where are the open sets? Comparing HoTT with Classical Topology*, 6 October 2022, ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Grossack-2022-10-06-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=syftUuf5Hpk))

* [[Max Zeuner]], *A univalent formalization of affine schemes*, 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zeuner-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nLP7GjL1Buc))

* [[Axel Ljungström]], *Calculating a Brunerie Number*, 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Ljungstrom-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=MIMeQ88YMKI))

* [[Evan Cavallo]], *Cubes with one connection and relative elegance*, 3 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Cavallo-2022-11-03-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=CrcExHaMDfk))

* [[Colin Zwanziger]], *Hofmann-Streicher Universes via Coalgebra*, 3 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zwanziger-2022-11-03-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=UG0xwTNUplo))

* [[Tom de Jong]], *Acyclic types and epimorphisms in HoTT*, 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Jong-2022-11-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=vohAYPAAWxs))

* [[Jarl G. Taxerås Flaten]], *Central H-spaces and banded types*, 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Flaten-2022-10-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=-7PG2bzseJI)).

* [[Jacob Neumann]], *(Co)ends and (Co)structure*, 1 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Neumann-2022-12-01-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=X4v5HnnF2-o))

* [[Astra Kolomatskaia]], *Semi-Simplicial Types*, 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kolomaskaia-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=fQv2FpeFxew))

* [[Philipp Joram]], *Final Coalgebras of Analytic Functors in Homotopy Type Theory*, 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Joram-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nsgztQ5UbLw))

### Spring 2023

* [[Johan Commelin]], *Liquid Tensor Experiment*, 26 January 2023 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Commelin-2023-01-26-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=gkeGLRrAkbY))

* [[Andrew Swan]], *Double negation stable h-propositions in cubical sets*, 9 February 2023 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Swan-2023-02-09-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=lNNrmuUQzkw))

## See also

* [[homotopy type theory events]]

## External Links

* [Website](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)

* [Youtube channel](https://www.youtube.com/user/jdchristensen123)

* [Discord server](https://discord.gg/tkhJ9zCGs9)

category: reference

[[!redirects HoTTEST]]


homotopy type theory events

This page collects programs, slides, and other resources of [[homotopy type theory]] workshops, meetings, and other events.

* [[CMU HoTT Research Group's local activities]].

## Perennial events ##

* [Seminar - Homotopy Type Theory at CMU](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)

* [HoTT/UF](https://hott-uf.github.io/) workshops

### Passed ###

* [Previous Seminars - Homotopy Type Theory at CMU](https://www.cmu.edu/dietrich/philosophy/hott/seminars/previous.html)

* [HoTT/UF'20](https://hott-uf.github.io/2020) 5-6 Jul 2020, online.

## Other events ##

* [Cortona](https://unimath.github.io/cortona2022/) UniMath school, July 17-23, 2022, Cortona, Italy

### Passed ###

* [ECM Minisymposium](https://www.8ecm.si/minisymposia) two days, sometime between 20-26 June 2021

* [EPIT](https://epit2020cnrs.inria.fr) April 12-16 2021,
Spring School, Fully online.

* [Memorial conference for Erik Palmgren](http://logic.math.su.se/palmgren-memorial/) Postponed, Stockholm, Sweden

* [Cortona](https://unimath.github.io/cortona2020/) UniMath school, 27-31 July 2020, Cortona, Italy

* [ICMS](https://univalent-math.github.io/) sometime between 13-16 July 2020, Braunschweig, Germany

* [HoTTest conference](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest_conference_2020.html) June 15-19

* [4th Midwest HoTT Seminar](https://paigenorth.github.io/midwesthott/) Cancelled.

* [TYPES](https://types2020.di.unito.it/) Cancelled, but abstracts available.

* [HoTT2019](https://hott.github.io/HoTT-2019/) Summer school and conference, 7-17 August 2019, at CMU.
Includes slides from the conference lectures and summer school courses.

* [Leeds](https://conferences.leeds.ac.uk/httuf/) The Summer School on Higher Topos Theory and Univalent Foundations will take place from June 24th to June 28th 2019

* [TYPES](https://cas.oslo.no/hott-uf/)
Oslo, Norway, 12 - 14 June 2019

* [6WFTop](http://www.cs.bham.ac.uk/~sjv/6WFTop/) Mon 8 to Fri 12 April 2019

* [Geometry in Modal Homotopy Type Theory](https://felix-cherubini.de/modal-workshop.html), CMU, March 11-15, 2019

* [[HoTT in Bonn2018]]:  Types, Homotopy Type Theory, and Verification,
Hausdorff Institute for Mathematics
[Videos of the talks are available here.](https://www.youtube.com/channel/UC2F-j2KMho0zVWIPFKWoXoA/videos/)
[The Hausdorff Institute page is here.](https://www.him.uni-bonn.de/programs/current-trimester-program/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)

* [[HoTT at DMV2015]]: Homotopy Type Theory and Univalent Foundations at DMV 2015

category: reference


homotopy type theory FAQ

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

This page is to provide non-technical or maybe semi-technical discussion of the nature and role of the [[foundation of mathematics|foundational system]] for [[mathematics]] known as _homotopy type theory_. For more technical details and further pointers see at [[homotopy type theory]].

#Contents#
* table of contents
{:toc}

## FAQ

### What is homotopy type theory?

#### For the person on the street

Homotopy type theory is a theory of objects, which we call terms, collections of objects, which we call types, and equalities between any two objects in a collection, which we call identity types.

One fundamental aspect of mathematics is that equalities between different collections of objects behave differently from each other. If we have a type where for all terms have exactly one identity between them, this is called a proposition or a truth value. The type with no terms is trivially a proposition, as every single one of the nonexistent terms have exactly one identity between them.

However, not all types are propositions: take the type with two distinct terms $a$ and $b$, there is no identity between $a$ and $b$. The identity types between two terms here are propositions, and the types are called sets.

Still, not all types are sets: take the type of sets with two terms, the first one with terms $a$ and $b$, and the second one with terms $c$ and $d$. There are two ways to say that these types are equal: by adding an identity between $a$ and $c$ and between $b$ and $d$, or by adding an identity between $a$ and $d$ and between $b$ and $c$. Thus, the identity types between the two sets is set-valued, rather than proposition-valued, and the type is a groupoid, rather than a set. In this way, we could say that, for example, the set of positive integers is equal to the set of negative integers, or that the set of complex numbers is equal to the set of ordered pairs of real numbers.

One could continue the process, yielding 2-groupoids whose identity types are groupoids, 3-groupoids whose identity types are 2-groupoids, and so forth. And finally, one could have types whose identity types are general types: these types are also known as $\infty$-groupoids.

#### For set theorists
 {#WhatIsHoTTForSetTheorists}

Homotopy type theory is a refinement of [[constructive set theory]] that takes fully seriously the constructive nature also of [[identity types|identity]]. (As with all [[constructive mathematics]], with the relevant [[axioms]] added, this subsumes, rather than excludes, classical mathematics, see below at _[Is HoTT limited to constructive mathematics?](#IsHoTTLimitedToConstructiveMathematics)_.)

It is this insistence on constructive witnesses for identity and on witnesses for identity of such witnesses, and so ever on, that makes what in traditional [[set theory]] are just [[sets]] (whose elements are either equal or not) in homotopy type theory instead be [[groupoids]] (whose elements may have non-trivial [[isomorphism]] between them) and indeed [[2-groupoids]] (with isomorphisms between these isomorphisms) and so ever on; hence what makes what used to be just [[sets]] be what is kown as _[[∞-groupoids]]_ or _[[homotopy types]]_.

More technically, restriction to the [[0-truncated types]] in HoTT (the "[[h-set]]s") gives a [[predicative topos]] "of sets", and a [[topos]] if one allows the [[resizing axiom]] (details are [here](http://ncatlab.org/nlab/show/set%20theory#ReferencesInHomotopyTypeTheory)). When adding the [[axiom of choice]] to HoTT, one obtains a model of [[ETCS]].
The iterative notion of set can also be captured. Aczel's sets-as-trees interpretation gives a model of constructive set theory [[CZF]]. Again by adding choice to HoTT, one obtains a model of [[ZFC]]; see Ch10 of the HoTT book.

Conversely HoTT has [[models]] in [[ZFC]] (+a number of [[Grothendieck universes]] to model the type theoretic universes), namely in structures called _[[(∞,1)-toposes]]_ which are presented by [[presheaves]] of [[simplicial sets]]. (See also at _[Does HoTT have models in infinity-toposes?](#InterpretationInInfinityToposes)_)

The nature and role of these higher toposes in HoTT may be understood by analogy with the familar [[forcing]] [[models]]:
When one proves something in [[ZF]], it is automatically also true in all [[forcing]] extensions.  The same is true for [[constructive set theory]], except that there are more forcing extensions since we don't have to force the [[law of excluded middle]]; those constructive notions of forcing (which also subsume [[permutation models]]) are called "[[sites]]" and their [[models]] are called "[[topos|1-toposes]]".  Now in HoTT we have an even more general sort of forcing appropriate for [[homotopy theory]], called [[(∞,1)-sites]], whose models are called [[(∞,1)-toposes]].

#### For type theorists

(...)

#### For homotopy theorists

(...)

### Why should I care?
 {#WhyShouldICare}

#### For the person on the street

Homotopy type theory is a foundation upon which all of mathematics is based upon. Every collection of objects in mathematics, such as the collection of natural numbers, collection of real numbers, collection of strings of characters in an alphabet, collection of propositions and the collection of all sets, is represented in homotopy type theory. It is also a language in which one could do mathematics in, from algebra to geometry to calculus.

#### For set theorists
 {#WhyShouldICareForSetTheorists}

Traditional [[set theory]] is formulated in [[first-order logic]]. Under the [[propositions as types]] interpretation, first order logic maps soundly and completely into [[type theory]] ([Martin-Löf 74, section 3.1](type+theory#MartinLoef74), [Barendregt 91, section 4](pure+type+system#Barendregt91)). But now the [[homotopy 0-types]] in [[homotopy type theory]] already give a [[constructive set theory]], natively, without further axiomatization. See at _[[h-set]]_ for more.

#### For type theorists

#### For homotopy theorists
 {#WhyShouldICareForHomotopyTheorists}

Most homotopy-theoretic theorems that are proven in the [[Homotopy Type Theory -- Univalent Foundations of Mathematics|HoTT textbook]] are, in their traditional informal version, material of a first-year course on homotopy theory. Experts who do not care about [[formal proof]] might not be impressed yet.

But the point is that there is significant prospect. By the discussion at _[Does HoTT have models in infinity-toposes](#InterpretationInInfinityToposes)_, it is a fact that homotopy type theory is the [[internal language]] of [[(∞,1)-toposes]], hence of the most powerful modern incarnation of [[homotopy theory]]. Whether or not anyone has already made impressive use of this fact (but see below), this is of genuine interest in homotopy theory irrespective of the issue of formal proof.

Historically, making use of the [[internal logic]]-perspective of [[elementary toposes]] led to substantial insight into all theory that uses [[Grothendieck toposes]], such as notably [[algebraic geometry]]. While [[Jacob Lurie|Lurie]]'s book _[[Higher Topos Theory]]_ based on [[Charles Rezk]]'s note _[Homotopy Topos Theory](infinity,1-topos#Rezk10)_ is an astounding piece of work, it falls short of saying anything about this crucial internal aspect of (higher) topos theory. Homotopy type theory is precisely what fills this gap. For more on how this works see at _[[HoTT methods for homotopy theorists]]_.

An illustration of the use of this is the proof of the _[[Blakers-Massey theorem]]_. This basic fact of homotopy theory has a classical proof in the classical homotopy category [[∞Grpd]] $\simeq$ $L_{whe}$ [[Top]], a proof that is however rather roundabout. From this follows a proof in all [[(∞,1)-toposes]] with an [[(∞,1)-site]] of definition by, essentially, reducing [[stalk|stalkwise]] to the classical theorem (details are [here](Blakers-Massey%20theorem#Rezk10)). But now the theorem was also proven in homotopy type theory (details are [here](Blakers-Massey%20theorem#LumsdaineFinsterLicata13)). Translating this formal proof back to ordinary language produces first of all a _new_ and more elegant proof of Blakers-Massey in [[∞Grpd]] $\simeq$ $L_{whe}$ [[Top]], second a new and elegant proof of the statement for [[(∞,1)-toposes]] with [[(∞,1)-site]] of definition. This is something that some homotopy theorists tried to find by classical means and failed (details are [here](Blakers-Massey%20theorem#Rezk14)). And moreover, the HoTT proof of Blakers-Massey is actually a new result when applied to [[elementary (∞,1)-toposes]], where the classical methods of proving it completely break down.

It seems rather plausible that this is just a simple first example of a future where HoTT methods allow to enter new territory in classical homotopy theory. Of course it will require some expert homotopy theorists to seriously look into the internal-language-of-$\infty$-toposes way of doing homotopy theory. This is now an open problem for homotopy theorists just as it is for type theorists.

### What role does the univalence axiom play?
 {#WhatRoleDoesTheUnivalenceAxiomPlay}

#### For the person on the street

The [[axiom]] of [[univalence]] may be thought of as a [[formal logic|formalization]] of what might be called the _[[principle of equivalence]]_ of [[mathematics]], which is the basic but important idea that [[mathematical structures]] which are [[equivalence|equivalent]] should behave the same, satisfy the same theorems and so forth.

In particular, in elementary [[Euclidean geometry]], two shapes are traditionally said to be equal or congruent to each other if there is an [[isometry]] (distance-preserving transformation) between the two shapes. Usually, this congruence requires the need for a quotient of the isomorphism classes in the category of geometric shapes, but with univalence, one does not need such a construction: equality of shapes is the isometries between the shapes.

Obvious as this may seem, this principle may be violated in other [[foundations of mathematics]] such as [[ZFC]]. On the other hand, while these models allow such violation, in practice one essentially never wants to use such violation. The univalence axiom hence serves to make formal mathematical foundation be closer to the actual "nature of mathematics", at least to the practice of mathematics.

#### For set theorists

#### For type theorists

The axiom of [[univalence]] added to [[Martin-Löf type theory]] implies all of the following:

1. [[functional extensionality]]

1. [[quotient types]]

1. [[higher inductive types]] with nontrivial homotopies,

   Which in turn imply a wealth of further structure such as (but not limited to)

   1. [[bracket types]]

   1. etc.

#### For homotopy theorists

In view of the answer at _[Does HoTT have interpretation in infinity-toposes?](#InterpretationInInfinityToposes)_
the [[axiom]] of [[univalence]] axiomatizes the presence of an [[object classifier]] in an [[(∞,1)-topos]] $\mathbf{H}$ (details are [here](univalence+axiom#GepnerKock12)).

In the default choice $\mathbf{H} = $[[∞Grpd]] $\simeq $ $L_{whe}$[[Top]] of traditonal [[homotopy theory]] this is the [[universal principal ∞-bundle|universal fibration]] over the disjoint union of [[classifying spaces]] $\coprod_{[F]} B Aut(F)$ of the [[automorphism ∞-groups]] of all (small) [[homotopy types]] $[F]$.

Therefore from the axiom of [[univalence]] follows for instance the theory of [[∞-actions]], [[associated ∞-bundles]] and of [[twisted cohomology]] (details are [[schreiber:Principal ∞-bundles -- theory, presentations and applications|here]]).

### What advantages does homotopy type theory have over set theory?

As explained at _[What is HoTT for set theorists?](#WhatIsHoTTForSetTheorists)_, HoTT subsumes [[set theory]]. It has all the advantages that [[structural set theory]] ([[Algebraic set theory]], [[ETCS]]) has over [[material set theory]] ([[ZFC]]), but moreover it allows us to natively capture [[higher category theory|higher categorical]] (more precisely, [[infinity-groupoid|higher groupoidal]]) and [[homotopy theory|homotopical]] reasoning. Moreover, as a practical foundation, set theory may be compared to the Turing machine model, or perhaps more generously, say, [ALGOL](http://en.wikipedia.org/wiki/ALGOL). Whereas, homotopy type theory is closely related to modern programming languages like Haskell and ML, or more accurately [[Coq]] and [[Agda]].

### Is homotopy type theory limited to constructive mathematics?
 {#IsHoTTLimitedToConstructiveMathematics}

No. On the [[0-types]] (the [[h-sets]]) the axioms of [[classical logic]] may be imposed, such as the [[law of excluded middle]] (details are [here](excluded+middle#ReferencesInHomotopyTypeTheory)) and the [[axiom of choice]] (details are [here](axiom+of+choice#ReferencesInHomotopy)).

One [[model]] that interprets the resulting axioms is the standard model in [[simplicial sets]] ([here](model%20structure%20on%20simplicial%20sets#KapulkinLumnsdaineVoevodsky12)) inside [[ZFC]]+[[inaccessible cardinals]]. The [[0-types]] in this model are precisely the ordinary sets in ZFC. See also the discussion at _[What is HoTT? For set theorists](#WhatIsHoTTForSetTheorists)_.

For more exposition see also

* [[Andrej Bauer]], _[Univalent foundations subsume classical mathematics](http://math.andrej.com/2014/01/13/univalent-foundations-subsume-classical-mathematics/)_ (2013)

### Does HoTT have interpretation/semantics/models in $\infty$-toposes?
 {#InterpretationInInfinityToposes}

The short answer is: Yes. Homotopy type theory has [[higher category theory|higher]] [[categorical semantics]] in every [[(∞,1)-topos]], in refinement of how plain [[dependent type theory]] has [[categorical semantics]] in [[toposes]].

The more detailed answer depends on which axiomatics precisely one subsumes under "homotopy type theory". A hierarchy of three main variants ("fragments") of homotopy type theory should be distinguished here:

1. HoTT without [[univalence|univalent]] [[type universes]];

1. HoTT with [[univalence|univalent]] weak [[type universes]]   [à la Tarski](type%20of%20types#TarskiStyle);

1. HoTT with [[univalence|univalent]] strict [[type universes]]  [à la Russell](type%20of%20types#RussellStyle) or [à la Tarski](type%20of%20types#TarskiStyle);

In all three cases the $\infty$-categorical semantics is induced by ordinary [[categorical semantics]] in a [[type-theoretic model category]] which in turn [[locally presentable (infinity,1)-category|presents]] an [[(∞,1)-category]]. The way this works is reviewed also at _[[HoTT methods for homotopy theorists]]_. This relies on standard techniques for interpreting any [[dependent type theory]] in a [[locally cartesian closed category]] (see [here](relation+between+type+theory+and+category+theory#DependentTypeTheory) for details).

With that understood, the higher categorical semantics for the above three cases is as follows:

1. HoTT without univalent universes has semantics in every [[locally presentable (∞,1)-category|locally presentable]] [[locally cartesian closed (∞,1)-category]] (details are [here](locally+cartesian+closed+%28infinity,1%29-category#InternalLogic)).

1. HoTT with strict univalent universes has semantics at least in [[(∞,1)-category of (∞,1)-presheaves|(∞,1)-presheaf]] [[(∞,1)-toposes]] over [[elegant Reedy categories]]. This includes in particular the standard [[base (∞,1)-topos]] [[∞Grpd]] as well as for instance the [[Sierpinski (∞,1)-topos]] (details are [here](relation+between+type+theory+and+category+theory#Shulman13)).

1. HoTT with weak univalent universes has semantics in every [[(∞,1)-sheaf (∞,1)-topos]] (details are [[homotopytypetheory:model of type theory in an (infinity,1)-topos|here]]).

Since we are assuming [[locally presentable (∞,1)-category|local presentability]] here, all these models have enough [[(∞,1)-colimits]] to interpret [[higher inductive types]]. (One might also ask for models of HoTT without univalence and without higher inductive types and would expect that this then also includes [[locally cartesian closed (∞,1)-categories]] which are not necessarily locally presentable.)

One may also ask about models in [[elementary (∞,1)-toposes]]. Their theory or even definition is however much in the making. On the other hand, existing proposals essentially turn the question around and define [[elementary (∞,1)-toposes]] as [[(∞,1)-categories]] with the minimum of properties such that HoTT may be modeled in them. This reflects the idea that what is "elementary" about elementary $(\infty,1)$-toposes is precisely the fact that there is an internal type theory language for them.

In conclusion, every [[proposition]] [[proof|proven]] in homtopy type theory yields (subject to the above qualification) a statement that holds true in every [[(∞,1)-topos]] (in every presentable [[locally cartesian closed (∞,1)-category]]), i.e. homotopy type theory is the [[internal language]] of $\infty$-toposes.

If one wishes to prove statements that hold only in some class of $\infty$-toposes, then one needs to add further [[axioms]] to HoTT that characterize these classes. For instance adding [[higher modalities]] that define [[cohesive homotopy type theory]] make it a language that proves theorems which hold in all [[cohesive (∞,1)-toposes]].

### What is meant by a "computational interpretation of univalence"?

### What are higher inductive types?

### What do we gain by a proof in homotopy type theory of $\pi_1(S^1) = \mathbb{Z}$ over ordinary proofs?

### Is it possible to define higher coinductive types?

### In what sense does homotopy type theory already contain logic?

A [[proposition]] can be defined as a type such that the identity type between any two terms of the type has a term. One could add to homotopy type theory the [[higher inductive type]] of [[propositional truncation]], which takes any type and turns it into a proposition. The propositional truncation of [[product types]] correspond to [[conjunction]], the propositional truncation of [[sum types]] correspond to [[disjunction]], the propositional truncation of [[function types]] correspond to [[implication]], the propositional truncation of [[dependent product types]] correspond to [[universal quantification]], the propositional truncation of [[dependent sum types]] correspond to [[existential quantification]], the [[empty type]] corresponds to [[false]], a [[function type]] into the empty type corresponds to [[negation]], and [[contractible types]] correspond to true. This corresponds to [[constructive logic|constructive]] [[predicate logic]], and the addition of the axiom of [[excluded middle]] to the type theory yields [[classical logic|classical]] [[predicate logic]].

### Can category theory be carried out in homotopy type theory?

Yes. One could define a **category** as a type $C$ whose identity types $a =_C b$ for $a:C$ and $b:C$ are sets, with a set $M$ with functions $s:M \to C$, $t:M \to C$, $id:C \to M$ and function
$$c:\left(\sum_{f:M} \sum_{g:M} t(f) =_C s(g)\right) \to M$$
such that $s(c(g,f)) =_C s(f)$ and $t(c(g,f)) =_C t(g)$, for every $f:M$, $c(f, id(s(f))) =_M f$ and $c(id(t(f)), f) =_M f$, and for every $f:M$, $g:M$, and $h:M$ such that $t(f) =_C s(g)$ and $t(g) =_C s(h)$, $c(h,c(g,f)) =_M c(c(h,g),f)$.

Then one could define a **functor** between categories $(C,M)$ and $(D,N)$ as a function $F:C \to D$ with a function $G:M \to N$ such that for all $f:M$, $F(s_C(f)) =_D s_D(G(f))$ and $F(t_C(f)) =_D t_D(G(f))$, for all $a:C$, $G(id_C(a)) =_N id_D(F(a))$, and for terms $f:M$ and $g:M$ such that $t(f) =_D s(g)$, $G(g \circ_C f) =_N F(g) \circ_D F(h)$.

### Can $(\infty,1)$-categories be defined in homotopy type theory?

It is currently unknown how to define the [[coherence theorems]] necessary to define $(\infty,1)$-categories in vanilla homotopy type theory. There are other approaches to type theory such as [[directed homotopy type theory]] which might be able to define $(\infty,1)$-categories.

## Related articles

* [[homotopy theory FAQ]]

[[!redirects HoTT FAQ]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of a [[homotopy type]] ([[homotopy n-type]]) all of whose [[homotopy groups]] are [[finite groups]] does not have an established name. Sometimes it is called _$\pi$-finiteness_. In the context of [[groupoid cardinality]] "tameness" is used. In [[homological algebra]] "[[of finite type]]" is used, which in [[homotopy theory]] however badly clashes with the concept of [[finite homotopy type]] which is crucially different from _homotopy type with finite homotopy groups_. ([Anel 21](#Anel21)) uses "truncated coherent spaces".

## Properties

### (Co-)Limits indexed by homotopy types with finite homotopy groups

see at _[[K(n)-local stable homotopy theory]]_ (...)

### Relation to coherent objects
 {#RelationToCoherentObjects}

+-- {: .num_prop }
###### Proposition

[[∞Grpd]] is a [[coherent (∞,1)-topos]] and a [[locally coherent (∞,1)-topos]]. An [[object]] $X$, hence an [[∞-groupoid]], is an [[n-coherent object]] precisely if all its [[homotopy groups]] in degree $k \leq n$ are [[finite set|finite]]. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

=--

([[Spectral Schemes|Lurie SpecSchm, example 3.13]])

+-- {: .num_prop }
###### Proposition
The $(\infty, 1)$-category of homotopy types with finite homotopy groups is the initial [[(∞,1)-pretopos]].

=--

([Anel 21, Theorem 2.6.6](#Anel21)).

### Codensity monad of the inclusion into all homotopy types
 {#CodensityMonadOfInclusionIntoAllHomotopyTypes}

The inclusion of homotopy types with finite groups into all homotopy types generates a [[codensity monad]] whose algebras lie somewhere between [[totally disconnected]] compact Hausdorff [[condensed mathematics|condensed]] $\infty$-groupoids, and all compact Hausdorff condensed $\infty$-groupoids. (See [Scholze](#Scholze) for more details.)

## Related concepts

* [[finite homotopy type]]

* [[finite type]]

* [[pi-finite type]]

## References

* G. J. Ellis, _Spaces with finitely many non-trivial homotopy groups all of which are finite_, Topology, 36, (1997), 501–504, ISSN 0040-9383.

* [[Jean-Louis Loday]], _Spaces with finitely many non-trivial homotopy groups_ (pdf)

* {#Scholze} [[Peter Scholze]], _Infinity-categorical analogue of compact Hausdorff_, [MO answer](https://mathoverflow.net/a/380399/447)

Discussion as an [[elementary (∞,1)-topos]]:

* {#Anel21} [[Mathieu Anel]], _The elementary infinity-topos of truncated coherent spaces_ ([arXiv:2107.02082](https://arxiv.org/abs/2107.02082))

Formalization in [[homotopy type theory]] (via [[Agda]]):

* [[UniMath project]], *π-finite types* [[web](https://unimath.github.io/agda-unimath/univalent-combinatorics.pi-finite-types.html)]

[[!redirects homotopy types with finite homotopy groups]]

[[!redirects tame homotopy type]]
[[!redirects tame homotopy types]]

[[!redirects π-finite homotopy type]]
[[!redirects π-finite homotopy types]]

[[!redirects pi-finite homotopy type]]
[[!redirects pi-finite homotopy types]]


Homotopy Types

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

This page compiles pointers related to:

* [[Hans Baues]]:

  \linebreak

  **Homotopy Types**

  [pdf](https://archive.mpim-bonn.mpg.de/id/eprint/2171/1/preprint_1994_88.pdf), [[Baues-HomotopyTypes.pdf:file]]

  \linebreak

  in:

  [[Ioan M. James]] (ed):

  *[[Handbook of Algebraic Topology]]*

  Oxford 1995

  [ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2)

  [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)

on [[homotopy theory]] with focus on the notion of [[homotopy types]].

It starts from a basic position and attacks three main topics:

* Homotopy types with non-trivial [[fundamental group]] (Sections 2 - 5);

* Homotopy types with trivial fundamental group (Sections 6 - 9, 12);

* [[stable homotopy types]] (Sections 10 and 11).

#Contents#
* table of contents
{:toc}

## What are homotopy types?

## How to build homotopy types

## Whitehead's realization  problem

## Algebraic models of $n$-types

## Cohomology of groups and cohomology of categories

## Simply connectd homotopy types and $H\pi$-duality

## The Hurewicz homomorphism

## Postnikov invariants and boundary invariants

## The classification theorems

## Stable homotopy types

## Decomposition of stable homotopy types

## Localization

##References

category:reference


homotopy weighted colimit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Homotopy weighted colimits__ (alias __weighted homotopy colimits__)
are the analog of [[weighted colimits]] in [[homotopy theory]].

## In relative categories

\[ ... \]

## In model categories

For the special case of [[model categories]], we can define homotopy weighted colimits as follows.

Fix a [[monoidal model category]] $V$,
a $V$-[[enriched model category]] $C$,
and a small $V$-[[enriched category]] $J$.

For simplicity, assume all [[enriched hom]] objects of $J$ are [[cofibrant]].
If this is not the case, we can first cofibrantly replace $J$
in the [[Dwyer-Kan model structure on enriched categories]].

We have a [[left Quillen bifunctor]]
$$V^{J^{op}} \times C^J \to C$$
given by the ordinary [[weighted colimit]] functor.

The _homotopy weighted colimit_ can then be defined
as the [[left derived]] [[Quillen bifunctor]] of the weighted colimit functor.

## Related concept

* [[homotopy coend]]

* [[fat geometric realization]]

## References

See Section 9.2 in

* [[Emily Riehl]], _[[Categorical Homotopy Theory]]_, New Mathematical Monographs 24, Cambridge University Press, 2014.

and for simplicially based theories,

*  [[Jean-Marc Cordier]] and [[Timothy Porter]], _Homotopy Coherent Category Theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54, ([pdf](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf))

(That article uses the older terminology of 'indexed colimits' rather than the `weighted' one.)

Other references:

* [[Lukáš Vokřínek]], _Homotopy weighted colimits_, [arXiv:1201.2970](https://arxiv.org/abs/1201.2970).

* [[Nicola Gambino]], *Weighted limits in simplicial homotopy theory*, Journal of Pure and Applied Algebra **214** 7 (2010) 1193–1199 [[doi:10.1016/j.jpaa.2009.10.006](https://doi.org/10.1016/j.jpaa.2009.10.006)]

* [[Michael Shulman]], _Homotopy limits and colimits and enriched homotopy theory_, [arXiv:math/0610194](https://arxiv.org/abs/math/0610194).

* [[Sergey Arkhipov]], [[Sebastian Ørsted]], _Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories_ ([arXiv:1807.03266](https://arxiv.org/abs/1807.03266)).

[[!redirects homotopy weighted colimits]]
[[!redirects weighted homotopy colimit]]
[[!redirects weighted homotopy colimits]]


homotopy-homology-cohomology

| | [[homotopy (as an operation)|homotopy]] | [[cohomology]] | [[homology]] |
|--|--|--|--|
|  | $[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ |
| [[category theory]] | [[covariant functor|covariant]] [[hom functor|hom]] | [[contravariant functor|contravariant]] [[hom functor|hom]] | [[tensor product]] |
| [[homological algebra]] | [[Ext]] | [[Ext]] | [[Tor]] |
| [[enriched category theory]] | [[end]] | [[end]] | [[coend]] |
| [[homotopy theory]] | [[derived hom space]] $\mathbb{R}Hom(S^n,-)$ | [[cocycles]] $\mathbb{R}Hom(-,A)$ | [[derived tensor product]] $(-) \otimes^{\mathbb{L}} A$ |


homotopy.io

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Introduction ##

### Overview ###

This page describes _homotopy.io_, a web-based [[proof assistant]] for finitely-presented [[globular set|globular]] [[n-category|_n_-categories]], for arbitrary $n$. It is available at the following address:

* [homotopy.io](https://homotopy.io)

* [github.com/homotopy-io](https://github.com/homotopy-io)

It is based on the theory of [[associative n-categories|associative _n_-categories (ANCs)]], which provides a strictly associative and unital model of [[composition]]. This reduces proof bureaucracy, with all the remaining weak structure in [[homotopies]] of composites, which have a direct geometrical interpretation. The proof assistant allows the user to define [[generators and relations|generators]] for a [[higher category]], [[composition|compose]] them, apply [[homotopies]], and visualize the resulting composites as [[string diagrams]] in 2 dimensions or [[surface diagrams]] in 3 dimensions. Interaction with the proof assistant is entirely by direct manipulation using the mouse.

It can be considered a successor to the existing proof assistant [[Globular]], which allows the construction of formal proofs in a strictly associative and unital model of [[4-categories]].

Homotopy construction in _homotopy.io_ is enabled by two basic mechanisms: _contraction_, which makes the geometry locally more singular; and _expansion_, which makes the geometry locally more generic. These local moves 'drag' neighbouring parts of the composite, and allow the construction of nontrivial [[homotopies]]. It is an open question whether these contraction and expansion moves are capable of generating all possible homotopies in every dimension.

The image at the top-right shows a 3d surface diagram of one side of the [[pentagon equation]] ([_live workspace_](https://homotopy.io/?id=kzmugNEGMPBF8BbB5FAm)).

### Contributors ###

The proof assistant is based on the theory of [[associative n-categories|associative _n_-categories]], which has been worked on by [Christoph Dorn](https://www.cs.ox.ac.uk/people/christoph.dorn/), [Christopher Douglas](https://www.maths.ox.ac.uk/people/christopher.douglas), [David Reutter](https://www.cs.ox.ac.uk/people/david.reutter/) and [Jamie Vicary](https://www.cs.bham.ac.uk/~vicaryjo/). The proof assistant has been implemented by [Lukas Heidemann](https://github.com/zrho), [Nick Hu](https://github.com/NickHu) and [Jamie Vicary](https://www.cs.bham.ac.uk/~vicaryjo/).

## Overview of functionality
 {#OverviewOfFunctionality}

Using _homotopy.io_ involves building a signature listing the types available to work with, composing them to form diagrams, and then deforming these diagrams with homotopies. The fundamental idea is that an $n$-diagram corresponds to an $n$-morphism in the free associative $n$-category over the signature.

**Signatures.** In dimension 0, generating cells can be constructed by clicking the "+" symbol next to the heading "0-cells". In dimension $n+1$, a cell is defined by two pieces of data: its source $n$-diagram, and its target $n$-diagram. We add such a cell by building an $n$-diagram $S$, clicking 'Source' to choose it as the source of the new generator, then building a $n$-diagram $T$, and clicking 'Target' to choose it as the target of the new generator. The new cell will then appear in the signature. A pair of $n$-diagrams $S,T$ can serve as the source and target of a new generator just when they satisfy the globularity condition, saying that their sources are the same, and their targets are the same.

**Diagrams.** An $n$-diagram is a composite of $k$-cells from the signature, for $k \leq n$. To build a diagram, we first click on the thumbnail of a cell in the signature, which will display it in the central window. We can _compose_ it with other cells by clicking near the boundary. These compositions are always [[whiskering|whiskerings]], with a $k$-cell generator attached to an $n$-diagram along a common $\text{min}(k,n)$-dimensional boundary. Non-whiskered composites can be obtained by first building whiskered composites, and then rewriting the interior of the diagram using homotopies. We can _rewrite_ parts of an $n$-diagram by clicking in the diagram interior, which will apply an $n+1$-cell, or clicking and dragging, which will apply a homotopy.

**Homotopies.** Once a diagram has been constructed, we can modify it by applying a _homotopy_, which we trigger by clicking and dragging on the diagram. If this is performed on the boundary of the diagram, it will attach the homotopy to the boundary; if this is performed in the interior of the diagram, it will rewrite it.

**Renderer (experimental).** Diagrams can be rendered in both 2d and 3d. At the moment, only the 2d renderer is interactive, allowing diagrams to be manipulated directly using the mouse; the 3d renderer is for display only, and is currently quite buggy, only working reliably for small diagrams with a connected geometry. It is hoped that _homotopy.io_ will soon allow rendering and smooth animation of arbitrary 3d composites.

**Projection.** Diagrams can be high-dimensional, and hence hard to display on a computer screen. _Projection_ causes lower dimensions of the diagram to be ignored; for example, given a 5-dimensional diagram with projection set to 3, you will see a single $(5-3=2)$-dimensional diagram, in which the vertices represent 5-cells, the wires represent 4-cells, and the regions represent 3-cells. Note that projection loses information in general.

**Slice.** For an $n$-diagram with $n \gt 2$, projection will always lose some information about the diagram. _Slicing_ allows an $(n+1)$-dimensional diagram to be viewed as a movie of $n$-dimensional diagrams, in a way which preserves all information about the diagram. For example, for a 9-dimensional diagram with projection set to 3, there are still 6 remaining dimensions to display. With the renderer set to 2d, this leaves 4 dimensions to be handled with slice controls, which allows us to investigate the 6d displayed geometry as a movie of movies of movies of movies of 2d diagrams.

**Undo and redo.** Use the browser back and forward buttons to undo and redo your interactions with the proof assistant.

**Server.** The proof assistant has a backend that lets you log in and save proofs to the server. You can also make proofs shareable, and obtain a short URL suitable for sharing with others, or embedding in a research paper. In the future you will also be able to publish proofs, obtaining a permanent date-stamped URL.

## Homotopies

A composite diagram can be deformed by clicking and dragging in its interior. If the indicated deformation would be geometrically invalid, an error message is displayed in the lower-right corner, and the diagram does not change. Otherwise, a new diagram is displayed, which is homotopic to the original diagram.

Conjecturally, homotopies can be used to construct arbitrary progressive deformations of the $n$-diagram as a subcomplex of $\mathbb{R}^n$. Informally, homotopies can move parts of the diagram around, but they can't change the local neighbourhoods of the generating cells inhabiting the diagram.

Of course, the proof assistant manipulates finite combinatorial structures, and we use the word 'homotopic' with reference to their presumed geometrical realization. We do not make this precise here, nor has it yet been made precise in the literature; for this reason, our use of the term 'homotopy' is not yet well connected to the standard notion.

Every homotopy is either a _contraction_, which make the geometry locally more singular, or an _expansion_, which make it locally more generic. If we view our original diagram $D$ as a zigzag (see [Combinatorial foundation](#CombinatorialFoundation), a contraction is encoded as a zigzag map $D \to D'$, and an expansion as a zigzag map $D' \to D$, for some other diagram $D'$.

### Base cases ###

A homotopy is triggered by choosing an $n$-diagram for $n \geq 2$, choosing some vertex that it contains, and dragging that vertex _vertically_; that is, either up or down.

If the chosen vertex is in _generic position_, meaning that it is the unique vertex at its height, a contraction will be triggered; otherwise, an expansion will be triggered. In the base case, a contraction decreases the height of a diagram by 1, and an expansion increases it by 1. Expansions always succeed, but contractions will only succeed if the diagram is actually locally contractible, in the sense of homotopy theory. Mathematically, contractions are performed by taking the colimit of part of the zigzag diagram (see the [Combinatorial Foundation](#CombinatorialFoundation) section.)

To illustrate this, we suppose we have the following diagram, where for convenience we label the vertices with numbers ([live workspace](https://homotopy.io/?id=QMw1sG1p4yCpsgL3Xh7K)):

Vertices 1 and 4 are in generic position, so dragging them will trigger contractions, while dragging vertices 2 and 3, which are in singular position, would trigger expansions. We illustrate the resulting diagrams as follows:

Expansion	Expansion	Contraction	Contraction

2 up or 3 down
2 down or 3 up
1 down
4 up

Dragging vertex 1 up, or vertex 4 down, has no effect, as vertices cannot be dragged off the top or bottom edge of a diagram.

The contractions illustrated above are unique. However, in some cases, contractions are non-unique, as illustrated here ([live workspace](https://homotopy.io/?id=9DDbCH0V7xdKULkqroIz)):

→

←

Starting with the central diagram, we produce the leftmost diagram by dragging the upper vertex south-east or the lower vertex north-west, and produce the rightmost diagram by dragging the upper vertex south-west or the lower vertex north-east.

Contraction is not always possible. The following images illustrate non-contractible scenarios:

(live workspace)

(live workspace)

In both of these cases, attempting to perform the contraction will give an error message. In the first case, the contraction fails because there is no canonical ordering on the rear wires; this is analogous to the nonunique contraction case given above, but here, because of the projection, we cannot give extra data to break the symmetry. In the second case, the contraction is successful at the level of the untyped diagram, but it is rejected by the type checker, since it merges two distinct generating cells, changing their neighbourhoods in a way which is not allowed by the type system.

If there are no slice controls displayed, then the homotopy is being performed at codimension 0 (that is, at the top level), and the diagram will be rewritten according to the homotopy. If every slice control shows S or T, then we are viewing a boundary of the main diagram, and the homotopy we are building will be attached to that boundary. Otherwise, we are in the interior of the diagram, working in some positive codimension, and the homotopy must now be propagated to lower codimension, using the recursive scheme detailed in the next section.

### Recursive cases ###

When we apply a homotopy to an interior slice of the main diagram, it is propagated up to the top level. A homotopy can be either a contraction or an expansion, and we can be in a regular or singular subslice; this gives 4 cases, which we consider separately.

## Combinatorial foundation
 {#CombinatorialFoundation}

The combinatorial data stored by the proof assistant can be understood in terms of zigzags.

+-- {: .num_defn}
###### Definition
In a category $C$, a _zigzag_ is a finite diagram of the following sort:

We write $Z_r = \{ r_0, \ldots, r_n \}$ for the set of _regular slices_, and $Z_s = \{ s_0, \ldots, s_{n-1} \}$ for the set of _singular slices_.
=--

+-- {: .num_defn}
###### Definition
Given zigzags $Z,Z'$ in a category $C$, a _zigzag map_ $f:Z \to Z'$ comprises a monotone function $f_\text{s} : Z_\text{s} \to Z'_\text{s}$, and for each $i \in Z_\text{s}$ a morphism $f_i : s_i \to s' _{f_\text{s}(i)}$, such that the diagram built as follows is commutative:

* Take the disjoint union of $Z$ and $Z'$ as diagrams in $C$.
* Add the arrows $f_i$ between singular objects.
* In between these arrows, add equalities between regular objects.

The idea is illustrated in the following picture:

=--

+-- {: .num_defn}
###### Definition
Given a category $C$, its _zigzag category_ $Z_C$ has zigzags as objects, and zigzag maps as morphisms.
=--

+-- {: .num_defn}
###### Definition
An _untyped $n$-diagram_ is an object of the category obtained by starting with the terminal category $1$, and applying the zigzag category construction $n$ times.
=--

For a signature $\Sigma$, we write $|\Sigma|$ for its set of generating types, which is equipped with a dimension function $\text{dim}:|\Sigma| \to \mathbb{N}$. We write $\mathcal{P}_\Sigma$ for the poset with objects given by elements of $|\Sigma|$, and a nonidentity morphism $x \to y$ just when $\text{dim}(x) \lt \text{dim}(y)$.

+-- {: .num_defn}
###### Definition
A _$|\Sigma|$-typed $n$-diagram_ is an object of the category obtained by starting with the category $\mathcal{P}_\Sigma$, and applying the zigzag category construction $n$ times.
=--

## Type checking and normalization

Given a $|\Sigma|$-typed $n$-diagram $D$, we can then ask if it is _well-typed_ with respect to $\Sigma$. This involves identifying the neighbourhoods of every point, and checking that they agree with the standard type neighbourhoods defined in $\Sigma$. This procedure is explained in Christoph Dorn's PhD thesis referenced below.

## Related entries

* [[associative n-category]]

* [[Globular]]

[[!include proof assistants and formalization projects -- list]]

## References

Background on [[associative n-categories]]:

* [[Christoph Dorn]], _Associative $n$-categories_, talk at _[103rd Peripatetic Seminar on Sheaves and Logic](https://www.math.muni.cz/~loregianf/PSSL103/PSSL103.php)_ [[pdf](https://www.math.muni.cz/~loregianf/PSSL103/slides/Dorn.pdf)]

* [[Christoph Dorn]], _Associative $n$-categories_, PhD thesis [[arXiv:1812.10586](https://arxiv.org/abs/1812.10586)]

In relation to `homotopy.io`:

* [[Jamie Vicary]], _Strictly associative and unital higher categories_, talk at the _[2018 Midland Graduate School Christmas Seminar](http://www.cs.bham.ac.uk/~axj/2018-mgs-xmas.html)_ ([pdf](http://www.cs.bham.ac.uk/~axj/files/mgs-xmas-2018-Jamie.pdf)).

* [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]

* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

category: software


homset > history

< [[homset]]
[[!redirects homset -- history]]


Honda formal group

A [[formal group law]].

## References

* Taira Honda, _On the theory of commutative formal groups_, J. Math. Soc. Japan Volume 22, Number 2 (1970), 213-246. ([Euclid](http://projecteuclid.org/euclid.jmsj/1259942752))

[[!redirects Honda formal groups]]

[[!redirects Honda formal group law]]
[[!redirects Honda formal group laws]]


Hong Liu

* [webpage](https://web.mit.edu/physics/people/faculty/liu_hong.html)

## Selected writings

On the [[AdS/CFT correspondence]] and [[multi-trace operators]]:

* {#Liu98} [[Hong Liu]], _Scattering in Anti-de Sitter Space and Operator Product Expansion_, Phys. Rev. D 60, 106005 (1999) ([arXiv:hep-th/9811152](https://arxiv.org/abs/hep-th/9811152))

On the [[AdS-CFT in condensed matter physics]]:

* [[Hong Liu]], [[Julian Sonner]], _Quantum many-body physics from a gravitational lens_, Nature Rev. Phys. **2** (2020)  615-633 [[arXiv:2004.06159](https://arxiv.org/abs/2004.06159), [doi:10.1038/s42254-020-0225-1](https://doi.org/10.1038/s42254-020-0225-1)]

category: people


Hongde Hu

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=67911)

## Selected writings

On [[free coproduct completions]]:

* {#HuTholen95} [[Hongde Hu]], [[Walter Tholen]], *Limits in  free  coproduct completions*, Journal of  Pure  and  Applied  Algebra 105 (1995) 277-291 (doi:10.1016/0022-4049(94)00153-7, [pdf](https://core.ac.uk/download/pdf/82604415.pdf))

On a generalization of [[coproducts]] to [[colimits]] over certain [[diagrams]] of the shape of [[skeletal groupoids]]:

* [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 [[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)]

category: people


hook length and content formulas -- table

| [[hook length formula]] | [[hook-content formula]] |
|-------------------------|--------------------------|
| [[number]] of [[standard Young tableaux]] | [[number]] of [[semistandard Young tableaux]] |
| [[dimension]] of [[irreps]] [[Specht module|of]] [[symmetric group|Sym(n)]] | [[dimension]] of [[irreps]] of [[SL(n)]] |