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<!DOCTYPE html>
<html lang="en">
<head>
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<meta http-equiv="Content-type" content="text/html;charset=UTF-8">
<title>Nicolai Kraus</title>
</head>
<body>
<header>
<H1>Nicolai Kraus</H1>
<p>
Professor of Theoretical Computer Science<br>
Royal Society University Research Fellow<br>
University of Nottingham
</p>
</header>
<div class="profile">
<div class="picture">
<img width="200" title= "Nicolai Kraus" alt="Nicolai Kraus" src="pictures/nicolaikraus.jpg">
</div>
<div class="info">
<div>
<p>
Welcome to by website! I am a Royal Society University Research Fellow and Professor of Theoretical Computer Science at the <a href="http://www.nottingham.ac.uk/">University of Nottingham</a>, in the <a href="https://nottingham.ac.uk/research/groups/fp-lab/index.aspx">Functional Programming Lab</a>.
I was previously a member of the <a href="http://www.cs.bham.ac.uk/research/groupings/theory/">Birmingham Theory Group</a> and at <a href="http://www.elte.hu/en">Eötvös Loránd University</a>.
</p>
<p>
I am working in the area of <a href="https://en.wikipedia.org/wiki/Intuitionistic_type_theory">dependent type theory</a> (favourite proof assistant: <a href="https://wiki.portal.chalmers.se/agda/pmwiki.php">Agda</a>), for which I will from 2025 lead an <a href="https://www.nottingham.ac.uk/news/type-theories-erc-grant">ERC (European Research Council) project</a>.
My main focus is <a href="https://homotopytypetheory.org/book/">homotopy type theory</a> and <a href="https://ncatlab.org/nlab/show/higher+category+theory">(higher) categories</a>, but I like to think about topics and questions in constructive or non-constructive mathematics in general.
</p>
<p>
Feel free to contact me:
<EM>[email protected]</EM>.
</p>
</div>
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<img src="https://tcs4f.org/themes/tcs4f/img/logo_TCS4F_initiative.svg"
alt="Theoretical Computer Scientists for Future" width="150">
</a>
</div>
-->
</div>
</div>
<div class="quick-links">
<p>
<strong>Quick links:</strong>
[<a href="#people">People</a>]
[<a href="#papers">Papers</a>]
[<a href="#workshops">Workshop Contributions</a>]
[<a href="#talks">Talks</a>]
[<a href="#events">Events</a>]
[<a href="#teaching">Teaching</a>]
[<a href="#notes">Notes</a>]
[<a href="#funding">Funding Acknowledgement</a>]
<br>
and a list of bibtex entries for my papers [<a href="bibliography/nicolaikraus_bib.html">html</a>].
</p>
</div>
<!--
<p>
<a href="logTopoi.txt">Log for the topos theory seminar</a>,
<a href="log.txt">earlier</a>
</p>
-->
<p style="clear: both;">
<h3 id="people">People</h3>
I am currently supervising and working with:
<ol class="nonumber">
<li>
<a href="https://tdejong.com/">Tom de Jong</a>, who started as a postdoc with me in October 2022.
</li>
<li>
<a href="https://www.stiephenpradal.com">Stiéphen Pradal</a>, who started his PhD with me in October 2022.
</li>
<li>
<a href="https://joshchen.io/">Joshua Chen</a>, working on his PhD with me since October 2020.
</li>
</ol>
I am also the second supervisor of:
<ol class="nonumber">
<li>
<a href="https://github.com/Schippmunk">Johannes Schipp von Branitz</a>, working on his PhD with <a href="https://ulrikbuchholtz.dk/">Ulrik Buchholtz
</a> since October 2022.
</li>
<li>
<a href="https://stefaniatadama.com/">Stefania Damato</a>, who started her PhD with <a href="http://www.cs.nott.ac.uk/~psztxa/">Thorsten Altenkirch</a> in October 2021.
</li>
<!--
<li>
<a href="https://github.com/brandonhewer">Brandon Hewer</a>, supervised by <a href="http://www.cs.nott.ac.uk/~pszgmh/">Graham Hutton</a>, who started in October 2019.
</li>
-->
</ol>
<h3 id="papers">Papers</h3>
<ol reversed>
<li>
<em>On symmetries of spheres in univalent foundations</em>
[<a href="https://doi.org/10.1145/3661814.3662115">ACM</a>]
[<a href="https://arxiv.org/abs/2401.15037">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R25">bibtex</a>]
<ul>
<li>
LICS'24.
</li>
<li>
With Pierre Cagne, Ulrik Buchholtz, and Marc Bezem.
</li>
</ul>
</li>
<li>
<em>Set-Theoretic and Type-Theoretic Ordinals Coincide</em>
[<a href="https://doi.org/10.1109/LICS56636.2023.10175762">IEEE</a>]
[<a href="https://arxiv.org/abs/2301.10696">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R24">bibtex</a>]
[<a href="https://tdejong.com/agda-html/st-tt-ordinals/index.html">Agda (html)</a>]
<ul>
<li>
LICS'23.
</li>
<li>
With Tom de Jong, Fredrik Nordvall Forsberg, and Chuangjie Xu.</li>
<li>
Note on the Agda source code: Part 1 is included in <a href="https://github.com/martinescardo/TypeTopology">Martín Escardó's TypeTopology library</a>, and Part 2 is included in our <a href="https://bitbucket.org/nicolaikraus/constructive-ordinals-in-hott/src/master/MEWO/index.agda">constructive ordinals in HoTT project</a>.
</li>
</ul>
</li>
<li>
<em>Type-Theoretic Approaches to Ordinals</em>
[<a href="https://doi.org/10.1016/j.tcs.2023.113843">TCS</a>]
[<a href="https://arxiv.org/abs/2208.03844">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R23">bibtex</a>]
[<a href="https://bitbucket.org/nicolaikraus/constructive-ordinals-in-hott/src">Agda (source)</a>]
[<a href="https://cj-xu.github.io/agda/type-theoretic-approaches-to-ordinals/">Agda (html)</a>]
<ul>
<li>
<a href="https://www.sciencedirect.com/science/article/abs/pii/S0304397523001561">Theoretical Computer Science.</a>
</li>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
</li>
<li>
This paper improves and expands on the constructions of "Connecting Constructive Notions of Ordinals in Homotopy Type Theory".
</li>
<li>
Caveat: In the published version (TCS), Theorem 73 uses an unnecessary assumption and Theorem 74 is incorrect. This is corrected in the arXiv version.
</li>
</ul>
</li>
<li>
<em>A Rewriting Coherence Theorem with Applications in Homotopy Type Theory
</em>
[<a href="https://arxiv.org/abs/2107.01594">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R22">bibtex</a>]
[<a href="https://doi.org/10.1017/S0960129523000026">journal</a>]
<ul>
<li>
<a href="https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/rewriting-coherence-theorem-with-applications-in-homotopy-type-theory/8AFEEEFAC1B7EEE61A187CFB0249AA1D">Mathematical Structures in Computer Science.</a>
</li>
<li>
With Jakob von Raumer.
</li>
<li>
Note: This paper presents some of the ideas of "Coherence via Wellfoundedness" (see below) in the language of higher-dimensional rewriting.
</li>
</ul>
</li>
<li>
<em>Connecting Constructive Notions of Ordinals in Homotopy Type Theory</em>
[<a href="https://drops.dagstuhl.de/opus/volltexte/2021/14510/">LIPIcs</a>]
[<a href="https://arxiv.org/abs/2104.02549">arxiv (includes proofs)</a>]
[<a href="bibliography/nicolaikraus_bib.html#R21">bibtex</a>]
[<a href="https://bitbucket.org/nicolaikraus/constructive-ordinals-in-hott/src">Agda (source)</a>]
[<a href="docs/html-ordinals/INDEX.html">Agda (html)</a>]
<ul>
<li>
<a href="https://compose.ioc.ee/mfcs/">MFCS'21</a>.
</li>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
Note: This paper is superseded by "Type-Theoretic Approaches to Ordinals".
</li>
</ul>
</li>
<li>
<em>Internal ∞-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT</em>
[<a href="https://ieeexplore.ieee.org/document/9470667">IEEE (published)</a>]
[<a href="docs/2021078147.pdf">lics version (pdf preprint)</a>]
[<a href="https://arxiv.org/abs/2009.01883">arxiv (longer version)</a>]
[<a href="bibliography/nicolaikraus_bib.html#R20">bibtex</a>]
[<a href="docs/html-idempotentequivalences/Identities.html">Agda (html)</a>]
[<a href="https://github.com/nicolaikraus/idempotentEquivalences">Agda (source)</a>]
<ul>
<li>
<a href="http://easyconferences.eu/lics2021/">LICS'21</a>.
</li>
<li>
The Agda formalisation linked above contains a mechanisation of <em>idempotent equivalences</em>.
</li>
</ul>
</li>
<li>
<em>Coherence via Wellfoundedness: Taming Set-Quotients in Homotopy Type Theory</em>
[<a href="https://dl.acm.org/doi/10.1145/3373718.3394800">acm</a>]
[<a href="https://arxiv.org/abs/2001.07655">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R19">bibtex</a>]
<ul>
<li>
<a href="https://lics.siglog.org/lics20/">LICS'20</a>.
</li>
<li>
With Jakob von Raumer. <a href="https://gitlab.com/fplab/freealgstr">Lean formalisation on GitLab</a>.
</li>
<li>
Note: A possibly better presentation of the ideas in this paper can be found in "A Rewriting Coherence Theorem with Applications in Homotopy Type Theory" (see above).
</li>
</ul>
</li>
<li>
<em>Two-Level Type Theory and Applications</em>
[<a href="https://doi.org/10.1017/S0960129523000130">journal</a>]
[<a href="https://arxiv.org/abs/1705.03307">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R18">bibtex</a>]
<ul>
<li>
<a href="https://doi.org/10.1017/S0960129523000130">Mathematical Structures in Computer Science.</a>
</li>
<li>
With Danil Annenkov, Paolo Capriotti, and Christian Sattler. <a href="https://github.com/annenkov/two-level">Lean formalisation on GitHub</a>.
</li>
</ul>
</li>
<li>
<em>Shallow Embedding of Type Theory is Morally Correct</em>
[<a href="https://doi.org/10.1007/978-3-030-33636-3_12">Springer</a>]
[<a href="https://arxiv.org/abs/1907.07562">arxiv</a>]
[<a href="https://bitbucket.org/akaposi/shallow/src/master/Agda">Agda (bitbucket)</a>]
[<a href="bibliography/nicolaikraus_bib.html#R17">bibtex</a>]
<ul>
<li>
<a href="https://www.cs.nott.ac.uk/~pszgmh/mpc19.html">MPC'19</a>.
</li>
<li>
With Ambrus Kaposi and András Kovács.
</li>
</ul>
</li>
<li>
<em>From Cubes to Twisted Cubes via Graph Morphisms in Type Theory</em>
[<a href="https://doi.org/10.4230/LIPIcs.TYPES.2019.5">LIPIcs</a>]
[<a href="https://arxiv.org/abs/1902.10820">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R16">bibtex</a>]
<ul>
<li>
With Gun Pinyo.
</li>
</ul>
</li>
<li>
<em>Path Spaces of Higher Inductive Types in Homotopy Type Theory</em>
[<a href="https://doi.org/10.1109/LICS.2019.8785661">IEEE</a>]
[<a href="https://arxiv.org/abs/1901.06022">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R15">bibtex</a>]
<ul>
<li>
<a href="http://lics.siglog.org/lics19/">LICS'19</a>.
</li>
<li>
With Jakob von Raumer.
</li>
</ul>
</li>
<li>
<em>Free Higher Groups in Homotopy Type Theory</em>
[<a href="https://dl.acm.org/doi/abs/10.1145/3209108.3209183">ACM</a>]
[<a href="https://arxiv.org/abs/1805.02069">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R14">bibtex</a>]
<ul>
<li>
<a href="https://lics.siglog.org/lics18">LICS'18</a>.
</li>
<li>
With Thorsten Altenkirch.
</li>
</ul>
</li>
<li>
<em>Quotient inductive-inductive types</em>
[<a href="https://link.springer.com/chapter/10.1007/978-3-319-89366-2_16">Springer</a>]
[<a href="https://arxiv.org/abs/1612.02346">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R13">bibtex</a>]
<ul>
<li>
<a href="https://www.etaps.org/index.php/2018/fossacs">FoSSaCS'18</a>.
</li>
<li>
With Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, and Fredrik Nordvall Forsberg.
</li>
</ul>
</li>
<li>
<em>Univalent Higher Categories via Complete Semi-Segal Types</em>
[<a href="https://dl.acm.org/citation.cfm?id=3158132">ACM</a>]
[<a href="https://arxiv.org/abs/1707.03693">arxiv</a> (full version)]
[<a href="bibliography/nicolaikraus_bib.html#R12">bibtex</a>]
<ul>
<li>
<a href="https://popl18.sigplan.org/home">POPL'18</a>.
</li>
<li>
With Paolo Capriotti. <a href="https://gitlab.com/pcapriotti/agda-segal">Agda formalisation on GitLab</a> and <a href="docs/html-segaltypes/README.html">as html</a>.
</li>
</ul>
</li>
<li>
<em>Space-Valued Diagrams, Type-Theoretically (Extended Abstract)</em>
[<a href="https://arxiv.org/abs/1704.04543">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R11">bibtex</a>]
<ul>
<li>
With Christian Sattler.
</li>
</ul>
</li>
<li>
<em>Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type</em>
[<a href="http://link.springer.com/chapter/10.1007/978-3-662-54458-7_31">Springer</a>]
[<a href="https://arxiv.org/abs/1610.09254">arxiv</a>]
[<a href="https://dl.acm.org/doi/10.1007/978-3-662-54458-7_31">ACM</a>]
[<a href="bibliography/nicolaikraus_bib.html#R10">bibtex</a>]
<ul>
<li>
With Thorsten Altenkirch and Nils Anders Danielsson. <a href="http://www.etaps.org/index.php/2017/fossacs">FoSSaCS'17</a>.
</li>
<li>
The final publication is available at Springer via <a href="http://dx.doi.org/10.1007/978-3-662-54458-7_31">http://dx.doi.org/10.1007/978-3-662-54458-7_31</a>.
</li>
</ul>
</li>
<li>
<em>Extending Homotopy Type Theory With Strict Equality</em>
[<a href="http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=6561">Dagstuhl</a>]
[<a href="http://arxiv.org/abs/1604.03799">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R9">bibtex</a>]
<ul>
<li>
With Thorsten Altenkirch and Paolo Capriotti. <a href="http://csl16.lif.univ-mrs.fr/">CSL'16</a>.
</li>
</ul>
</li>
<li>
<em>Constructions with Non-Recursive Higher Inductive Types</em>
[<a href="https://dl.acm.org/doi/abs/10.1145/2933575.2933586">ACM</a>]
[<a href="docs/pseudotruncations.pdf">pdf</a>]
[<a href="bibliography/nicolaikraus_bib.html#R8">bibtex</a>]
<ul>
<li>
<a href="http://lics.rwth-aachen.de/lics16/">LICS'16</a>.
</li>
<li>
You can find an <a href="https://github.com/nicolaikraus/HoTT-Agda/tree/master/nicolai/pseudotruncations">Agda formalisation on GitHub</a>, alternatively <a href="docs/html-pseudotruncs/nicolai.pseudotruncations.NONRECURSIVE-INDEX.html">as html here</a>.
</li>
</ul>
</li>
<li>
<em>Functions out of Higher Truncations</em>
[<a href="http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=5425">dagstuhl</a>]
[<a href="http://arxiv.org/abs/1507.01150">arXiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R7">bibtex</a>]
<ul>
<li>
With Paolo Capriotti and Andrea Vezzosi. <a href="http://logic.las.tu-berlin.de/csl2015/index.html">CSL'15</a>.
</li>
</ul>
</li>
<li>
<em>Truncation Levels in Homotopy Type Theory</em>
[<a href="docs/thesis_nicolai.pdf">pdf</a>]
[<a href="docs/thesisagda_nicolai/INDEX.html">formalisation (html)</a>]
[<a href="docs/agdasource_thesis.zip">formalisation (source, zip)</a>]
[<a href="bibliography/nicolaikraus_bib.html#R6">bibtex</a>]
<ul>
<li>
PhD thesis, University of Nottingham 2015, won the Ackermann award (<a href="http://drops.dagstuhl.de/opus/volltexte/2016/6541/">report</a> by Thierry Coquand and Anuj Dawar).
</li>
<li>
Advisor: Thorsten Altenkirch. Examiners: Julie Greemsmith (internal) and Steve Awodey (external).
</li>
</ul>
</li>
<li>
<em>The General Universal Property of the Propositional Truncation</em>
[<a href="http://drops.dagstuhl.de/opus/volltexte/2015/5494/">dagstuhl</a>]
[newest version: <a href="http://arxiv.org/abs/1411.2682">arXiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R5">bibtex</a>]
<ul>
<li>
Post-proceedings of <a href="http://www.pps.univ-paris-diderot.fr/types2014/PostProceedings">TYPES'14</a>.
</li>
</ul>
</li>
<li>
<em>Notions of Anonymous Existence in Martin-Löf Type Theory</em>
[<a href="https://doi.org/10.23638/LMCS-13(1:15)2017">LMCS</a>]
[<a href="https://arxiv.org/abs/1610.03346">arxiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R4">bibtex</a>]
<ul>
<li>
With Martín Escardó, Thierry Coquand and Thorsten Altenkirch. <a href="https://lmcs.episciences.org/3217">Logical Methods in Computer Science</a>, our contribution to the special issue of TLCA'13.
</li>
<li>
The <a href="https://github.com/nicolaikraus/HoTT-Agda/tree/master/nicolai/anonymousExistence">formalisation is on GitHub</a>, but it is also contained in my <a href="docs/thesisagda_nicolai/INDEX.html">thesis formalisation as html</a>.
</li>
</ul>
</li>
<li>
<em>Higher Homotopies in a Hierarchy of Univalent Universes</em>
[<a href="http://dl.acm.org/citation.cfm?id=2729979">ACM</a>]
[<a href="http://arxiv.org/abs/1311.4002">arXiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R3">bibtex</a>]
<ul>
<li>
With Christian Sattler. <a href="http://tocl.acm.org/index.cfm">Transactions on Computational Logic</a>.
</li>
</ul>
</li>
<li>
<em>Generalizations of Hedberg’s Theorem</em>
[<a href="https://doi.org/10.1007/978-3-642-38946-7_14">Springer</a>]
[<a href="docs/hedberg.pdf">pdf</a>]
[<a href="http://www.cs.bham.ac.uk/~mhe/GeneralizedHedberg/html/GeneralizedHedberg.html">html on ME's website</a>]
[<a href="http://homotopytypetheory.org/2012/11/27/on-h-propositional-reflection-and-hedbergs-theorem/">blog post</a>]
[<a href="bibliography/nicolaikraus_bib.html#R2">bibtex</a>]
<ul>
<li>
With Martín Escardó, Thierry Coquand and Thorsten Altenkirch. <a href="http://www.kurims.kyoto-u.ac.jp/tlca2013/">TLCA'13</a>.
</li>
<li>
The final publication is available at <a href="http://link.springer.com/">link.springer.com</a>.
</li>
</ul>
</li>
<li>
<em>A Lambda Term Representation Inspired by Linear Ordered Logic</em> [<a href="http://arxiv.org/abs/1111.0085v1">arXiv</a>]
[<a href="bibliography/nicolaikraus_bib.html#R1">bibtex</a>]
<ul>
<li>
With Andreas Abel. <a href="http://lfmtp11.cs.umn.edu/Site/Welcome.html">LFMTP 2011</a>.
</li>
</ul>
</li>
</ol>
<h3 id="workshops">Peer-Reviewed Workshop Contributions</h3>
<ol class="nonumber">
<li>
The ordinals in set theory and type theory are the same [<a href="docs/types2023_ordinals_abstract.pdf">pdf</a>] - at <a href="https://types2023.webs.upv.es/">TYPES 2023</a>.
<ul>
<li>
With Tom de Jong, Fredrik Nordvall Forsberg, and Chuangjie Xu.
</li>
</ul>
</li>
<li>
Categories as Semicategories with Identities [<a href="docs/short_abstract_semicats_with_identities.pdf">pdf</a>] - at <a href="https://types2023.webs.upv.es/">TYPES 2023</a>.
<ul>
<li>
With Joshua Chen, Tom de Jong, and Stiéphen Pradal.
</li>
</ul>
</li>
<li>
Relating ordinals in set theory to ordinals in type theory [<a href="docs/HoTT-UF-2023.pdf">pdf</a>] - at <a href="https://hott-uf.github.io/2023/">HoTT/UF 2023</a>
and
<a href="https://hott.github.io/HoTT-2023/">Homotopy Type Theory 2023</a>.
<ul>
<li>
With Tom de Jong, Fredrik Nordvall Forsberg, and Chuangjie Xu.
</li>
</ul>
</li>
<li>
Constructivity Aspects of Brouwer Tree Ordinals [<a href="docs/brouwer_ordinals_abstract.pdf">pdf</a>] - at <a href="https://events.math.unipd.it/ccc2022/">Continuity, Computability, Constructivity</a>.
<ul>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
</li>
</ul>
</li>
<li>
Decidability and Semidecidability via Ordinals [<a href="docs/decidabilityViaOrdinals.pdf">pdf</a>] - at <a href="https://types22.inria.fr/">TYPES 2022</a>.
<ul>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
</li>
</ul>
</li>
<li>
Wellfounded and Extensional Ordinals in Homotopy Type Theory [<a href="http://dcs.elte.hu/wp-content/uploads/2022/01/DCS_proceedings.pdf">conference proceedings pdf</a>] - at <a href="http://dcs.elte.hu/">DCS (Developments in Computer Science)</a>.
<ul>
<li>
Not peer-reviewed (invited contribution).
</li>
</ul>
</li>
<li>
A Certified Library of Ordinal Arithmetic [<a href="docs/CCC_2021_paper_10.pdf">pdf</a>] - at <a href="https://www.cs.bham.ac.uk/~axj/ccc2021/index.html">CCC 2021</a>.
<ul>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
</li>
</ul>
</li>
<li>
Syntax for two-level type theory [<a href="docs/HoTTUF_2021_paper_5.pdf">pdf</a>] - at HoTT/UF 2021.
<ul>
<li>
With Roberta Bonacina and Benedikt Ahrens.
</li>
</ul>
</li>
<li>
Semisimplicial Types in Internal Categories with Families [<a href="docs/internalsemisimp.pdf">pdf</a>] - at TYPES 2021.
<ul>
<li>
With Joshua Chen.
</li>
</ul>
</li>
<li>
Constructive Notions of Ordinals in Homotopy Type Theory [<a href="docs/connectingOrd.pdf">pdf</a>] - at TYPES 2021.
<ul>
<li>
With Fredrik Nordvall Forsberg and Chuangjie Xu.
</li>
</ul>
</li>
<li>
An Induction Principle for Cycles [<a href="docs/wellfounded_types.pdf">pdf</a>] - at TYPES 2020.
<ul>
<li>
With Jakob von Raumer.
</li>
</ul>
</li>
<li>
On Symmetries of Spheres in HoTT-UF [<a href="docs/symm_types.pdf">pdf</a>] - at TYPES 2020.
<ul>
<li>
With Pierre Cagne and Marc Bezem.
</li>
</ul>
</li>
<li>
Shallow Embedding of Type Theory is Morally Correct [<a href="docs/shallow_types.pdf">pdf</a>] - at TYPES 2020.
<ul>
<li>
With Ambrus Kaposi and András Kovács.
</li>
</ul>
</li>
<li>
Twisted Cubes via Graph Morphisms [<a href="docs/twcubes_types.pdf">pdf</a>, <a href="https://cas.oslo.no/types2019/programme/book-of-abstracts/">book of abstracts</a>] - at TYPES 2019.
<ul>
<li>
With Gun Pinyo.
</li>
</ul>
</li>
<li>
On the Role of Semisimplicial Types [<a href="docs/on_semisimplicial_types.pdf">pdf</a>, <a href="https://types2018.projj.eu/book-of-abstracts/">book of abstracts</a>] - at TYPES 2018.
<ul>
<li>
Some reasons why I believe that the open problem of defining semisimplicial types is important.
</li>
</ul>
</li>
<li>
Specifying Quotient Inductive-Inductive Types [<a href="https://types2018.projj.eu/book-of-abstracts/">book of abstracts</a>] - at TYPES 2018.
<ul>
<li>
With Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, and Fredrik Nordvall Forsberg.
</li>
</ul>
</li>
<li>
Formalisations Using Two-Level Type Theory [<a href="docs/oxford_twolevel_abstract.pdf">pdf</a>] - at HoTT/UF 2017.
<ul>
<li>
With Danil Annenkov and Paolo Capriotti.
</li>
</ul>
</li>
<li>
Type Theory with Weak J [<a href="docs/conservativityAbstract.pdf">pdf</a>] - at TYPES 2017.
<ul>
<li>
With Thorsten Altenkirch, Paolo Capriotti, Thierry Coquand, Nils Anders Danielsson, and Simon Huber.
</li>
</ul>
</li>
<li>
Non-Recursive Truncations [<a href="docs/nrt_types2016.pdf">pdf</a>] - at TYPES 2016.
<ul>
<li>
I mainly compare different constructions of the propositional truncation.
</li>
</ul>
</li>
<li>
Higher Categories in Homotopy Type Theory [<a href="docs/highercats_types16.pdf">pdf</a>] - at TYPES 2016.
<ul>
<li>
With Thorsten Altenkirch and Paolo Capriotti. We describe how we want to use a suitable notion of infinity-semicategory in order to organise towers of coherences.
</li>
</ul>
</li>
<li>
Infinite Structures in Type Theory: Problems and Approaches [<a href="docs/higherstruc_types.pdf">pdf</a>]
<!-- [<a href="http://cs.ioc.ee/types15/abstracts-book/contrib7.pdf">pdf</a>] -->
- at TYPES 2015.
<ul>
<li>
With Thorsten Altenkirch and Paolo Capriotti. We discuss constructions that are desirable but cannot be performed in HoTT.
</li>
</ul>
</li>
<li>
Eliminating out of Truncations [<a href="docs/warsaw_truncations_abstract.pdf">pdf</a>] - at <a href="http://hott-uf.gforge.inria.fr/2015/">HoTT/UF in Warsaw</a>, April 2015.
<ul>
<li>
The "Čech nerve formula" for the (-1)-truncation and related results for higher truncations.
</li>
</ul>
</li>
<li>
Eliminating Higher Truncations via Constancy [<a href="docs/elim_hottuf.pdf">pdf</a>]
<!-- [<a href="http://www.pps.univ-paris-diderot.fr/types2014/abstract-9.pdf">from TYPES website</a>] -->
- at TYPES 2014.
<ul>
<li>
With Paolo Capriotti. We generalise the universal property of n-truncations.
</li>
</ul>
</li>
<li>
Isomorphism of Finitary Inductive Types [<a href="docs/inductive-isomorphism.pdf">pdf</a>] - at TYPES 2014.
<ul>
<li>
With Christian Sattler. We present an algorithm for deciding isomorphism of finitary inductive types.
</li>
</ul>
</li>
</ol>
<h3 id="talks">Some of My Talks</h3>
Below are slides, abstracts, or descriptions of some of my talks. Many of my talks are board-based.
<ol class="nonumber">
<!-- This list is extremely incomplete. At some point, I could add some of the missing talks such as higher cats in tt (Budapest), ordinals in Nottingham (Oct'19), multiple earlier ones; the talks in Birmingham (one in 2018, one 2019, the 2020 one is added below); what else?
Also, add videos to the online talks. -->
<li>
Programs as Proofs (and why some computer scientists nowadays do homotopy theory) [<a href="docs/aberdeen2023talk.txt">abstract</a>]
<ul>
<li>
At the University of Aberdeen, 5 June 2023.
</li>
</ul>
</li>
<li>
Decidability and Semidecidability via ordinals [<a href="docs/nicolai_types_2022.pdf">slides</a>]
<ul>
<li>
At <a href="https://types22.inria.fr/">TYPES'22</a>, Nantes, 20-25 June 2022.
</li>
</ul>
</li>
<li>
Identities in higher categories
[<a href="docs/nicolai_structures_handout.pdf">slides</a>]
<ul>
<li>
At <a href="https://conferences.cirm-math.fr/2689.html">Logic and Higher Structures</a>, CIRM Luminy, 21-25 February 2022.
</li>
</ul>
</li>
<li>
Connecting Constructive Notions of Ordinals
in Homotopy Type Theory [<a href="docs/ordinals_MFCS_talk_annotated.pdf">slides</a>]
<ul>
<li>
At <a href="https://compose.ioc.ee/mfcs/">MFCS'21</a>, Tallinn/online, 23-27 August 2021.
</li>
</ul>
</li>
<li>
Internal ∞-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT [<a href="docs/nicolaikraus_lics2021_slides_with_annotations.pdf">slides</a>][<a href="https://www.youtube.com/watch?v=6feAittCfgw">youtube</a>]
<ul>
<li>
At <a href="http://easyconferences.eu/lics2021/">LICS'21</a>, Rome/online, 29 June - 2 July 2021.
</li>
</ul>
</li>
<li>
Wellfounded and Extensional Ordinals in Homotopy Type Theory [<a href="docs/2021-developments-in-CS-nicolaikraus-ordinals-budapest.pdf">annotated slides</a>]
<ul>
<li>
At <a href="http://dcs.elte.hu/">Developments in Computer Science</a>, Budapest/online, 17-19 June 2021.
</li>
</ul>
</li>
<li>
Constructive Notions of Ordinals in Homotopy Type Theory [<a href="docs/types2021_handout.pdf">slides</a>]
[<a href="https://www.youtube.com/watch?v=nduKY600mOo">youtube</a>]
<ul>
<li>
Joint talk with Fredrik Nordvall Forsberg and Chuangjie Xu. At <a href="https://types21.liacs.nl/">TYPES'21</a>, Leiden/online, 14-18 June 2021.
</li>
</ul>
</li>
<li>
Internal ∞-Categories with Families [<a href="docs/strathclyde_talk_with_annotations_nicolai.pdf">slides</a>]
<ul>
<li>
At <a href="http://msp.cis.strath.ac.uk/msp101.html">Strathclyde's MSP 101 Seminar</a>, 18 February 2021.
</li>
<li>
Note: I have given another talk on the topic at <a href="http://mathieu.anel.free.fr/seminar.html">Pittsburgh's HoTT Seminar</a> on 9 October 2020.
</li>
</ul>
</li>
<li>
Two-Level Type Theory (2LTT) - What is it, what can it do, and does Agda need it? [<a href="docs/AIM_Oct_2020_nicolai.pdf">annotated slides</a>]
<ul>
<li>
At the <a href="https://wiki.portal.chalmers.se/agda/Main/AIMXXXIII">AIM XXXIII</a>, 20 October 2020.
</li>
</ul>
</li>
<li>
Induction for Cycles [<a href="docs/nicolaikraus_munichonline.pdf">slides</a>]
<ul>
<li>
At the <a href="http://talks.bham.ac.uk/talk/index/4151">Birmingham Theory Seminar</a>, 31 January 2020;
</li>
<li>
also at <a href="https://cj-xu.github.io/tim20/">Types in Munich</a>, replaced by an online conference, 11 March 2020;
</li>
<li>
also at the <a href="https://bitbucket.org/akaposi/tipuselmelet/src/master/">Budapest type theory seminar</a>, held online, 17 March 2020.
</li>
<li>
A similar talk at FP lunch Nottingham, 24 April 2020: <a href="docs/fplunch_2020_04_24.png">"whiteboard" picture</a>
</li>
</ul>
</li>
<li>
Coherence via Wellfoundedness [<a href="https://cj-xu.github.io/faum/#kraus">conference abstract</a>]
<ul>
<li>
FAUM (Foundations and Applications of Univalent Mathematics), Herrsching, 18 December 2019.
</li>
</ul>
</li>
<li>
Internal higher-categorical models of dependent type theory
[<a href="http://talks.bham.ac.uk/talk/index/3962">abstract</a>]
<ul>
<li>
Lab Lunch, Birmingham, 24 October 2019.
</li>
</ul>
</li>
<li>
Constructive Ordinals
<ul>
<li>
FP Lunch, Nottingham, 18 October 2019.
</li>
</ul>
</li>
<li>
Higher Type Theory [<a href="https://bitbucket.org/akaposi/tipuselmelet/src/master/">repository</a>]
<ul>
<li>
Budapest Type Theory Seminar, 9 September 2019.
</li>
</ul>
</li>
<li>
Homotopical ideas in type theory [<a href="https://cj-xu.github.io/tim19/index.html">conference website and abstract</a>]
<ul>
<li>
Types in Munich, Munich, 7 June 2019.
</li>
</ul>
</li>
<li>
Deep and shallow embedding of groups [<a href="https://photos.google.com/share/AF1QipO7b06B5YyAfULZd_GUSaLmXCCjpUv4-Ejd_abRuZ9-74rIkqtQ-QXgA7gOD1FlWQ/photo/AF1QipMgdJ6PW5YPOuMRUPAk71l2hjFRo8S7OfsubCdH?key=V1c4Vk9mRVh0RGZncF8wOGRvT3IwbVMzTWdRQWJR">greenboard pictures</a>]
<ul>
<li>
Budapest Type Theory Seminar, 13 May 2019.
</li>
</ul>
</li>
<li>
Some connections between open problems [<a href="docs/nicolai_hottest2018.pdf">slides</a>][<a href="https://www.youtube.com/watch?v=l8S0Ll4PhoI">video</a>]
<ul>
<li>
At <a href="https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html">Homotopy Type Theory Electronic Seminar Talks</a> (HoTTEST), expert level seminar, online, hosted by Chris Kapulkin and Dan Christensen at the University of Western Ontario, 25 October 2018.
</li>
</ul>
</li>
<li>
Free Higher Groups in Homotopy Type Theory [<a href="docs/nicolaikraus_oxford_lics.pdf">slides</a>]
<ul>
<li>
At <a href="https://lics.siglog.org/lics18/">LICS'18</a>, Oxford, 9 July 2018.
</li>