@software{alex_rice_2024_10964565,
  author       = {Alex Rice},
  title        = {Agda formalisation of Catt},
  month        = apr,
  year         = 2024,
  publisher    = {Zenodo},
  version      = {thesis},
  doi          = {10.5281/zenodo.10964565},
  url          = {https://github.com/alexarice/catt-agda/tree/thesis}
}

@software{alex_rice_2024_10964705,
  author       = {Alex Rice},
  title        = {Semistrict Catt implementation},
  month        = apr,
  year         = 2024,
  publisher    = {Zenodo},
  version      = {thesis},
  doi          = {10.5281/zenodo.10966141},
  url          = {https://github.com/alexarice/catt-strict/tree/thesis}
}

@software{sd-visualiser,
  author = {Hu, Nick and Rice, Alex and Tataru, Calin},
  title = {\textsf{sd-visualiser}},
  year = 2024,
  url = {https://github.com/sd-visualiser/sd-visualiser}
}

@unpublished{andrastalk,
  title= {Efficient Evaluation with Controlled Definition Unfolding},
  author = {András Kovács},
  year = {2024},
  note= {Workshop on the Implementation of Type Systems},
  URL= {https://popl24.sigplan.org/details/wits-2024-papers/8/Efficient-Evaluation-with-Controlled-Definition-Unfolding}, % tex-fmt: skip
}

@inbook{selinger2011survey,
  title = {A Survey of Graphical Languages for Monoidal Categories},
  DOI = {10.1007/978-3-642-12821-9_4},
  booktitle = {New Structures for Physics},
  publisher = {Springer Berlin Heidelberg},
  author = {Selinger,  Peter},
  year = {2011},
  pages = {289-–355},
  isbn="978-3-642-12821-9",
  doi="10.1007/978-3-642-12821-9_4"
}

@article{forest2022unifying,
  title={Unifying notions of pasting diagrams},
  author={Forest, Simon},
  journal={Higher Structures},
  volume={6},
  number={1},
  pages={1--79},
  year={2022},
  doi={10.21136/HS.2022.01}
}

@unpublished{makkai2005word,
  title={The word problem for computads},
  author={Makkai, Michael},
  note={\url{https://www.math.mcgill.ca/makkai/WordProblem/WordProblemCombined.pdf}}, % tex-fmt: skip
  year={2005}
}

@phdthesis{forest2021computational,
  title={Computational descriptions of higher categories},
  author={Forest, Simon},
  year={2021},
  school={Institut Polytechnique de Paris}
}

@unpublished{douglas2016internal,
  title={Internal bicategories},
  author={Christopher L. Douglas and André G. Henriques},
  year={2016},
  eprint={1206.4284},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@book{leinster2004higher,
  title={Higher operads, higher categories},
  author={Leinster, Tom},
  volume={298},
  year={2004},
  publisher={Cambridge University Press}
}

@unpublished{simpson1998homotopy,
  title={Homotopy types of strict 3-groupoids},
  author={Carlos Simpson},
  year={1998},
  eprint={math/9810059},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@incollection {joyal2006weak,
  AUTHOR = {Joyal, Andr\'{e} and Kock, Joachim},
  TITLE = {Weak units and homotopy 3-types},
  BOOKTITLE = {Categories in algebra, geometry and mathematical physics},
  SERIES = {Contemp. Math.},
  VOLUME = {431},
  PAGES = {257--276},
  PUBLISHER = {Amer. Math. Soc., Providence, RI},
  YEAR = {2007},
  ISBN = {978-0-8218-3970-6},
  DOI = {10.1090/conm/431/08277},
  URL = {https://doi.org/10.1090/conm/431/08277},
}

@inproceedings{10.1145/237721.237728,
  author = {Jim, Trevor},
  title = {What are principal typings and what are they good for?},
  year = {1996},
  isbn = {0897917693},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  url = {https://doi.org/10.1145/237721.237728},
  doi = {10.1145/237721.237728},
  abstract = {We demonstrate the pragmatic value of the principal
    typing property, a property distinct from ML's principal type
    property, by studying a type system with principal typings. The
    type system is based on rank 2 intersection types and is closely
    related to ML. Its principal typing property provides elegant
    support for separate compilation, including "smartest
    recompilation" and incremental type inference. Moreover, it
    motivates a new rule for typing recursive definitions that can type
  some interesting examples of polymorphic recursion.},
  booktitle = {Proceedings of the 23rd ACM SIGPLAN-SIGACT Symposium
  on Principles of Programming Languages},
  pages = {42–53},
  numpages = {12},
  location = {St. Petersburg Beach, Florida, USA},
  series = {POPL '96}
}

@article{10.1145/3450952,
  author = {Dunfield, Jana and Krishnaswami, Neel},
  title = {Bidirectional Typing},
  year = {2021},
  issue_date = {June 2022},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {54},
  number = {5},
  issn = {0360-0300},
  url = {https://doi.org/10.1145/3450952},
  doi = {10.1145/3450952},
  abstract = {Bidirectional typing combines two modes of typing: type
    checking, which checks that a program satisfies a known type, and
    type synthesis, which determines a type from the program. Using
    checking enables bidirectional typing to support features for which
    inference is undecidable; using synthesis enables bidirectional
    typing to avoid the large annotation burden of explicitly typed
    languages. In addition, bidirectional typing improves error
    locality. We highlight the design principles that underlie
    bidirectional type systems, survey the development of bidirectional
    typing from the prehistoric period before Pierce and Turner’s local
    type inference to the present day, and provide guidance for future
  investigations.},
  journal = {ACM Comput. Surv.},
  month = {5},
  articleno = {98},
  numpages = {38},
  keywords = {Type checking, type inference}
}
@article{abel2013normalization,
  title={Normalization by evaluation: Dependent types and impredicativity},
  author={Abel, Andreas},
  journal={Habilitation. Ludwig-Maximilians-Universit{\"a}t M{\"u}nchen},
  year={2013}
}

@article{gratzer2019implementing,
  title={Implementing a modal dependent type theory},
  author={Gratzer, Daniel and Sterling, Jonathan and Birkedal, Lars},
  journal={Proceedings of the ACM on Programming Languages},
  volume={3},
  number={ICFP},
  pages={1--29},
  year={2019},
  publisher={ACM New York, NY, USA},
  doi = {10.1145/3341711}
}

@unpublished{hadzihasanovic2019representable,
  title={Representable diagrammatic sets as a model of weak higher categories},
  author={Amar Hadzihasanovic},
  year={2019},
  eprint={1909.07639},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@inproceedings{reutter2019high,
  title={High-level methods for homotopy construction in associative
  n-categories},
  author={Reutter, David and Vicary, Jamie},
  booktitle={Proceedings of the 34th Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={1--13},
  year={2019},
  doi={10.1109/LICS.2019.8785895}
}

@unpublished{corbyn2024homotopy,
  title={\textsf{homotopy.io}: a proof assistant for
  finitely-presented globular $n$-categories},
  author={Nathan Corbyn and Lukas Heidemann and Nick Hu and Chiara
  Sarti and Calin Tataru and Jamie Vicary},
  year={2024},
  eprint={2402.13179},
  archivePrefix={arXiv},
  primaryClass={cs.LO}
}

@unpublished{tataru2024theory,
  title={The theory and applications of anticolimits},
  author={Calin Tataru and Jamie Vicary},
  year={2024},
  eprint={2401.17076},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@incollection{MARTINLOF197573,
  title = {An Intuitionistic Theory of Types: Predicative Part},
  editor = {H.E. Rose and J.C. Shepherdson},
  series = {Studies in Logic and the Foundations of Mathematics},
  publisher = {Elsevier},
  volume = {80},
  pages = {73-118},
  year = {1975},
  booktitle = {Logic Colloquium '73},
  issn = {0049-237X},
  doi = {https://doi.org/10.1016/S0049-237X(08)71945-1},
  url = {https://www.sciencedirect.com/science/article/pii/S0049237X08719451},
  author = {Per Martin-Löf},
  abstract = {Publisher Summary
    The theory of types is intended to be a full-scale system for
    formalizing intuitionistic mathematics as developed. The language
    of the theory is richer than the languages of traditional
    intuitionistic systems in permitting proofs to appear as parts of
    propositions so that the propositions of the theory can express
    properties of proofs. There are axioms for universes that link the
    generation of objects and types and play somewhat the same role for
    the present theory as does the replacement axiom for
    Zermelo–Fraenkel set theory. The present theory is based on a
    strongly impredicative axiom that there is a type of all types in
    symbols. This axiom has to be abandoned, however, after it has been
    shown to lead to a contraction. This chapter discusses
    Normalization theorem, which can be strengthened in two ways: it
    can be made to cover open terms and it can be proved that every
    reduction sequence starting from an arbitrary term leads to a
    unique normal term after a finite number of steps. The definition
    of the notion of convertibility and the proof that an arbitrary
    term is convertible can no longer be separated because the type
  symbols and the terms are generated simultaneously.}
}

@article{lumsdaine2010weak,
  title = {Weak omega-categories from intensional type theory},
  volume = {Volume 6,  Issue 3},
  ISSN = {1860-5974},
  url = {http://dx.doi.org/10.2168/LMCS-6(3:24)2010},
  DOI = {10.2168/lmcs-6(3:24)2010},
  journal = {Logical Methods in Computer Science},
  publisher = {Centre pour la Communication Scientifique Directe (CCSD)},
  author = {Lumsdaine,  Peter LeFanu},
  year = {2010},
  month = sep
}

@article{garner2011types,
  title={Types are weak omega-groupoids},
  author={Garner, Richard and van den Berg, Benno},
  journal={Proceedings of the London Mathematical Society},
  volume={102},
  number={2},
  pages={370--394},
  year={2010},
  publisher={London Mathematical Society},
  DOI = {10.1112/plms/pdq026}
}

@unpublished{dorn2021framed,
  title={Framed combinatorial topology},
  author={Christoph Dorn and Christopher L. Douglas},
  year={2021},
  eprint={2112.14700},
  archivePrefix={arXiv},
  primaryClass={math.GT}
}

@unpublished{heidemann2023framed,
  title={Framed Combinatorial Topology with Labels in $\infty$-Categories},
  author={Lukas Heidemann},
  year={2023},
  eprint={2305.06288},
  archivePrefix={arXiv},
  primaryClass={math.AT}
}

@article{eckmann1962group,
  title={Group-like structures in general categories I
  multiplications and comultiplications},
  author={Eckmann, Beno and Hilton, Peter J},
  journal={Mathematische Annalen},
  volume={145},
  number={3},
  pages={227--255},
  year={1962}
}

@phdthesis{brunerie2016homotopy,
  title={On the homotopy groups of spheres in homotopy type theory},
  author={Brunerie, Guillaume},
  year={2016},
  school={Universit{\'e} Nice Sophia Antipolis}
}

@unpublished{shulman2019all,
  title={All $(\infty,1)$-toposes have strict univalent universes},
  author={Michael Shulman},
  year={2019},
  eprint={1904.07004},
  archivePrefix={arXiv},
  primaryClass={math.AT}
}

@Book{hottbook,
  author =    {The {Univalent Foundations Program}},
  title =     {Homotopy Type Theory: Univalent Foundations of Mathematics},
  publisher = {\url{https://homotopytypetheory.org/book}},
  address =   {Institute for Advanced Study},
year =      2013}

@incollection {hofmannstreicher,
  AUTHOR = {Hofmann, Martin and Streicher, Thomas},
  TITLE = {The groupoid interpretation of type theory},
  BOOKTITLE = {Twenty-five years of constructive type theory},
  VOLUME = {36},
  PAGES = {83--111},
  PUBLISHER = {Oxford University Press},
  YEAR = {1998},
  DOI = {10.1093/oso/9780198501275.003.0008},
}

@inproceedings{heidemann2022zigzag,
  title={Zigzag normalisation for associative n-categories},
  author={Heidemann, Lukas and Reutter, David and Vicary, Jamie},
  booktitle={Proceedings of the 37th Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={1--13},
  year={2022},
  doi = {10.1145/3531130.3533352}
}

@article{Batanin2013,
  author = {Michael Batanin and Denis-Charles Cisinski and Mark Weber},
  title = {Multitensor lifting and strictly unital higher category theory},
  year = {2013},
  journal = {Theory and Applications of Categories},
  volume = 28,
  pages = {804--856}
}

@phdthesis{dorn2018associative,
  title={Associative n-categories},
  author={Dorn, C},
  year={2018},
  school={University of Oxford}
}

@article {joyalcoherence,
  AUTHOR = {Joyal, Andr\'{e} and Kock, Joachim},
  TITLE = {Coherence for weak units},
  JOURNAL = {Documenta Mathematica},
  VOLUME = {18},
  YEAR = {2013},
  PAGES = {71--110},
  ISSN = {1431-0635,1431-0643},
}

@incollection {cheng2007periodic,
  AUTHOR = {Cheng, Eugenia and Gurski, Nick},
  TITLE = {The periodic table of {$n$}-categories for low dimensions
  {I}. {D}egenerate categories and degenerate bicategories},
  BOOKTITLE = {Categories in algebra, geometry and mathematical physics},
  SERIES = {Contemp. Math.},
  VOLUME = {431},
  PAGES = {143--164},
  PUBLISHER = {Amer. Math. Soc., Providence, RI},
  YEAR = {2007},
  ISBN = {978-0-8218-3970-6},
  DOI = {10.1090/conm/431/08270},
  URL = {https://doi.org/10.1090/conm/431/08270}
}

@unpublished{cheng2007periodic2,
  title={The periodic table of $n$-categories for low dimensions II:
  degenerate tricategories},
  author={Eugenia Cheng and Nick Gurski},
  year={2007},
  eprint={0706.2307},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@article{Baez1995,
  title = {Higher-dimensional algebra and topological quantum field theory},
  volume = {36},
  ISSN = {1089-7658},
  url = {http://dx.doi.org/10.1063/1.531236},
  DOI = {10.1063/1.531236},
  number = {11},
  journal = {Journal of Mathematical Physics},
  publisher = {AIP Publishing},
  author = {Baez,  John C. and Dolan,  James},
  year = {1995},
  month = nov,
  pages = {6073–6105}
}

@BOOK{Heunen2019-jt,
  title     = "Categories for quantum theory",
  author    = "Heunen, Chris and Vicary, Jamie",
  publisher = "Oxford University Press",
  series    = "Oxford Graduate Texts in Mathematics",
  month     =  nov,
  year      =  2019,
  address   = "London, England",
  doi = {10.1093/oso/9780198739623.001.0001}
}
@article{Barr1991,
  title = {*-Autonomous categories and linear logic},
  volume = {1},
  ISSN = {1469-8072},
  url = {http://dx.doi.org/10.1017/S0960129500001274},
  DOI = {10.1017/s0960129500001274},
  number = {2},
  journal = {Mathematical Structures in Computer Science},
  publisher = {Cambridge University Press (CUP)},
  author = {Barr,  Michael},
  year = {1991},
  month = jul,
  pages = {159–178}
}

@book{riehl2022elements,
  title={Elements of \(\infty\)-Category Theory},
  author={Riehl, Emily and Verity, Dominic},
  volume={194},
  year={2022},
  publisher={Cambridge University Press},
  DOI = {10.1017/9781108936880}
}

@article{Street2012,
  title = {Monoidal categories in,  and linking,  geometry and algebra},
  volume = {19},
  ISSN = {1370-1444},
  url = {http://dx.doi.org/10.36045/bbms/1354031551},
  DOI = {10.36045/bbms/1354031551},
  number = {5},
  journal = {Bulletin of the Belgian Mathematical Society - Simon Stevin},
  publisher = {The Belgian Mathematical Society},
  author = {Street,  Ross},
  year = {2012},
  month = dec
}

@article{mellies2009categorical,
  title={Categorical semantics of linear logic},
  author={Mellies, Paul-Andr{\'e}},
  journal={Panoramas et syntheses},
  volume={27},
  pages={15--215},
  year={2009}
}

@inproceedings{ghani2018compositional,
  title={Compositional game theory},
  author={Ghani, Neil and Hedges, Jules and Winschel, Viktor and Zahn, Philipp},
  booktitle={Proceedings of the 33rd annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={472--481},
  year={2018},
  doi = {10.1145/3209108.3209165}
}

@book{Bourbaki2016,
  title = {Topologie algébrique},
  ISBN = {9783662493618},
  url = {http://dx.doi.org/10.1007/978-3-662-49361-8},
  DOI = {10.1007/978-3-662-49361-8},
  publisher = {Springer Berlin Heidelberg},
  author = {Bourbaki,  N.},
  year = {2016}
}

@article{Weber2004,
  title = {Generic Morphisms, Parametric Representations and Weakly
  Cartesian Monads.},
  author = {Weber, Mark},
  date = {2004},
  journaltitle = {Theory and Applications of Categories},
  volume = {13},
  pages = {191--234},
  publisher = {{Mount Allison University, Department of Mathematics
  and Computer Science, Sackville}},
  url = {http://eudml.org/doc/124614},
  langid = {english},
  keywords = {braiding,centre,descent,endofunctor,generic
    morphism,higher category theory,monad,operand,parametric
  representation,pseudofunctor}
}

@article{lipparini16,
  title={An infinite natural sum},
  author={Lipparini, Paolo},
  journal={Mathematical Logic Quarterly},
  DOI = {10.1002/malq.201500017},
  volume={62},
  number={3},
  pages={249--257},
  year={2016},
  publisher={Wiley Online Library}
}

@article{newman1942theories,
  title={On theories with a combinatorial definition of equivalence},
  author={Newman, Maxwell and Herman, Alexander},
  journal={Annals of mathematics},
  pages={223--243},
  year={1942},
  publisher={JSTOR}
}

@unpublished{maltsiniotis2010grothendieck,
  title={Grothendieck $\infty$-groupoids, and still another
  definition of $\infty$-categories},
  author={Georges Maltsiniotis},
  year={2010},
  eprint={1009.2331},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@unpublished{leinster2001survey,
  title={A Survey of Definitions of n-Category},
  author={Tom Leinster},
  year={2001},
  eprint={math/0107188},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@inproceedings{finster2017type,
  title={A type-theoretical definition of weak $\omega$-categories},
  author={Finster, Eric and Mimram, Samuel},
  booktitle={Proceedings of the 32nd Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={1--12},
  year={2017},
  doi={10.1109/LICS.2017.8005124}
}

@phdthesis{Ara,
  author={Dimitri Ara},
  title={Sur les $\infty$-groupoides de {G}rothendieck et une variante
  $\infty$-cat\'egorique},
  school={Universit\'e Paris Diderot},
  year={2010}
}

@unpublished{PursuingStacks,
  author={Alexander Grothendieck},
  year=1983,
  title={Pursuing stacks}
}
@phdthesis{gurski2006algebraic,
  title={An algebraic theory of tricategories},
  author={Gurski, Michael Nicholas},
  year={2006},
  school={University of Chicago, Department of Mathematics}
}

@inproceedings{bar2017data,
  title={Data structures for quasistrict higher categories},
  author={Bar, Krzysztof and Vicary, Jamie},
  booktitle={Proceedings of the 32nd Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={1--12},
  year={2017},
  doi={10.1109/LICS.2017.8005147}
}

@book{gordon1995coherence,
  title={Coherence for tricategories},
  author={Gordon, Robert and Power, Anthony John and Street, Ross},
  volume={558},
  year={1995},
  publisher={American Mathematical Soc.}
}

@inproceedings{finster2022type,
  title={A type theory for strictly unital ∞-categories},
  author={Finster, Eric and Reutter, David and Vicary, Jamie and Rice, Alex},
  booktitle={Proceedings of the 37th Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pages={1--12},
  year={2022},
  doi = {10.1145/3531130.3533363}
}

@inproceedings{finster2023strictly,
  title={A Syntax for Strictly Associative and Unital ∞-categories},
  author={Finster, Eric and Rice, Alex and Vicary, Jamie},
  booktitle={Proceedings of the 39th Annual ACM/IEEE Symposium on
  Logic in Computer Science (LICS)},
  pubstate={forthcoming},
  year={2024}
}

@InProceedings{cwf,
  author="Dybjer, Peter",
  editor="Berardi, Stefano
  and Coppo, Mario",
  title="Internal type theory",
  booktitle="Types for Proofs and Programs",
  year="1996",
  publisher="Springer Berlin Heidelberg",
  address="Berlin, Heidelberg",
  pages="120--134",
  abstract="We introduce categories with families as a new notion of
  model for a basic framework of dependent types. This notion is
  close to ordinary syntax and yet has a clean categorical
  description. We also present categories with families as a
  generalized algebraic theory. Then we define categories with
  families formally in Martin-L{\"o}f's intensional intuitionistic
  type theory. Finally, we discuss the coherence problem for these
  internal categories with families.",
  isbn="978-3-540-70722-6"
}

@article{batanin1998computads,
  title={Computads for finitary monads on globular sets},
  author={Batanin, Michael A},
  journal={Contemporary Mathematics},
  volume={230},
  pages={37--58},
  year={1998},
  issn = {0271-4132},
  publisher={American Mathematical Society}
}

@article{street1976limits,
  title={Limits indexed by category-valued 2-functors},
  author={Street, Ross},
  journal={Journal of Pure and Applied Algebra},
  volume={8},
  number={2},
  pages={149--181},
  year={1976},
  publisher={Elsevier},
  doi={10.1016/0022-4049(76)90013-X}
}

@article{burroni1993higher,
  title={Higher-dimensional word problems with applications to
  equational logic},
  author={Burroni, Albert},
  journal={Theoretical computer science},
  volume={115},
  number={1},
  pages={43--62},
  year={1993},
  publisher={Elsevier}
}

@unpublished{dean2022computads,
  title={Computads for weak $\omega$-categories as an inductive type},
  author={Christopher J. Dean and Eric Finster and Ioannis Markakis
  and David Reutter and Jamie Vicary},
  year={2024},
  eprint={2208.08719},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@unpublished{benjamin2021globular,
  title={Globular weak $\omega$-categories as models of a type theory},
  author={Thibaut Benjamin and Eric Finster and Samuel Mimram},
  year={2024},
  eprint={2106.04475},
  archivePrefix={arXiv},
  primaryClass={cs.LO}
}

@phdthesis{benjamin2020type,
  title={A type theoretic approach to weak w-categories and related
  higher structures},
  author={Benjamin, Thibaut},
  year={2020},
  school={Institut polytechnique de Paris}
}

@unpublished{benjamin2024duamity,
  title={Opposites of weak $\omega$-categories and the suspension and
  hom adjunction},
  author={Thibaut Benjamin and Ioannis Markakis},
  year={2024},
  eprint={2402.01611},
  archivePrefix={arXiv},
  primaryClass={math.CT}
}

@article{batanin1998monoidal,
  title={Monoidal globular categories as a natural environment for
  the theory of weak n-categories},
  author={Batanin, Michael A},
  journal={Advances in Mathematics},
  volume={136},
  number={1},
  pages={39--103},
  year={1998},
  publisher={Academic Press}
}
