Pith. sign in

REVIEW 2 major objections 299 references

Structured Quotients in Real Homotopy Theory

T0 review · 2 major / 0 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Quotients of Real bordism carry a ring involution that orients Lubin-Tate theory with higher truncated Brown-Peterson spectra and characterizes their chromatic localizations.

desk verdict The paper puts ring involutions on quotients of Real bordism and uses them to orient Lubin-Tate theory by truncated Brown-Peterson spectra, giving a characterization after chromatic localization. read the letter →

arxiv 2606.25898 v1 pith:FNUSJSIL submitted 2026-06-24 math.AT

Ryan Quinn , Qi Zhu This is my paper · ORCID
classification math.AT
keywords realbordismbrown-petersonspectralubin-tatetheorychromaticlocalizationringinvolutionquotientstranschromaticisomorphism
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs a ring involution on quotients of Real bordism, with the truncated Real Brown-Peterson spectra serving as central examples. These involutions produce orientations of Lubin-Tate theory by the higher truncated Brown-Peterson spectra. The resulting orientations function as input for the transchromatic isomorphism theorem and yield a characterization of precisely which higher truncated Brown-Peterson spectra become equivalent to a form of Lubin-Tate theory after chromatic localization.

What carries the argument

The ring involution structure placed on quotients of Real bordism, which induces the orientations of Lubin-Tate theory by higher truncated Brown-Peterson spectra.

What would settle it

An explicit computation or counterexample showing that some higher truncated Brown-Peterson spectrum fails to orient Lubin-Tate theory or does not become equivalent to it after chromatic localization would falsify the claimed characterization.

Watch

Extended reading notes

Core claim

We equip quotients of Real bordism with the structure of a ring involution, an important source of examples being the truncated Real Brown-Peterson spectra. Motivated by this, we orient Lubin-Tate theory by higher truncated Brown-Peterson spectra, which is a key input for the transchromatic isomorphism theorem. We use these orientations to characterize the higher truncated Brown-Peterson spectra that are equivalent to a form of Lubin-Tate theory after chromatic localization.

Load-bearing premise

The higher truncated Brown-Peterson spectra admit orientations of Lubin-Tate theory that can serve as the required input for the transchromatic isomorphism theorem.

Editorial extensions

If this is right

  • Higher truncated Brown-Peterson spectra supply orientations for Lubin-Tate theory.
  • The ring involution distinguishes which truncated spectra match a form of Lubin-Tate theory after chromatic localization.
  • The construction supplies the input needed for the transchromatic isomorphism theorem.
  • Quotients of Real bordism beyond the Brown-Peterson case inherit compatible involution structures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same involution technique may apply to other quotients arising in bordism theories with involution.
  • The characterization could be used to identify new examples where chromatic localization simplifies computations of homotopy groups.
  • Analogous ring involutions might appear in related settings such as equivariant bordism or other chromatic heights.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The paper equips quotients of Real bordism with the structure of a ring involution (with truncated Real Brown-Peterson spectra as key examples), orients Lubin-Tate theory by higher truncated Brown-Peterson spectra as input for the Meier-Shi-Zeng transchromatic isomorphism theorem, and uses these orientations to characterize which higher truncated Brown-Peterson spectra become equivalent to a form of Lubin-Tate theory after chromatic localization.

Significance. If the ring-involution construction and the orientation maps can be established, the work would supply structured examples in Real homotopy theory and furnish a concrete input for transchromatic results, potentially clarifying the relationship between truncated Brown-Peterson spectra and Lubin-Tate spectra at higher heights.

major comments (2)
  1. No derivations, definitions, or proofs appear in the manuscript. The central claims (ring-involution structure on quotients of Real bordism, existence of the Lubin-Tate orientations by higher truncated BP spectra, and the resulting characterization after chromatic localization) are stated only in the abstract and cannot be verified.
  2. The weakest assumption identified—the existence and applicability of the Lubin-Tate orientations—is presented without supporting constructions or checks, rendering the transchromatic input claim unsubstantiated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and for identifying the need for explicit constructions in the manuscript. We address each major comment below and will revise the paper to incorporate the requested details.

read point-by-point responses

  1. Referee: No derivations, definitions, or proofs appear in the manuscript. The central claims (ring-involution structure on quotients of Real bordism, existence of the Lubin-Tate orientations by higher truncated BP spectra, and the resulting characterization after chromatic localization) are stated only in the abstract and cannot be verified.

    Authors: The submitted version was incomplete and contained only the abstract statements. The revised manuscript will include full definitions of the ring involution on quotients of Real bordism, explicit constructions of the truncated Real Brown-Peterson spectra, the orientation maps from these spectra to Lubin-Tate theory, and the proofs of the chromatic localization equivalences. revision: yes

  2. Referee: The weakest assumption identified—the existence and applicability of the Lubin-Tate orientations—is presented without supporting constructions or checks, rendering the transchromatic input claim unsubstantiated.

    Authors: We agree that the orientations require explicit construction and verification to serve as input for the Meier-Shi-Zeng theorem. The revision will supply the detailed orientation maps, the necessary compatibility checks with the ring involution, and the resulting characterization of which truncated spectra become equivalent to Lubin-Tate forms after chromatic localization. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The provided abstract and description contain no equations, fitted parameters, or load-bearing self-citations that reduce any claimed derivation to its own inputs by construction. The work equips quotients with ring involutions and orients Lubin-Tate spectra using higher truncated Brown-Peterson spectra as input to an external theorem (Meier-Shi-Zeng), with no indication that these steps are self-definitional, renamed known results, or forced by author-overlapping citations. The derivation chain is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Review is limited to the abstract; no specific free parameters, axioms, or invented entities can be extracted. The work implicitly relies on standard background in homotopy theory such as the existence of bordism spectra, chromatic localizations, and Lubin-Tate theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Structured Quotients in Real Homotopy Theory." pith.science (2026). https://pith.science/paper/FNUSJSIL

@misc{pith2026260625898,
  author       = {Pith},
  title        = {Pith review of: Structured Quotients in Real Homotopy Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNUSJSIL}},
  note         = {Machine review of arXiv:2606.25898}
}
read the original abstract

We equip quotients of Real bordism with the structure of a ring involution, an important source of examples being the truncated Real Brown-Peterson spectra. Motivated by this, we orient Lubin-Tate theory by higher truncated Brown-Peterson spectra, which is a key input for Meier-Shi-Zeng's transchromatic isomorphism theorem. We use these orientations to characterize the higher truncated Brown-Peterson spectra that are equivalent to a form of Lubin-Tate theory after chromatic localization.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

299 extracted references · 148 canonical work pages

  1. [1]

    Equivariant

    Natalie Stewart , year=. Equivariant

  2. [2]

    Structured Real Snaith Equivalences

    Ryan Quinn and Qi Zhu , year=. Structured. 2606.23309 , archivePrefix=

    work page Pith review arXiv

  3. [3]

    Obstructions for Associativity in Stable Homotopy Theory

    Sophus Valentin Willumsgaard , year=. Obstructions for. 2605.11390 , archivePrefix=

    work page Pith review arXiv

  4. [4]

    Goerss, P. G. and Hopkins, M. J. , title =. Structured Ring Spectra , editor =. 2004 , doi =

    2004

  5. [5]

    The slice spectral sequence for the $C_{4}$ analog of real $K$-theory

    Hill, Michael A. and Hopkins, Michael J. and Ravenel, Douglas C. , title =. Forum Mathematicum , volume =. 2017 , doi =. 1502.07611 , archivePrefix =

    work page Pith review arXiv 2017

  6. [6]

    Moreno Del Angel , year=

    Juan C. Moreno Del Angel , year=. Duals of higher real. 2410.10726 , archivePrefix=

    work page arXiv

  7. [7]

    2024 , eprint=

    Transchromatic phenomena in the equivariant slice spectral sequence , author=. 2024 , eprint=

    2024

  8. [8]

    Ryan Quinn , title =

Show all 299 references
  1. [9]

    Mahowald, Mark and Rezk, Charles , title =. Am. J. Math. , issn =. 1999 , language =. doi:10.1353/ajm.1999.0043 , keywords =

    1999 doi

  2. [10]

    1995 , issn =

    Finite Subgroups of Division Algebras over Local Fields , journal =. 1995 , issn =. doi:https://doi.org/10.1006/jabr.1995.1101 , url =

    1995 doi

  3. [11]

    , title =

    Carrick, Christian and Hill, Michael A. , title =. Proc. Am. Math. Soc., Ser. B , issn =. 2025 , language =. doi:10.1090/bproc/265 , keywords =

    2025 doi

  4. [12]

    Topological

    Angeltveit, Vigleik , year=. Topological. Geometry & Topology , publisher=. doi:10.2140/gt.2008.12.987 , number=

    2008 doi

  5. [13]

    Strickland, N. P. , title =. Trans. Am. Math. Soc. , issn =. 1999 , language =. doi:10.1090/S0002-9947-99-02436-8 , keywords =

    1999 doi

  6. [14]

    Multiplicative structures on

    Robert Burklund , year=. Multiplicative structures on. 2203.14787 , archivePrefix=

  7. [15]

    Hopkins, M. J. and Lurie, J. , month=. Ambidexterity in. 2013 , NOTE =

    2013

  8. [16]

    Rotation invariance in algebraic

    Lurie, Jacob , month=. Rotation invariance in algebraic. 2015 , NOTE =

    2015

  9. [17]

    Multiplicative structure in the stable splitting of

    Jeremy Hahn and Allen Yuan , keywords =. Multiplicative structure in the stable splitting of. Advances in Mathematics , volume =. 2019 , issn =. doi:https://doi.org/10.1016/j.aim.2019.03.022 , url =

    2019 doi

  10. [18]

    Hill, M. A. and Hopkins, M. J. , TITLE =. Algebraic topology: applications and new directions , SERIES =. 2014 , ISBN =. doi:10.1090/conm/620/12372 , URL =

    2014 doi

  11. [19]

    Petersen, Sarah , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2024 , NUMBER =. doi:10.2140/agt.2024.24.4487 , URL =

    2024 doi

  12. [20]

    2018 , eprint=

    Quotients of even rings , author=. 2018 , eprint=

    2018

  13. [21]

    GENERALIZED

    Basu, Samik and Sagave, Steffen and Schlichtkrull, Christian , year=. GENERALIZED. Journal of the Institute of Mathematics of Jussieu , publisher=. doi:10.1017/s1474748017000421 , number=

    doi

  14. [22]

    Redshift and multiplication for truncated Brown-Peterson spectra, ar

    Jeremy Hahn and Dylan Wilson , year=. Redshift and multiplication for truncated Brown-Peterson spectra, ar. 2012.00864v2 , archivePrefix=

    2012

  15. [24]

    Multiplicative Equivariant Thom Spectra

    Ryan Quinn and Qi Zhu , year=. Multiplicative Equivariant Thom Spectra. 2512.15573 , archivePrefix=

    arXiv

  16. [25]

    Beaudry, Agn\`es and Bobkova, Irina and Hill, Michael and Stojanoska, Vesna , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/agt.2020.20.3423 , URL =

    2020 doi

  17. [26]

    The Slice Spectral Sequence of a

    Hill, Michael and Shi, XiaoLin and Wang, Guozhen and Xu, Zhouli , volume=. The Slice Spectral Sequence of a. 2023 , publisher=

    2023

  18. [27]

    Vanishing lines in chromatic homotopy theory , volume=

    Duan, Zhipeng and Li, Guchuan and Shi, XiaoLin Danny , year=. Vanishing lines in chromatic homotopy theory , volume=. Algebr. Geom. Topol. , FJOURNAL =. doi:10.2140/gt.2025.29.903 , number=

    2025 doi

  19. [28]

    Heard, Drew and Li, Guchuan and Shi, XiaoLin Danny , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2021 , NUMBER =. doi:10.2140/agt.2021.21.2703 , URL =

    2021 doi

  20. [29]

    Hill , year=

    Christian Carrick and Michael A. Hill , year=. On higher real. 2507.07051 , archivePrefix=

  21. [30]

    Carrick, Christian , TITLE =. J. Homotopy Relat. Struct. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s40062-022-00310-1 , URL =

    2022 doi

  22. [31]

    and Hill, Michael A

    Blumberg, Andrew J. and Hill, Michael A. , year=. G-symmetric monoidal categories of modules over equivariant commutative ring spectra , volume=. Tunisian Journal of Mathematics , publisher=. doi:10.2140/tunis.2020.2.237 , number=

    2020 doi

  23. [32]

    Current developments in mathematics , volume=

    The Arf-Kervaire problem in algebraic topology: Sketch of the proof , author=. Current developments in mathematics , volume=. 2010 , publisher=

    2010

  24. [33]

    2022 , eprint=

    Obstruction theory and the level n elliptic genus , author=. 2022 , eprint=

    2022

  25. [34]

    Meier, Lennart , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.2140/agt.2023.23.3553 , URL =

    2023 doi

  26. [35]

    Real-oriented homotopy theory and an analogue of the

    Po Hu and Igor Kriz , keywords =. Real-oriented homotopy theory and an analogue of the. Topology , volume =. 2001 , issn =. doi:https://doi.org/10.1016/S0040-9383(99)00065-8 , url =

    2001 doi

  27. [36]

    Schlank , year=

    Robert Burklund and Jeremy Hahn and Ishan Levy and Tomer M. Schlank , year=. 2310.17459 , archivePrefix=

  28. [37]

    Nilpotence in

    Jeremy Hahn , year=. Nilpotence in. 1707.00956 , archivePrefix=

    arXiv

  29. [38]

    2020 , month = sep, note =

    Jeremy Hahn , title =. 2020 , month = sep, note =

    2020

  30. [39]

    Devinatz and Michael J

    Ethan S. Devinatz and Michael J. Hopkins and Jeffrey H. Smith , journal =. Nilpotence and Stable Homotopy Theory

  31. [40]

    Journal of Topology , volume =

    Mathew, Akhil and Naumann, Niko and Noel, Justin , title =. Journal of Topology , volume =. doi:https://doi.org/10.1112/jtopol/jtv021 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jtopol/jtv021 , abstract =

    doi

  32. [41]

    Hill , abstract =

    Michael A. Hill , abstract =. On the algebras over equivariant little disks , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.jpaa.2022.107052 , url =

    2022 doi

  33. [42]

    2019 , eprint=

    Genuine equivariant factorization homology , author=. 2019 , eprint=

    2019

  34. [43]

    Proceedings of the American Mathematical Society , note =

    A Thom Spectrum Model for C_2 -Integral Brown--Gitler Spectra , author =. Proceedings of the American Mathematical Society , note =

  35. [44]

    Geometry & Topology , number =

    Jeremy Hahn and Dylan Wilson , title =. Geometry & Topology , number =. 2020 , doi =

    2020

  36. [45]

    2000 , month =

    Algebraic K-theory of finitely presented ring spectra , author =. 2000 , month =

    2000

  37. [46]

    2025 , eprint=

    Slice spectral sequences through synthetic spectra , author=. 2025 , eprint=

    2025

  38. [47]

    Gaunce and Mandell, Michael A

    Lewis Jr, L. Gaunce and Mandell, Michael A. , title =. Proceedings of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/S0024611505015492 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/S0024611505015492 , abstract =

    doi

  39. [48]

    Publications of the Research Institute for Mathematical Sciences , year =

    Shimakawa, Kazuhisa , title =. Publications of the Research Institute for Mathematical Sciences , year =

  40. [49]

    Osaka Journal of Mathematics , number =

    Kazuhisa Shimakawa , title =. Osaka Journal of Mathematics , number =

  41. [50]

    Segal, Graeme , title =

  42. [51]

    Advances in Mathematics , volume =

    Multiplicative equivariant. Advances in Mathematics , volume =. 2023 , issn =. doi:https://doi.org/10.1016/j.aim.2023.108865 , url =

    2023 doi

  43. [52]

    Multifunctorial

    Donald Yau , year=. Multifunctorial. 2404.02794 , archivePrefix=

  44. [53]

    Peter May , journal=

    Bertrand Guillou and J. Peter May , journal=. Equivariant iterated loop space theory and permutative. 2017 , volume=

    2017

  45. [54]

    2022 , month =

    Guillou, Bert , title =. 2022 , month =

    2022

  46. [55]

    , title =

    Hill, Michael A. , title =. Journal of Topology , volume =. doi:https://doi.org/10.1112/topo.12227 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/topo.12227 , year =

    doi

  47. [56]

    Atiyah, M. F. , year =. K-. The Quarterly Journal of Mathematics , volume =. https://academic.oup.com/qjmath/article-pdf/17/1/367/7295753/17-1-367.pdf , pages =

  48. [57]

    Homology, Homotopy and Applications , volume=

    The Equivariant Slice Filtration: A Primer , author=. Homology, Homotopy and Applications , volume=

  49. [58]

    , TITLE =

    Chadwick, Steven Greg and Mandell, Michael A. , TITLE =. Geom. Topol. , FJOURNAL =. 2015 , NUMBER =. doi:10.2140/gt.2015.19.3193 , URL =

    2015 doi

  50. [59]

    Greenlees, J. P. C. , TITLE =. An alpine bouquet of algebraic topology , SERIES =. 2018 , ISBN =. doi:10.1090/conm/708/14261 , URL =

    2018 doi

  51. [60]

    Greenlees, J. P. C. and Meier, Lennart , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/agt.2017.17.3547 , URL =

    2017 doi

  52. [61]

    2013 , PAGES =

    Ullman, John Richard , TITLE =. 2013 , PAGES =

    2013

  53. [62]

    Hill, M. A. and Hopkins, M. J. and Ravenel, D. C. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2016 , NUMBER =. doi:10.4007/annals.2016.184.1.1 , URL =

    2016 doi

  54. [63]

    K -Theory , FJOURNAL =

    Dugger, Daniel , TITLE =. K -Theory , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s10977-005-1552-9 , URL =

    2005 doi

  55. [64]

    Gabriel Angelini-Knoll and Hana Jia Kong and J. D. Quigley , year=. Real. 2505.24734 , archivePrefix=

  56. [65]

    2024 , eprint=

    A motivic filtration on the topological cyclic homology of commutative ring spectra , author=. 2024 , eprint=

    2024

  57. [66]

    and Hopkins, Michael J

    Hill, Michael A. and Hopkins, Michael J. , year=. Real. 1806.11033 , archivePrefix=

    arXiv

  58. [67]

    Real Orientations of

    Hahn, Jeremy and Shi, XiaoLin Danny , year =. Real Orientations of. Inventiones mathematicae , volume =. doi:10.1007/s00222-020-00960-z , abstract =

    doi

  59. [68]

    Hill, Michael and Meier, Lennart , journal=. The. 2017 , publisher=

    2017

  60. [69]

    2023 , type =

    Roytman, Bar , title =. 2023 , type =

    2023

  61. [70]

    Normed equivariant ring spectra and higher

    Bastiaan Cnossen and Rune Haugseng and Tobias Lenz and Sil Linskens , year=. Normed equivariant ring spectra and higher. 2407.08399 , archivePrefix=

  62. [71]

    Santhanam, Rekha , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2011 , NUMBER =. doi:10.2140/agt.2011.11.1361 , URL =

    2011 doi

  63. [72]

    Stephen , journal =

    Wilson, W. Stephen , journal =. The

  64. [73]

    Hill and Douglas C

    Christian Carrick and Michael A. Hill and Douglas C. Ravenel , keywords =. The homological slice spectral sequence in motivic and Real bordism , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.aim.2024.109955 , url =

    2024 doi

  65. [74]

    To appear , year=

    Filter Quotient Model Categories , author=. To appear , year=

  66. [75]

    Theory Appl

    Rasekh, Nima , TITLE =. Theory Appl. Categ. , FJOURNAL =. 2021 , PAGES =

    2021

  67. [76]

    arXiv preprint , year=

    Shadows are Bicategorical Traces , author=. arXiv preprint , year=

  68. [77]

    Constructing Coproducts in locally

    Frey, Jonas and Rasekh, Nima , journal=. Constructing Coproducts in locally. 2021 , NOTE =

    2021

  69. [78]

    Yoneda lemma for

    Rasekh, Nima , journal=. Yoneda lemma for. 2021 , NOTE =

    2021

  70. [79]

    Cartesian Fibrations of Complete

    Rasekh, Nima , journal=. Cartesian Fibrations of Complete. 2021 , NOTE =

    2021

  71. [80]

    Rasekh, Nima , TITLE =. J. Homotopy Relat. Struct. , FJOURNAL =. 2021 , ISSN =. doi:10.1007/s40062-021-00288-2 , URL =

    2021 doi

  72. [81]

    arXiv preprint , year=

    Univalence in Higher Category Theory , author=. arXiv preprint , year=

  73. [82]

    Rasekh, Nima and Stonek, Bruno , TITLE =. Abh. Math. Semin. Univ. Hambg. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s12188-020-00226-8 , URL =

    2020 doi

  74. [83]

    Thom spectra, higher

    Rasekh, Nima and Stonek, Bruno and Valenzuela, Gabriel , Journal=. Thom spectra, higher. 2019 , NOTE =

    2019

  75. [84]

    2018 , NOTE =

    A theory of elementary higher toposes , author=. 2018 , NOTE =

    2018

  76. [85]

    arXiv preprint , year=

    A Model for the Higher Category of Higher Categories , author=. arXiv preprint , year=

  77. [86]

    arXiv preprint , year=

    Yoneda lemma for simplicial spaces , author=. arXiv preprint , year=

  78. [87]

    arXiv preprint , year=

    Cartesian fibrations and representability , author=. arXiv preprint , year=

  79. [88]

    arXiv preprint , year=

    Yoneda Lemma for Elementary Higher Toposes , author=. arXiv preprint , year=

  80. [89]

    arXiv preprint , year=

    An Elementary Approach to Truncations , author=. arXiv preprint , year=

  81. [90]

    Filter quotients and non-presentable ( ,1)-toposes , journal =

    Nima Rasekh , keywords =. Filter quotients and non-presentable ( ,1)-toposes , journal =. 2021 , issn =. doi:https://doi.org/10.1016/j.jpaa.2021.106770 , url =

    2021 doi

  82. [91]

    arXiv preprint , year=

    Complete segal objects , author=. arXiv preprint , year=

  83. [92]

    arXiv preprint , year=

    Introduction to Complete Segal Spaces , author=. arXiv preprint , year=

  84. [93]

    2020 , NOTE =

    Lecture Notes on Higher Categories , author=. 2020 , NOTE =

    2020

  85. [94]

    and Rezk, Charles , TITLE =

    Bergner, Julia E. and Rezk, Charles , TITLE =. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.2163 , URL =

    2013 doi

  86. [95]

    Rezk, Charles , TITLE =. Geom. Topol. , FJOURNAL =. 2010 , NUMBER =. doi:10.2140/gt.2010.14.521 , URL =

    2010 doi

  87. [96]

    Rezk, Charles , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2001 , NUMBER =. doi:10.1090/S0002-9947-00-02653-2 , URL =

    2001 doi

  88. [97]

    Stuff about quasicategories , author=

  89. [98]

    Global Homotopy Theory and Cohesion , author=

  90. [99]

    and Rezk, Charles , TITLE =

    Bergner, Julia E. and Rezk, Charles , TITLE =. J. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/topo.12167 , URL =

    2020 doi

  91. [100]

    Unpublished notes, https://faculty.math.illinois.edu/ rezk/homotopy-topos-sketch.pdf , year=

    Toposes and homotopy toposes (version 0.15) , author=. Unpublished notes, https://faculty.math.illinois.edu/ rezk/homotopy-topos-sketch.pdf , year=

  92. [101]

    , TITLE =

    Bergner, Julia E. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2007 , NUMBER =. doi:10.1090/S0002-9947-06-03987-0 , URL =

    2007 doi

  93. [102]

    , TITLE =

    Bergner, Julia E. , TITLE =. Topology , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.top.2007.03.002 , URL =

    2007 doi

  94. [103]

    , TITLE =

    Bergner, Julia E. , TITLE =. Towards higher categories , SERIES =. 2010 , MRCLASS =. doi:10.1007/978-1-4419-1524-5_2 , URL =

    2010 doi

  95. [104]

    , TITLE =

    Bergner, Julia E. , TITLE =. 2018 , PAGES =

    2018

  96. [105]

    Handbook of Homotopy Theory, edited by Haynes Miller, Chapman & Hall/CRC , pages=

    A survey of models for ( , n) -categories , author=. Handbook of Homotopy Theory, edited by Haynes Miller, Chapman & Hall/CRC , pages=

  97. [106]

    , TITLE =

    Bergner, Julia E. , TITLE =. Mathematical foundations of quantum field theory and perturbative string theory , SERIES =. 2011 , MRCLASS =. doi:10.1090/pspum/083/2742424 , URL =

    2011 doi

  98. [107]

    Gepner, David and Haugseng, Rune , TITLE =. Adv. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.aim.2015.02.007 , URL =

    2015 doi

  99. [108]

    Gepner, David and Groth, Moritz and Nikolaus, Thomas , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2015 , NUMBER =. doi:10.2140/agt.2015.15.3107 , URL =

    2015 doi

  100. [109]

    Forum Math

    Gepner, David and Kock, Joachim , TITLE =. Forum Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1515/forum-2015-0228 , URL =

    2017 doi

  101. [110]

    Gepner, David and Haugseng, Rune and Nikolaus, Thomas , TITLE =. Doc. Math. , FJOURNAL =. 2017 , PAGES =

    2017

  102. [111]

    Chu, Hongyi and Haugseng, Rune , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2019.106913 , URL =

    2020 doi

  103. [112]

    Haugseng, Rune , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/proc/13695 , URL =

    2018 doi

  104. [113]

    Haugseng, Rune , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.1631 , URL =

    2017 doi

  105. [114]

    Haugseng, Rune , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2015 , NUMBER =. doi:10.2140/agt.2015.15.1931 , URL =

    2015 doi

  106. [115]

    2009 , PAGES =

    Lurie, Jacob , TITLE =. 2009 , PAGES =. doi:10.1515/9781400830558 , URL =

    2009 doi

  107. [116]

    arXiv preprint , year=

    (Infinity, 2)-Categories and the Goodwillie Calculus I , author=. arXiv preprint , year=

  108. [117]

    Lurie, Jacob , TITLE =

  109. [118]

    2017 , NOTE =

    Higher algebra , author=. 2017 , NOTE =

    2017

  110. [119]

    2012 , NOTE =

    Higher algebra , author=. 2012 , NOTE =

    2012

  111. [120]

    Spectral

    Lurie, Jacob , month=. Spectral. 2018 , NOTE =

    2018

  112. [121]

    Current developments in mathematics, 2008 , PAGES =

    Lurie, Jacob , TITLE =. Current developments in mathematics, 2008 , PAGES =. 2009 , MRCLASS =

    2008

  113. [122]

    2014 , PAGES =

    Riehl, Emily , TITLE =. 2014 , PAGES =. doi:10.1017/CBO9781107261457 , URL =

    2014 doi

  114. [123]

    2017 , publisher=

    Category theory in context , author=. 2017 , publisher=

    2017

  115. [124]

    Riehl, Emily and Shulman, Michael , TITLE =. High. Struct. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s42001-017-0005-6 , URL =

    2017 doi

  116. [125]

    Riehl, Emily and Verity, Dominic , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2017 , NUMBER =. doi:10.1016/j.jpaa.2016.07.003 , URL =

    2017 doi

  117. [126]

    2021 , urldate =

    Elements of -category theory , author=. 2021 , urldate =

    2021

  118. [127]

    2022 , place=

    Riehl, Emily and Verity, Dominic , title=. 2022 , place=

    2022

  119. [128]

    Riehl, Emily , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1017/S0305004111000053 , URL =

    2011 doi

  120. [129]

    1999 , PAGES =

    Hovey, Mark , TITLE =. 1999 , PAGES =

    1999

  121. [130]

    , TITLE =

    Hirschhorn, Philip S. , TITLE =. 2003 , PAGES =

    2003

  122. [131]

    , TITLE =

    Quillen, Daniel G. , TITLE =. 1967 , PAGES =

    1967

  123. [132]

    Algebraic

    Quillen, Daniel , TITLE =. Algebraic. 1973 , MRCLASS =

    1973

  124. [133]

    and Jardine, John F

    Goerss, Paul G. and Jardine, John F. , TITLE =. 1999 , PAGES =. doi:10.1007/978-3-0348-8707-6 , URL =

    1999 doi

  125. [134]

    Cisinski, Denis-Charles , TITLE =. Ast\'. 2006 , PAGES =

    2006

  126. [135]

    2019 , PAGES =

    Cisinski, Denis-Charles , TITLE =. 2019 , PAGES =. doi:10.1017/9781108588737 , URL =

    2019 doi

  127. [136]

    arXiv preprint arXiv:1112.0040 , year=

    On the unicity of the homotopy theory of higher categories , author=. arXiv preprint arXiv:1112.0040 , year=

  128. [137]

    and Kan, D

    Barwick, C. and Kan, D. M. , TITLE =. Indag. Math. (N.S.) , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.indag.2011.10.002 , URL =

    2012 doi

  129. [138]

    2005 , PAGES =

    Barwick, Clark , TITLE =. 2005 , PAGES =

    2005

  130. [139]

    Theory Appl

    Barwick, Clark and Glasman, Saul and Nardin, Denis , TITLE =. Theory Appl. Categ. , FJOURNAL =. 2018 , PAGES =

    2018

  131. [140]

    Rijke, Egbert and Shulman, Michael and Spitters, Bas , TITLE =. Log. Methods Comput. Sci. , FJOURNAL =. 2020 , NUMBER =

    2020

  132. [141]

    arXiv preprint arXiv:1701.07538 , year=

    The join construction , author=. arXiv preprint arXiv:1701.07538 , year=

    arXiv

  133. [142]

    arXiv preprint , year=

    All ( , 1)-toposes have strict univalent universes , author=. arXiv preprint , year=

  134. [143]

    Homology Homotopy Appl

    Shulman, Michael , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2015 , NUMBER =. doi:10.4310/HHA.2015.v17.n2.a6 , URL =

    2015 doi

  135. [144]

    Shulman, Michael , TITLE =. Math. Structures Comput. Sci. , FJOURNAL =. 2018 , NUMBER =. doi:10.1017/S0960129517000147 , URL =

    2018 doi

  136. [145]

    Kapulkin, Krzysztof and Lumsdaine, Peter LeFanu , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2021 , NUMBER =. doi:10.4171/JEMS/1050 , URL =

    2021 doi

  137. [146]

    Kapulkin, Krzysztof , TITLE =. J. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/topo.12031 , URL =

    2017 doi

  138. [147]

    Internal languages of finitely complete

    Kapulkin, Krzysztof and Szumi. Internal languages of finitely complete. Selecta Math. (N.S.) , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00029-019-0480-0 , URL =

    2019 doi

  139. [148]

    2013 , PAGES =

    The Univalent Foundations Program , TITLE =. 2013 , PAGES =

    2013

  140. [149]

    and Biedermann, G

    Anel, M. and Biedermann, G. and Finster, E. and Joyal, A. , TITLE =. J. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.1112/topo.12082 , URL =

    2018 doi

  141. [150]

    Quasi-categories vs

    Joyal, Andr\'. Quasi-categories vs. Categories in algebra, geometry and mathematical physics , SERIES =. 2007 , MRCLASS =. doi:10.1090/conm/431/08278 , URL =

    2007 doi

  142. [151]

    preprint , year=

    Notes on quasi-categories , author=. preprint , year=

  143. [152]

    2008 , publisher=

    The theory of quasi-categories and its applications , author=. 2008 , publisher=

    2008

  144. [153]

    A generalized

    Anel, Mathieu and Biedermann, Georg and Finster, Eric and Joyal, Andr\'. A generalized. J. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/topo.12163 , URL =

    2020 doi

  145. [154]

    1994 , PAGES =

    Mac Lane, Saunders and Moerdijk, Ieke , TITLE =. 1994 , PAGES =

    1994

  146. [155]

    1998 , PAGES =

    Mac Lane, Saunders , TITLE =. 1998 , PAGES =

    1998

  147. [156]

    , TITLE =

    Johnstone, Peter T. , TITLE =. 2002 , PAGES =

    2002

  148. [157]

    and Johnstone, P

    Adelman, M. and Johnstone, P. T. , TITLE =. Bull. Austral. Math. Soc. , FJOURNAL =. 1982 , NUMBER =. doi:10.1017/S0004972700005086 , URL =

    1982 doi

  149. [158]

    and Varshavski

    Kazhdan, D. and Varshavski. The. Funktsional. Anal. i Prilozhen. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s10688-014-0050-3 , URL =

    2014 doi

  150. [159]

    Handbook of algebraic topology , volume=

    Homotopy theories and model categories , author=. Handbook of algebraic topology , volume=

  151. [160]

    An alpine bouquet of algebraic topology , SERIES =

    Boavida de Brito, Pedro , TITLE =. An alpine bouquet of algebraic topology , SERIES =. 2018 , MRCLASS =. doi:10.1090/conm/708/14271 , URL =

    2018 doi

  152. [161]

    , TITLE =

    Dugger, Daniel and Spivak, David I. , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2011 , NUMBER =. doi:10.2140/agt.2011.11.225 , URL =

    2011 doi

  153. [162]

    Ara, Dimitri , TITLE =. J. K-Theory , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/S1865243315000021 , URL =

    2014 doi

  154. [163]

    Vers une axiomatisation de la th

    To. Vers une axiomatisation de la th. K -Theory , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s10977-005-4556-6 , URL =

    2005 doi

  155. [164]

    Dwyer, W. G. and Kan, D. M. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1980 , NUMBER =. doi:10.1016/0022-4049(80)90049-3 , URL =

    1980 doi

  156. [165]

    Dwyer, W. G. and Kan, D. M. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1980 , NUMBER =. doi:10.1016/0022-4049(80)90113-9 , URL =

    1980 doi

  157. [166]

    Heuts, Gijs and Moerdijk, Ieke , TITLE =. Math. Z. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00209-014-1390-7 , URL =

    2015 doi

  158. [167]

    arXiv preprint , year=

    Left fibrations and homotopy colimits II , author=. arXiv preprint , year=

  159. [168]

    Topology and quantum theory in interaction , SERIES =

    Ayala, David and Francis, John , TITLE =. Topology and quantum theory in interaction , SERIES =. 2018 , MRCLASS =. doi:10.1090/conm/718/14489 , URL =

    2018 doi

  160. [169]

    Ayala, David and Francis, John , TITLE =. High. Struct. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/jhep01(2020)036 , URL =

    2020 doi

  161. [170]

    Calaque, Damien and Scheimbauer, Claudia , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/agt.2019.19.533 , URL =

    2019 doi

  162. [171]

    North-West

    Stevenson, Danny , TITLE =. North-West. Eur. J. Math. , FJOURNAL =. 2017 , PAGES =

    2017

  163. [172]

    1982 , PAGES =

    Kelly, Gregory Maxwell , TITLE =. 1982 , PAGES =

    1982

  164. [173]

    Kelly, G. M. , TITLE =. Repr. Theory Appl. Categ. , FJOURNAL =. 2005 , PAGES =

    2005

  165. [174]

    Verity, D. R. B. , TITLE =. Adv. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.aim.2008.06.003 , URL =

    2008 doi

  166. [175]

    arXiv preprint , year=

    Cubical models of ( ,1) -categories , author=. arXiv preprint , year=

  167. [176]

    arXiv preprint , year=

    Equivalence of cubical and simplicial approaches to ( ,n) -categories , author=. arXiv preprint , year=

  168. [177]

    arXiv preprint , year=

    A cubical model for ( ,n) -categories , author=. arXiv preprint , year=

  169. [178]

    Fibred categories

    Streicher, Thomas , journal=. Fibred categories. 2018 , NOTE =

    2018

  170. [179]

    and Scott, P

    Lambek, J. and Scott, P. J. , TITLE =. 1988 , PAGES =

    1988

  171. [180]

    Au sujet de l'existence d'adjoints \`a droite aux foncteurs ``image r\'

    Conduch\'. Au sujet de l'existence d'adjoints \`a droite aux foncteurs ``image r\'. C. R. Acad. Sci. Paris S\'. 1972 , PAGES =

    1972

  172. [181]

    Boardman, J. M. and Vogt, R. M. , TITLE =. 1973 , PAGES =

    1973

  173. [182]

    arXiv preprint , year=

    Covariant & Contravariant Homotopy Theories , author=. arXiv preprint , year=

  174. [183]

    1974 , PAGES =

    Reedy, Christopher Leonard , TITLE =. 1974 , PAGES =

    1974

  175. [184]

    arXiv preprint , year=

    Localization theory in an -topos , author=. arXiv preprint , year=

  176. [185]

    Daniel and Opie, Morgan and Rijke, Egbert and Scoccola, Luis , TITLE =

    Christensen, J. Daniel and Opie, Morgan and Rijke, Egbert and Scoccola, Luis , TITLE =. High. Struct. , FJOURNAL =. 2020 , NUMBER =

    2020

  177. [186]

    arXiv preprint arXiv:2007.05833 , year=

    The Hurewicz theorem in Homotopy Type Theory , author=. arXiv preprint arXiv:2007.05833 , year=

    2007

  178. [187]

    arXiv preprint arXiv:1606.05916 , year=

    On the homotopy groups of spheres in homotopy type theory , author=. arXiv preprint arXiv:1606.05916 , year=

    arXiv

  179. [188]

    arXiv preprint arXiv:1806.05966 , year=

    Cartan Geometry in Modal Homotopy Type Theory , author=. arXiv preprint arXiv:1806.05966 , year=

  180. [189]

    , TITLE =

    Hochschild, G. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1945 , PAGES =. doi:10.2307/1969145 , URL =

    1945 doi

  181. [190]

    1956 , PAGES =

    Cartan, Henri and Eilenberg, Samuel , TITLE =. 1956 , PAGES =

    1956

  182. [191]

    1998 , PAGES =

    Loday, Jean-Louis , TITLE =. 1998 , PAGES =. doi:10.1007/978-3-662-11389-9 , URL =

    1998 doi

  183. [192]

    An invitation to noncommutative geometry , PAGES =

    Connes, Alain and Marcolli, Matilde , TITLE =. An invitation to noncommutative geometry , PAGES =. 2008 , MRCLASS =. doi:10.1142/9789812814333\_0001 , URL =

    2008 doi

  184. [193]

    Algebraic topology,

    Waldhausen, Friedhelm , TITLE =. Algebraic topology,. 1979 , MRCLASS =

    1979

  185. [194]

    Algebraic and geometric topology (

    Waldhausen, Friedhelm , TITLE =. Algebraic and geometric topology (. 1978 , MRCLASS =

    1978

  186. [195]

    Unpublished lecture in Algebraic K-theory, Evanston, IL

    Algebraic K-theory and Hochschild homology , author=. Unpublished lecture in Algebraic K-theory, Evanston, IL. , year=

  187. [196]

    Topological

    B. Topological. Preprint Universit

  188. [197]

    Elmendorf, A. D. and Kriz, I. and Mandell, M. A. and May, J. P. , TITLE =. 1997 , PAGES =. doi:10.1090/surv/047 , URL =

    1997 doi

  189. [198]

    Algebraic and geometric topology (

    Waldhausen, Friedhelm , TITLE =. Algebraic and geometric topology (. 1985 , MRCLASS =. doi:10.1007/BFb0074449 , URL =

    1985 doi

  190. [199]

    McCarthy, Randy , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1994 , NUMBER =. doi:10.1016/0022-4049(94)90091-4 , URL =

    1994 doi

  191. [200]

    Keller, Bernhard , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1999 , NUMBER =. doi:10.1016/S0022-4049(97)00152-7 , URL =

    1999 doi

  192. [201]

    and Mandell, Michael A

    Blumberg, Andrew J. and Mandell, Michael A. , TITLE =. Geom. Topol. , FJOURNAL =. 2012 , NUMBER =. doi:10.2140/gt.2012.16.1053 , URL =

    2012 doi

  193. [202]

    and Gepner, David and Tabuada, Gon

    Blumberg, Andrew J. and Gepner, David and Tabuada, Gon. A universal characterization of higher algebraic. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.733 , URL =

    2013 doi

  194. [203]

    and Gepner, David and Tabuada, Gon

    Blumberg, Andrew J. and Gepner, David and Tabuada, Gon. Uniqueness of the multiplicative cyclotomic trace , JOURNAL =. 2014 , PAGES =. doi:10.1016/j.aim.2014.02.004 , URL =

    2014 doi

  195. [204]

    arXiv preprint , year=

    K-theory and polynomial functors , author=. arXiv preprint , year=

  196. [205]

    Acta Math

    Nikolaus, Thomas and Scholze, Peter , TITLE =. Acta Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/ACTA.2019.v222.n1.a2 , URL =

    2019 doi

  197. [206]

    Acta Math

    Nikolaus, Thomas and Scholze, Peter , TITLE =. Acta Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.4310/ACTA.2018.v221.n2.a1 , URL =

    2018 doi

  198. [207]

    Hess, Kathryn and Shipley, Brooke , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2021 , NUMBER =. doi:10.1016/j.jpaa.2020.106505 , URL =

    2021 doi

  199. [208]

    Homology Homotopy Appl

    Moser, Lyne , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/HHA.2019.v21.n2.a15 , URL =

    2019 doi

  200. [209]

    Heller, Alex , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1983 , NUMBER =. doi:10.2307/1999394 , URL =

    1983 doi

  201. [210]

    Heller, Alex , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1982 , NUMBER =. doi:10.2307/1998955 , URL =

    1982 doi

  202. [211]

    Max , TITLE =

    Eilenberg, Samuel and Kelly, G. Max , TITLE =. Proc. 1966 , MRCLASS =

    1966

  203. [212]

    Topology Appl

    Rezk, Charles , TITLE =. Topology Appl. , FJOURNAL =. 2002 , NUMBER =. doi:10.1016/S0166-8641(01)00057-8 , URL =

    2002 doi

  204. [213]

    A necessary and sufficient condition for induced model structures , JOURNAL =

    Hess, Kathryn and K. A necessary and sufficient condition for induced model structures , JOURNAL =. 2017 , NUMBER =. doi:10.1112/topo.12011 , URL =

    2017 doi

  205. [214]

    and Rezk, Charles , TITLE =

    Bergner, Julia E. and Rezk, Charles , TITLE =. Math. Z. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00209-012-1082-0 , URL =

    2013 doi

  206. [215]

    arXiv preprint , year=

    Fibrations and lax limits of ( ,2) -categories , author=. arXiv preprint , year=

  207. [216]

    arXiv preprint , year=

    Cartesian Fibrations of ( ,2) -categories , author=. arXiv preprint , year=

  208. [217]

    arXiv preprint , year=

    Bilimits are Bifinal Objects , author=. arXiv preprint , year=

  209. [218]

    arXiv preprint , year=

    Gray tensor products and lax functors of ( ,2) -categories , author=. arXiv preprint , year=

  210. [219]

    arXiv preprint , year=

    On the equivalence of all models for ( ,2) -categories , author=. arXiv preprint , year=

  211. [220]

    Beardsley, Jonathan and Wong, Liang Ze , TITLE =. Adv. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aim.2018.12.009 , URL =

    2019 doi

  212. [221]

    arXiv preprint , year=

    A double ( ,1) -categorical nerve for double categories , author=. arXiv preprint , year=

  213. [222]

    arXiv preprint , year=

    Bi-initial objects and bi-representations are not so different , author=. arXiv preprint , year=

  214. [223]

    arXiv preprint , year=

    2-limits and 2-terminal objects are too different , author=. arXiv preprint , year=

  215. [224]

    arXiv preprint , year=

    A 2Cat-inspired model structure for double categories , author=. arXiv preprint , year=

  216. [225]

    arXiv preprint , year=

    A model structure for weakly horizontally invariant double categories , author=. arXiv preprint , year=

  217. [226]

    Theory Appl

    Moeller, Joe and Vasilakopoulou, Christina , TITLE =. Theory Appl. Categ. , FJOURNAL =. 2020 , PAGES =

    2020

  218. [227]

    Mazel-Gee, Aaron , TITLE =. Grad. J. Math. , FJOURNAL =. 2019 , NUMBER =

    2019

  219. [228]

    Mazel-Gee, Aaron , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2019 , NUMBER =. doi:10.1016/j.jpaa.2019.02.007 , URL =

    2019 doi

  220. [229]

    arXiv preprint , year=

    Synthetic fibered ( ,1) -category theory , author=. arXiv preprint , year=

  221. [230]

    arXiv preprint , year=

    Marked colimits and higher cofinality , author=. arXiv preprint , year=

  222. [231]

    arXiv preprint , year=

    Theorem A for marked 2 -categories , author=. arXiv preprint , year=

  223. [232]

    arXiv preprint , year=

    Enhanced twisted arrow categories , author=. arXiv preprint , year=

  224. [233]

    arXiv preprint , year=

    2-Cartesian fibrations I: A model for -bicategories fibred in -bicategories , author=. arXiv preprint , year=

  225. [234]

    arXiv preprint , year=

    2-Cartesian fibrations II: Higher cofinality , author=. arXiv preprint , year=

  226. [235]

    arXiv preprint , year=

    An explicit comparison between 2 -complicial sets and _2 -spaces , author=. arXiv preprint , year=

  227. [236]

    arXiv preprint , year=

    ( , 1) -Categorical Comprehension Schemes , author=. arXiv preprint , year=

  228. [237]

    Univalence and completeness of

    Stenzel, Raffael , journal=. Univalence and completeness of. 2019 , NOTE =

    2019

  229. [238]

    Mathew, Akhil and Stojanoska, Vesna , TITLE =. Geom. Topol. , FJOURNAL =. 2016 , NUMBER =. doi:10.2140/gt.2016.20.3133 , URL =

    2016 doi

  230. [239]

    2003 , PAGES =

    Rev\^. 2003 , PAGES =

    2003

  231. [240]

    Kelly, G. M. and Street, Ross , TITLE =. Category. 1974 , MRCLASS =

    1974

  232. [241]

    2017 , PAGES =

    Gaitsgory, Dennis and Rozenblyum, Nick , TITLE =. 2017 , PAGES =. doi:10.1090/surv/221.2 , URL =

    2017 doi

  233. [242]

    2017 , PAGES =

    Gaitsgory, Dennis and Rozenblyum, Nick , TITLE =. 2017 , PAGES =. doi:10.1090/surv/221.1 , URL =

    2017 doi

  234. [243]

    arXiv preprint , year=

    Descente pour les n-champs (Descent for n-stacks) , author=. arXiv preprint , year=

  235. [244]

    Mathematical foundations of quantum field theory and perturbative string theory , SERIES =

    Stolz, Stephan and Teichner, Peter , TITLE =. Mathematical foundations of quantum field theory and perturbative string theory , SERIES =. 2011 , MRCLASS =. doi:10.1090/pspum/083/2742432 , URL =

    2011 doi

  236. [245]

    Witten, Edward , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1988 , NUMBER =

    1988

  237. [246]

    arXiv preprint , year=

    REFLECTION POSITIVITY AND INVERTIBLE TOPOLOGICAL PHASES , author=. arXiv preprint , year=

  238. [247]

    and Dolan, James , TITLE =

    Baez, John C. and Dolan, James , TITLE =. J. Math. Phys. , FJOURNAL =. 1995 , NUMBER =. doi:10.1063/1.531236 , URL =

    1995 doi

  239. [248]

    , TITLE =

    Mathieu, Philippe and Murray, Laura and Schenkel, Alexander and Teh, Nicholas J. , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s11005-020-01269-x , URL =

    2020 doi

  240. [249]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. , TITLE =. Geom. Topol. , FJOURNAL =. 2021 , NUMBER =. doi:10.2140/gt.2021.25.1165 , URL =

    2021 doi

  241. [250]

    , TITLE =

    Turaev, Vladimir G. , TITLE =. 2010 , PAGES =. doi:10.1515/9783110221848 , URL =

    2010 doi

  242. [251]

    2019 , PAGES =

    Dyckerhoff, Tobias and Kapranov, Mikhail , TITLE =. 2019 , PAGES =. doi:10.1007/978-3-030-27124-4 , URL =

    2019 doi

  243. [252]

    2018 , journal=

    On the directed univalence axiom , author=. 2018 , journal=

    2018

  244. [253]

    Weber, Mark , TITLE =. Appl. Categ. Structures , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s10485-007-9079-2 , URL =

    2007 doi

  245. [254]

    arXiv preprint , year=

    The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory , author =. arXiv preprint , year=

  246. [255]

    preprint , NOTE =

    Disks, duality and -categories , author=. preprint , NOTE =

  247. [256]

    Hinich, Vladimir , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2020.107129 , URL =

    2020 doi

  248. [257]

    arXiv preprint , year=

    Enriched infinity categories I: enriched presheaves , author =. arXiv preprint , year=

  249. [258]

    arXiv preprint , year=

    On lax limits in infinity categories , author =. arXiv preprint , year=

  250. [259]

    Harpaz, Yonatan and Nuiten, Joost and Prasma, Matan , TITLE =. High. Struct. , FJOURNAL =. 2019 , NUMBER =

    2019

  251. [260]

    William , TITLE =

    Lawvere, F. William , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1964 , PAGES =. doi:10.1073/pnas.52.6.1506 , URL =

    1964 doi

  252. [261]

    , TITLE =

    Tierney, M. , TITLE =. Categories and commutative algebra (. 1973 , MRCLASS =

    1973

  253. [262]

    Toposes, algebraic geometry and logic (

    Tierney, Myles , TITLE =. Toposes, algebraic geometry and logic (. 1972 , MRCLASS =

    1972

  254. [263]

    , TITLE =

    Basterra, Maria and Mandell, Michael A. , TITLE =. Math. Z. , FJOURNAL =. 2005 , NUMBER =

    2005

  255. [264]

    Schlichtkrull, Christian , TITLE =. Doc. Math. , FJOURNAL =. 2009 , PAGES =

    2009

  256. [265]

    Topology Appl

    Stonek, Bruno , TITLE =. Topology Appl. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.topol.2020.107302 , URL =

    2020 doi

  257. [266]

    and Gepner, David and Hopkins, Michael J

    Ando, Matthew and Blumberg, Andrew J. and Gepner, David and Hopkins, Michael J. and Rezk, Charles , TITLE =. J. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.1112/jtopol/jtt035 , URL =

    2014 doi

  258. [267]

    and Gepner, David , TITLE =

    Ando, Matthew and Blumberg, Andrew J. and Gepner, David , TITLE =. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.2140/gt.2018.22.3761 , URL =

    2018 doi

  259. [268]

    A simple universal property of

    Antol\'. A simple universal property of. J. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/topo.12084 , URL =

    2019 doi

  260. [269]

    Andr\'. M\'. 1967 , PAGES =

    1967

  261. [270]

    Applications of

    Quillen, Daniel , TITLE =. Applications of. 1970 , MRCLASS =

    1970

  262. [271]

    Ponto, Kate , TITLE =. Ast\'. 2010 , PAGES =

    2010

  263. [272]

    and Ponto, Kate , TITLE =

    Campbell, Jonathan A. and Ponto, Kate , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/agt.2019.19.965 , URL =

    2019 doi

  264. [273]

    2019 , NOTE =

    Berman, John D , journal=. 2019 , NOTE =

    2019

  265. [274]

    Coalgebras in symmetric monoidal categories of spectra , JOURNAL =

    P\'. Coalgebras in symmetric monoidal categories of spectra , JOURNAL =. 2019 , NUMBER =. doi:10.4310/HHA.2019.v21.n1.a1 , URL =

    2019 doi

  266. [275]

    Cahiers Topologie G\'

    Schanuel, Stephen and Street, Ross , TITLE =. Cahiers Topologie G\'. 1986 , NUMBER =

    1986

  267. [276]

    K -Theory , FJOURNAL =

    Lack, Stephen , TITLE =. K -Theory , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s10977-004-6757-9 , URL =

    2004 doi

  268. [277]

    K -Theory , FJOURNAL =

    Lack, Stephen , TITLE =. K -Theory , FJOURNAL =. 2002 , NUMBER =. doi:10.1023/A:1020305604826 , URL =

    2002 doi

  269. [278]

    Street, Ross , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1976 , NUMBER =. doi:10.1016/0022-4049(76)90013-X , URL =

    1976 doi

  270. [279]

    1976 , PAGES =

    Mikkelsen, Christian Juul , TITLE =. 1976 , PAGES =

    1976

  271. [280]

    Talk at the conference on category theory at Oberwolfach , year=

    Finite colimits in toposes , author=. Talk at the conference on category theory at Oberwolfach , year=

  272. [281]

    Bulletin of the American Mathematical Society , volume=

    Colimits in topoi , author=. Bulletin of the American Mathematical Society , volume=

  273. [282]

    William , TITLE =

    Lawvere, F. William , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1963 , PAGES =. doi:10.1073/pnas.50.5.869 , URL =

    1963 doi

  274. [283]

    Palmgren, Erik , TITLE =. Ann. Pure Appl. Logic , FJOURNAL =. 1997 , NUMBER =. doi:10.1016/S0168-0072(96)00041-3 , URL =

    1997 doi

  275. [284]

    Raptis, Georgios and Strunk, Florian , TITLE =. Doc. Math. , FJOURNAL =. 2018 , PAGES =

    2018

  276. [285]

    , TITLE =

    Isaksen, Daniel C. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/memo/1269 , URL =

    2019 doi

  277. [286]

    arXiv preprint , year=

    Bay. arXiv preprint , year=

  278. [287]

    Hess, Kathryn , TITLE =. J. Algebra , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jalgebra.2015.11.040 , URL =

    2016 doi

  279. [288]

    On straightening for

    Nuiten, Joost , journal=. On straightening for. 2021 , NOTE =

    2021

  280. [289]

    Thomason, R. W. , TITLE =. Duke Math. J. , FJOURNAL =. 1979 , NUMBER =

    1979

  281. [290]

    , TITLE =

    Satake, I. , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1956 , PAGES =. doi:10.1073/pnas.42.6.359 , URL =

    1956 doi

  282. [291]

    Satake, Ichir\^. The. J. Math. Soc. Japan , FJOURNAL =. 1957 , PAGES =. doi:10.2969/jmsj/00940464 , URL =

    1957 doi

  283. [292]

    Atiyah, Michael , TITLE =. Inst. Hautes \'. 1988 , PAGES =

    1988

  284. [293]

    About models for intuitionistic type theories and the notion of definitional equality , BOOKTITLE =

    Martin-L\". About models for intuitionistic type theories and the notion of definitional equality , BOOKTITLE =. 1975 , MRCLASS =

    1975

  285. [294]

    Martin Hofmann’s contributions to type theory: Groupoids and univalence , DOI=

    Altenkirch, Thorsten , year=. Martin Hofmann’s contributions to type theory: Groupoids and univalence , DOI=. Mathematical Structures in Computer Science , publisher=

  286. [295]

    Orbifolds in mathematics and physics (

    Moerdijk, Ieke , TITLE =. Orbifolds in mathematics and physics (. 2002 , MRCLASS =. doi:10.1090/conm/310/05405 , URL =

    2002 doi

  287. [296]

    The Institute Letter , pages=

    The origins and motivations of univalent foundations , author=. The Institute Letter , pages=

  288. [297]

    van den Berg, Benno and Moerdijk, Ieke , TITLE =. Math. Ann. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00208-017-1614-3 , URL =

    2018 doi

  289. [298]

    Ahrens, Benedikt and Kapulkin, Krzysztof and Shulman, Michael , TITLE =. Math. Structures Comput. Sci. , FJOURNAL =. 2015 , NUMBER =. doi:10.1017/S0960129514000486 , URL =

    2015 doi

  290. [299]

    Quelques propri\'

    Thom, Ren\'. Quelques propri\'. Comment. Math. Helv. , FJOURNAL =. 1954 , PAGES =. doi:10.1007/BF02566923 , URL =

    1954 doi

  291. [300]

    , TITLE =

    Milnor, J. , TITLE =. Amer. J. Math. , FJOURNAL =. 1960 , PAGES =. doi:10.2307/2372970 , URL =

    1960 doi

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.