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REVIEW 3 major objections 2 minor 299 references

On Galois categories and condensed contractible schemes

T0 review · 3 major / 2 minor · reviewed 2026-05-12 · grok-4.3

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Pith's one-line read The condensed fundamental group of Spec(Z) is non-trivial, so the spectrum of the integers is not condensed contractible.

desk verdict The paper classifies schemes with initial or terminal Galois objects to get trivial condensed homotopy types and gives an explicit formula showing the condensed fundamental group of Spec(Z) is non-trivial. read the letter →

arxiv 2605.10358 v1 submitted 2026-05-11 math.AG math.ATmath.CT

Catrin Mair This is my paper · ORCID
classification math.AGmath.ATmath.CT
keywords condensedGaloiscategoryhomotopytypefundamentalgroupDedekinddomainSpec(Z)w-contractibleringsultracategories
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 extends the condensed Galois category of a scheme, first introduced in work on exodromy, and relates it to Lurie's ultracategories through a description in terms of w-contractible rings. It classifies schemes whose Galois category admits an initial or terminal object; such schemes have trivial condensed homotopy type. For any Dedekind domain the underlying group of the condensed fundamental group is given by an explicit formula. In the special case of the integers this group is shown to be non-trivial, proving that Spec(Z) fails to be condensed contractible.

What carries the argument

The condensed Galois category of a scheme, extended from prior work and described in terms of w-contractible rings and Lurie's ultracategories, which classifies objects with initial or terminal objects and computes the condensed fundamental group.

What would settle it

An explicit calculation of the condensed fundamental group of Spec(Z) that returns the trivial group rather than a non-trivial one.

Watch

Extended reading notes

Core claim

Extending the condensed Galois category of a scheme and describing it via w-contractible rings and ultracategories yields a classification of schemes with initial or terminal objects in their Galois category, implying trivial condensed homotopy type, together with an explicit formula for the condensed fundamental group of a general Dedekind domain; this group is non-trivial when the domain is the integers, so Spec(Z) is not condensed contractible.

Load-bearing premise

The extension of the condensed Galois category to schemes and Dedekind domains continues to satisfy the expected formal properties of Galois categories.

Editorial extensions

If this is right

  • Schemes whose condensed Galois category has an initial or terminal object have trivial condensed homotopy type.
  • The condensed fundamental group of any Dedekind domain is given by a concrete formula.
  • Spec(Z) is not condensed contractible.
  • The condensed homotopy type of a scheme is determined by the existence of initial or terminal objects in its Galois category.

Reading between the lines

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

  • The distinction between condensed and classical homotopy types may be visible already at the level of the fundamental group for arithmetic schemes.
  • The same methods could be applied to compute condensed fundamental groups of other number rings or to compare with pro-etale homotopy types.
  • Non-contractibility of Spec(Z) suggests that condensed homotopy theory retains arithmetic information that would be lost if every scheme were treated as contractible.
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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

3 major / 2 minor

Summary. The manuscript extends the condensed Galois category of schemes from Barwick-Glasman-Haine by connecting it to Lurie's ultracategories and providing a description in terms of w-contractible rings. It classifies schemes whose Galois categories have initial or terminal objects (implying trivial condensed homotopy types) and derives an explicit formula for the underlying group of the condensed fundamental group of a general Dedekind domain, proving this group is non-trivial for Spec(Z) and hence that Spec(Z) is not condensed contractible.

Significance. If the w-contractible ring description faithfully reproduces the original condensed Galois category (including morphisms and Galois actions) and the formula is correctly derived, the work would supply a concrete computational tool for condensed fundamental groups of Dedekind domains and establish a non-trivial condensed homotopy type for Spec(Z). This strengthens the structural theory of condensed Galois categories and provides a concrete counterexample to contractibility for the most basic arithmetic scheme, with potential implications for exodromy and condensed homotopy in arithmetic geometry.

major comments (3)
  1. [Description of the condensed Galois category via w-contractible rings] The section introducing the description of the condensed Galois category via w-contractible rings and Lurie's ultracategories must explicitly verify that the new category reproduces the objects, morphisms, and Galois actions of the Barwick-Glasman-Haine category when restricted to Dedekind domains. The non-triviality result for Spec(Z) is load-bearing on this equivalence; any mismatch would render the formula an artifact of the new presentation rather than a property of the original condensed Galois category.
  2. [Formula for the condensed fundamental group of Dedekind domains] In the computation of the formula for the underlying group of the condensed fundamental group of a general Dedekind domain, the derivation should include a direct reduction or comparison showing that the formula agrees with the profinite completion extracted from the original Barwick-Glasman-Haine category (rather than solely from the ultracategory description). The non-triviality claim for Spec(Z) requires an explicit example or computation confirming the group is non-trivial under the original definition.
  3. [Classification of schemes with initial/terminal objects] The classification of schemes with initial or terminal objects in the Galois category is used to deduce trivial condensed homotopy types. The paper should clarify how this classification interacts with the condensed structure when applied to Spec(Z) or other Dedekind domains, particularly whether the initial/terminal object property survives the extension to w-contractible rings.
minor comments (2)
  1. [Introduction and notation] Notation for the 'underlying group' of the condensed fundamental group should be defined more explicitly, including how it is extracted from the ultracategory or Galois category.
  2. [Throughout] The manuscript would benefit from additional cross-references to specific results in Barwick-Glasman-Haine and Lurie to make the extension steps easier to follow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and indicate the revisions we will make to strengthen the presentation and ensure the results are firmly grounded in the original Barwick-Glasman-Haine framework.

read point-by-point responses

  1. Referee: [Description of the condensed Galois category via w-contractible rings] The section introducing the description of the condensed Galois category via w-contractible rings and Lurie's ultracategories must explicitly verify that the new category reproduces the objects, morphisms, and Galois actions of the Barwick-Glasman-Haine category when restricted to Dedekind domains. The non-triviality result for Spec(Z) is load-bearing on this equivalence; any mismatch would render the formula an artifact of the new presentation rather than a property of the original condensed Galois category.

    Authors: We agree that an explicit verification of the equivalence is required. In the revised manuscript we will add a dedicated subsection that constructs functors in both directions between the w-contractible ring description and the Barwick-Glasman-Haine condensed Galois category (restricted to Dedekind domains), verifies that these functors are inverse equivalences, and checks that they preserve morphisms and Galois actions. This will confirm that the non-triviality result for Spec(Z) holds with respect to the original category. revision: yes

  2. Referee: [Formula for the condensed fundamental group of Dedekind domains] In the computation of the formula for the underlying group of the condensed fundamental group of a general Dedekind domain, the derivation should include a direct reduction or comparison showing that the formula agrees with the profinite completion extracted from the original Barwick-Glasman-Haine category (rather than solely from the ultracategory description). The non-triviality claim for Spec(Z) requires an explicit example or computation confirming the group is non-trivial under the original definition.

    Authors: We will insert a direct comparison proposition that reduces the derived formula to the profinite completion obtained from the original Barwick-Glasman-Haine category via the equivalence established in the preceding subsection. For the non-triviality on Spec(Z) we will add an explicit computation under the original definition, for instance by exhibiting a specific condensed cover arising from an arithmetic extension whose Galois action remains non-trivial after condensation. revision: yes

  3. Referee: [Classification of schemes with initial/terminal objects] The classification of schemes with initial or terminal objects in the Galois category is used to deduce trivial condensed homotopy types. The paper should clarify how this classification interacts with the condensed structure when applied to Spec(Z) or other Dedekind domains, particularly whether the initial/terminal object property survives the extension to w-contractible rings.

    Authors: We will add a short clarifying paragraph immediately after the classification statement. It will note that the condensed homotopy type is functorially determined by the Galois category, so the existence of an initial or terminal object directly yields a trivial condensed homotopy type. Because the equivalence with the w-contractible ring description preserves categorical limits and colimits, the initial/terminal object property is invariant under the extension; consequently the classification applies unchanged to Dedekind domains such as Spec(Z), which lack such objects. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; new formula and non-triviality result for Spec(Z) are independent computations

full rationale

The paper extends the condensed Galois category from Barwick-Glasman-Haine using Lurie's ultracategories and a description via w-contractible rings, then classifies schemes with initial/terminal objects to deduce trivial homotopy type (referencing a prior study) and derives an explicit formula for the condensed fundamental group of Dedekind domains. The non-triviality for Spec(Z) follows from applying this formula to Z, without any reduction of the output to a fitted parameter, self-definition, or load-bearing self-citation chain. The framework is applied to standard properties of Dedekind domains and schemes, making the central claims externally falsifiable and not equivalent to the inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Relies on prior definitions of condensed Galois categories and ultracategories; no new free parameters or invented entities introduced in the abstract.

assumptions (1)
  • domain assumption Standard properties of schemes, Dedekind domains, and Galois categories as in Barwick-Glasman-Haine and Lurie.
    The extensions and classifications presuppose these background structures.

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Cite this review

Pith. "Pith review of On Galois categories and condensed contractible schemes." pith.science (2026). https://pith.science/paper/2605.10358

@misc{pith2026260510358,
  author       = {Pith},
  title        = {Pith review of: On Galois categories and condensed contractible schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2605.10358}},
  note         = {Machine review of arXiv:2605.10358}
}
read the original abstract

We extend the study of the condensed Galois category of a scheme introduced by Barwick, Glasman and Haine in their work on Exodromy. We elaborate its connection to Lurie's work on Ultracategories and provide a description in terms of w-contractible rings. We give a classification of schemes whose Galois category has an initial, respectively, a terminal object. This implies the condensed homotopy type of the scheme, which was studied in more detail in [arXiv:2510.07443v1], to be trivial. Furthermore, we compute a formula for the (underlying group of the) condensed fundamental group of a general Dedekind domain and show that it is non-trivial for the spectrum of the integers Spec(Z).This means that Spec(Z) is not condensed contractible.

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Works this paper leans on

299 extracted references · 299 canonical work pages

  1. [1]

    Haine and Sebastian Wolf , year=

    Magnus Carlson and Peter J. Haine and Sebastian Wolf , year=. Reconstruction of schemes from their \'. 2407.19920 , archivePrefix=

    work page arXiv

  2. [2]

    , TITLE =

    Haine, Peter J. , TITLE =. 2025 , NOTE =

    work page 2025

  3. [3]

    Hebestreit, Fabian and Steinebrunner, Jan , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2025 , NUMBER =. doi:10.1093/imrn/rnaf021 , URL =

    work page doi:10.1093/imrn/rnaf021 2025

  4. [4]

    Vechtomov, E. M. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1992 , NUMBER =. doi:10.1070/RM1992v047n05ABEH000961 , URL =

    work page doi:10.1070/rm1992v047n05abeh000961 1992

  5. [5]

    Vechtomov, E. M. , TITLE =. J. Math. Sci. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/BF02363066 , URL =

    work page doi:10.1007/bf02363066 1996

  6. [6]

    Vechtomov, E. M. , TITLE =. Mat. Zametki , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF02110350 , URL =

    work page doi:10.1007/bf02110350 1994

  7. [7]

    , TITLE =

    Munkres, James R. , TITLE =. 2000 , PAGES =

    work page 2000

  8. [8]

    Galois Cohomology and Class Field Theory , url =

    Harari, David , year =. Galois Cohomology and Class Field Theory , url =

    work page

Show all 299 references
  1. [9]

    1976 , PAGES =

    Gillman, Leonard and Jerison, Meyer , TITLE =. 1976 , PAGES =

    1976

  2. [10]

    Jean-Pierre , TITLE =

    Serre. Jean-Pierre , TITLE =. 1980 , PAGES =

    1980

  3. [11]

    and Prestel, Alexander , year =

    Engler, Antonio J. and Prestel, Alexander , year =. Valued fields , url =

  4. [12]

    Algebraic number theory , address =

    Neukirch, Jürgen , year =. Algebraic number theory , address =

  5. [13]

    Séminaire

    Tate, John , TITLE =. Séminaire. 1995 , MRCLASS =

    1995

  6. [14]

    Feng, Tony , TITLE =. Compos. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/s0010437x20007216 , URL =

    2020 doi

  7. [15]

    2018 , PAGES =

    Fujiwara, Kazuhiro and Kato, Fumiharu , TITLE =. 2018 , PAGES =

    2018

  8. [16]

    Martini, Louis and Wolf, Sebastian , TITLE =. High. Struct. , FJOURNAL =. 2024 , NUMBER =

    2024

  9. [17]

    Krull, Wolfgang and Neukirch, Jürgen , TITLE =. Math. Ann. , FJOURNAL =. 1971 , PAGES =. doi:10.1007/BF02052391 , URL =

    1971 doi

  10. [18]

    Wolf, Sebastian , TITLE =. Doc. Math. , FJOURNAL =. 2022 , PAGES =

    2022

  11. [19]

    2025 , NOTE =

    Wolf, Sebastian , TITLE =. 2025 , NOTE =

    2025

  12. [20]

    , TITLE =

    Haine, Peter J. , TITLE =. 2024 , NOTE =

    2024

  13. [21]

    , TITLE =

    Haine, Peter J. , TITLE =. 2022 , NOTE =

    2022

  14. [22]

    2024 , NOTE =

    Aoki, Ko , TITLE =. 2024 , NOTE =

    2024

  15. [23]

    Selecta Math

    Clausen, Dustin and Jansen, Mikala Ørsnes , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00029-023-00900-8 , URL =

    2024 doi

  16. [24]

    Douady, Adrien , TITLE =. C. R. Acad. Sci. Paris , FJOURNAL =. 1964 , PAGES =

    1964

  17. [25]

    Pacific J

    Haran, Dan and Jarden, Moshe , TITLE =. Pacific J. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.2140/pjm.2000.196.445 , URL =

    2000 doi

  18. [26]

    Recent developments in the inverse

    Jarden, Moshe , TITLE =. Recent developments in the inverse. 1995 , ISBN =. doi:10.1090/conm/186/02192 , URL =

    1995 doi

  19. [27]

    Moerdijk, Ieke and Nuiten, Joost , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/agt.2020.20.1769 , URL =

    2020 doi

  20. [28]

    Animated Condensed Sets and Their Homotopy Groups , Year =

    Mair, Catrin , Month =. Animated Condensed Sets and Their Homotopy Groups , Year =

  21. [29]

    2026 , eprint=

    Classifying anima of condensed -categories of points , author=. 2026 , eprint=

    2026

  22. [31]

    2022 , NOTE =

    Martini, Louis , TITLE =. 2022 , NOTE =

    2022

  23. [32]

    2022 , NOTE =

    Ramzi, Maxime , TITLE =. 2022 , NOTE =

    2022

  24. [33]

    and Ramzi, Maxime and Steinebrunner, Jan , TITLE =

    Haine, Peter J. and Ramzi, Maxime and Steinebrunner, Jan , TITLE =. 2022 , NOTE =

    2022

  25. [34]

    Yoneda's lemma for internal higher categories , Month =

    Martini, Louis , Note =. Yoneda's lemma for internal higher categories , Month =

  26. [35]

    Constructible sheaves on schemes and a categorical

    Hemo, Tamir and Richarz, Timo and Scholbach, Jakob , Month =. Constructible sheaves on schemes and a categorical

  27. [36]

    A categorical Künneth formula for constructible Weil sheaves , volume=

    Hemo, Tamir and Richarz, Timo and Scholbach, Jakob , year=. A categorical Künneth formula for constructible Weil sheaves , volume=. Algebra & Number Theory , publisher=. doi:10.2140/ant.2024.18.499 , number=

    2024 doi

  28. [37]

    Hemo, Tamir and Richarz, Timo and Scholbach, Jakob , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.109179 , URL =

    2023 doi

  29. [38]

    Cohomology of number fields , SERIES =

    Neukirch, J\". Cohomology of number fields , SERIES =. 2008 , PAGES =. doi:10.1007/978-3-540-37889-1 , URL =

    2008 doi

  30. [39]

    Infinite stable fields are

    Scanlon, Thomas , Note =. Infinite stable fields are. 1999 , Month =

    1999

  31. [40]

    , TITLE =

    Kaplan, Itay and Scanlon, Thomas and Wagner, Frank O. , TITLE =. Israel J. Math. , FJOURNAL =. 2011 , PAGES =. doi:10.1007/s11856-011-0104-7 , URL =

    2011 doi

  32. [41]

    Zargar, Masoud , TITLE =. Adv. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.aim.2019.106744 , URL =

    2019 doi

  33. [42]

    Geometrization of the local Langlands correspondence , Year =

    Fargues, Laurent and Scholze, Peter , Month =. Geometrization of the local Langlands correspondence , Year =

  34. [43]

    The tame site of a scheme , JOURNAL =

    H\". The tame site of a scheme , JOURNAL =. 2021 , NUMBER =. doi:10.1007/s00222-020-00993-4 , URL =

    2021 doi

  35. [44]

    Answer to

    Landesman, Aaron , Month =. Answer to

  36. [45]

    Compositio Math

    Schmidt, Alexander , TITLE =. Compositio Math. , FJOURNAL =. 1996 , NUMBER =

    1996

  37. [46]

    Temkin, Michael , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2017 , NUMBER =. doi:10.4007/annals.2017.186.1.3 , URL =

    2017 doi

  38. [47]

    On Gabber's refined uniformization , Year =

    Illusie, Luc , Note =. On Gabber's refined uniformization , Year =

  39. [48]

    Hoyois, Marc and Kelly, Shane and Østvær, Paul Arne , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2017 , NUMBER =. doi:10.4171/JEMS/754 , URL =

    2017 doi

  40. [49]

    Elmanto, Elden and Hoyois, Marc and Iwasa, Ryomei and Kelly, Shane , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1515/crelle-2021-0040 , URL =

    2021 doi

  41. [50]

    Elmanto, Elden and Hoyois, Marc and Iwasa, Ryomei and Kelly, Shane , TITLE =. Math. Ann. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00208-020-02083-5 , URL =

    2021 doi

  42. [51]

    Duke Math

    Bhatt, Bhargav and Mathew, Akhil , TITLE =. Duke Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1215/00127094-2020-0088 , URL =

    2021 doi

  43. [52]

    Anel, Mathieu and Biedermann, Georg and Finster, Eric and Joyal, André , TITLE =. J. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/topo.12163 , URL =

    2020 doi

  44. [53]

    Orgogozo, Fabrice , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 2003 , NUMBER =. doi:10.24033/bsmf.2438 , URL =

    2003 doi

  45. [54]

    Invariance of the fundamental group under base change between algebraically closed fields , Year =

    Landesman, Aaron , Month =. Invariance of the fundamental group under base change between algebraically closed fields , Year =

  46. [55]

    Lara, Marcin , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2022 , NUMBER =. doi:10.1093/imrn/rnab101 , URL =

    2022 doi

  47. [56]

    Algebra Number Theory , FJOURNAL =

    Lara, Marcin , TITLE =. Algebra Number Theory , FJOURNAL =. 2024 , NUMBER =. doi:10.2140/ant.2024.18.631 , URL =

    2024 doi

  48. [57]

    2019 , PAGES =

    Lara, Marcin , TITLE =. 2019 , PAGES =

    2019

  49. [58]

    and Holzschuh, Tim and Wolf, Sebastian , TITLE =

    Haine, Peter J. and Holzschuh, Tim and Wolf, Sebastian , TITLE =. J. Topol. , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/topo.70009 , URL =

    2024 doi

  50. [59]

    and Holzschuh, Tim and Wolf, Sebastian , TITLE =

    Haine, Peter J. and Holzschuh, Tim and Wolf, Sebastian , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2024 , NUMBER =. doi:10.1093/imrn/rnad018 , URL =

    2024 doi

  51. [60]

    Duke Math

    Lu, Qing and Zheng, Weizhe , TITLE =. Duke Math. J. , FJOURNAL =. 2019 , NUMBER =. doi:10.1215/00127094-2019-0057 , URL =

    2019 doi

  52. [61]

    2019 , NOTE =

    Carchedi, David and Scherotzke, Sarah and Sibilla, Nicolò and Talpo, Mattia , TITLE =. 2019 , NOTE =

    2019

  53. [62]

    Carchedi, David and Scherotzke, Sarah and Sibilla, Nicol\`o and Talpo, Mattia , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.3093 , URL =

    2017 doi

  54. [63]

    and Skorobogatov, Alexei N

    Schlank, Tomer M. and Skorobogatov, Alexei N. , TITLE =. Torsors, étale homotopy and applications to rational points , SERIES =. 2013 , MRCLASS =

    2013

  55. [64]

    2009 , Month =

    Lurie, Jacob , Note =. 2009 , Month =

    2009

  56. [65]

    Devalapurkar, Sanath and Haine, Peter , TITLE =. Doc. Math. , FJOURNAL =. 2021 , PAGES =

    2021

  57. [66]

    , TITLE =

    Friedlander, Eric M. , TITLE =. Inst. Hautes Études Sci. Publ. Math. , FJOURNAL =. 1973 , PAGES =

    1973

  58. [67]

    , TITLE =

    Friedlander, Eric M. , TITLE =. Manuscripta Math. , FJOURNAL =. 1973 , PAGES =. doi:10.1007/BF01332767 , URL =

    1973 doi

  59. [68]

    1970 , PAGES =

    Friedlander, Eric Mark , TITLE =. 1970 , PAGES =

    1970

  60. [69]

    Quick, Gereon , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.jpaa.2010.07.008 , URL =

    2011 doi

  61. [70]

    , TITLE =

    Cox, David A. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/2042908 , URL =

    1979 doi

  62. [71]

    Forum Math

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

    2017 doi

  63. [72]

    Neeman, Amnon , TITLE =. Doc. Math. , FJOURNAL =. 2001 , PAGES =

    2001

  64. [73]

    Peking Math

    Hesselholt, Lars and Pstrągowski, Piotr , TITLE =. Peking Math. J. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s42543-023-00072-6 , URL =

    2025 doi

  65. [74]

    , Month =

    Haine, Peter J. , Month =. From nonabelian basechange to basechange with coefficients , Year =

  66. [75]

    Aoki, Ko , TITLE =. Math. Z. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00209-023-03210-z , URL =

    2023 doi

  67. [76]

    and Porta, Mauro and Teyssier, Jean-Baptiste , TITLE =

    Haine, Peter J. and Porta, Mauro and Teyssier, Jean-Baptiste , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2023 , NUMBER =. doi:10.4310/hha.2023.v25.n2.a6 , URL =

    2023 doi

  68. [77]

    Categorical topology (

    Dyckhoff, Roy , TITLE =. Categorical topology (. 1976 , MRCLASS =

    1976

  69. [78]

    Synthetic spectra and the cellular motivic category , Year =

    Pstrągowski, Piotr , Month =. Synthetic spectra and the cellular motivic category , Year =

  70. [79]

    Six functor formalism for sheaves with non-presentable coefficients , Year =

    Volpe, Marco , Month =. Six functor formalism for sheaves with non-presentable coefficients , Year =

  71. [80]

    Hoyois, Marc , Month =

  72. [81]

    Relative de

    Aizenbud, Avraham and Carmeli, Shachar , Month =. Relative de

  73. [82]

    Hoyois, Marc and Scherotzke, Sarah and Sibilla, Nicol\`o , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2017.01.008 , URL =

    2017 doi

  74. [83]

    2017 , PAGES =

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

    2017 doi

  75. [84]

    2017 , PAGES =

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

    2017 doi

  76. [85]

    and Shah, Jay , Month =

    Quigley, J.D. and Shah, Jay , Month =. On the parametrized

  77. [86]

    , TITLE =

    Berman, John D. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2020 , NUMBER =. doi:10.1016/j.jpaa.2020.106362 , URL =

    2020 doi

  78. [87]

    , TITLE =

    Berman, John D. , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.2140/agt.2018.18.2963 , URL =

    2018 doi

  79. [88]

    Bénabou, Jean and Roubaud, Jacques , TITLE =. C. R. Acad. Sci. Paris Sér. A-B , FJOURNAL =. 1970 , PAGES =

    1970

  80. [89]

    and Yanovski, Lior , Month =

    Carmeli, Shachar and Schlank, Tomer M. and Yanovski, Lior , Month =. Ambidexterity in Chromatic Homotopy Theory , Year =

  81. [90]

    Ambidexterity in

    Hopkins, Michael and Lurie, Jacob , Note =. Ambidexterity in

  82. [91]

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

    1974

  83. [92]

    Cheng, Eugenia and Gurski, Nick and Riehl, Emily , TITLE =. J. K-Theory , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/is013012007jkt250 , URL =

    2014 doi

  84. [93]

    Hoyois, Marc , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2016.09.031 , URL =

    2017 doi

  85. [94]

    Hoyois, Marc , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/agt.2014.14.3603 , URL =

    2014 doi

  86. [95]

    Clausen, Dustin and Mathew, Akhil , TITLE =. Invent. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00222-021-01043-3 , URL =

    2021 doi

  87. [96]

    Answer to

    Hoyois, Marc , Month =. Answer to

  88. [97]

    Tohoku Math

    Grothendieck, Alexander , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 1957 , PAGES =. doi:10.2748/tmj/1178244839 , URL =

    1957 doi

  89. [98]

    The stratified homotopy type of the reductive

    Ørsnes Jansen, Mikala , Month =. The stratified homotopy type of the reductive

  90. [99]

    Ben-Zvi, David and Francis, John and Nadler, David , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0894-0347-10-00669-7 , URL =

    2010 doi

  91. [100]

    Antieau, Benjamin and Gepner, David and Heller, Jeremiah , TITLE =. Invent. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00222-018-00847-0 , URL =

    2019 doi

  92. [101]

    , Month =

    Arad, Edo and Carmeli, Shachar and Schlank, Tomer M. , Month =. Étale Homotopy Obstructions of Arithmetic Spheres , Year =

  93. [102]

    Lax monoidal adjunctions, two-variable fibrations and the calculus of mates , Year =

    Haugseng, Rune and Hebestreit, Fabian and Linskens, Sil and Nuiten, Joost , Month =. Lax monoidal adjunctions, two-variable fibrations and the calculus of mates , Year =

  94. [103]

    , TITLE =

    Spaltenstein, N. , TITLE =. Compositio Math. , FJOURNAL =. 1988 , NUMBER =

    1988

  95. [104]

    Carlsson, Gunnar , TITLE =. Invent. Math. , FJOURNAL =. 1991 , NUMBER =. doi:10.1007/BF01239524 , URL =

    1991 doi

  96. [105]

    Proceedings of the

    Miller, Haynes , TITLE =. Proceedings of the. 1987 , MRCLASS =

    1987

  97. [106]

    Miller, Haynes , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/1971212 , URL =

    1985 doi

  98. [107]

    Miller, Haynes , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1984 , NUMBER =. doi:10.2307/2007071 , URL =

    1984 doi

  99. [108]

    Miller, Haynes , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 1983 , NUMBER =. doi:10.1090/S0273-0979-1983-15159-5 , URL =

    1983 doi

  100. [109]

    Lannes, Jean , TITLE =. Inst. Hautes Études Sci. Publ. Math. , FJOURNAL =. 1992 , PAGES =

    1992

  101. [110]

    Néron models , SERIES =

    Bosch, Siegfried and L\". Néron models , SERIES =. 1990 , PAGES =. doi:10.1007/978-3-642-51438-8 , URL =

    1990 doi

  102. [111]

    Weil restriction for schemes and beyond , Year =

    Ji, Lena and Li, Shizhang and McFaddin, Patrick and Moore, Drew and Stevenson, Matthew , Month =. Weil restriction for schemes and beyond , Year =

  103. [112]

    1994 , PAGES =

    Scheiderer, Claus , TITLE =. 1994 , PAGES =. doi:10.1007/BFb0074269 , URL =

    1994 doi

  104. [113]

    , TITLE =

    Raynaud, M. , TITLE =. Inst. Hautes Études Sci. Publ. Math. , FJOURNAL =. 1970 , PAGES =

    1970

  105. [114]

    Bégueri, Lucile , TITLE =. Mém. Soc. Math. France (N.S.) , FJOURNAL =. 1980/81 , PAGES =

    1980

  106. [115]

    Purity for flat cohomology , Year =

    Česnavičius, Kęstutis and Scholze, Peter , Month =. Purity for flat cohomology , Year =

  107. [116]

    Algebraic Theories and

    Cranch, James , Month =. Algebraic Theories and

  108. [117]

    Ambidexterity and the universality of finite spans , Year =

    Harpaz, Yonatan , Month =. Ambidexterity and the universality of finite spans , Year =

  109. [118]

    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

  110. [119]

    Algebraic topology,

    Kraines, David and Lada, Thomas , TITLE =. Algebraic topology,. 1979 , MRCLASS =

    1979

  111. [120]

    An alpine expedition through algebraic topology , SERIES =

    Noel, Justin , TITLE =. An alpine expedition through algebraic topology , SERIES =. 2014 , MRCLASS =. doi:10.1090/conm/617/12285 , URL =

    2014 doi

  112. [121]

    Brazelton, Thomas , Month =

  113. [122]

    Classifying spaces of infinity-sheaves , Year =

    Berwick-Evans, Daniel and Boavida de Brito, Pedro and Pavlov, Dmitri , Month =. Classifying spaces of infinity-sheaves , Year =

  114. [123]

    2006 , PAGES =

    Mazza, Carlo and Voevodsky, Vladimir and Weibel, Charles , TITLE =. 2006 , PAGES =

    2006

  115. [124]

    Associative algebras and broken lines , Year =

    Lurie, Jacob and Tanaka, Hiro Lee , Note =. Associative algebras and broken lines , Year =

  116. [125]

    Porta, Mauro and Yu, Tony Yue , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2016.07.017 , URL =

    2016 doi

  117. [126]

    , TITLE =

    Brown, Kenneth S. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1973 , PAGES =. doi:10.2307/1996573 , URL =

    1973 doi

  118. [127]

    Jardine, J. F. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1987 , NUMBER =. doi:10.1016/0022-4049(87)90100-9 , URL =

    1987 doi

  119. [128]

    1994 , PAGES =

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

    1994

  120. [129]

    Surveys on recent developments in algebraic geometry , SERIES =

    Antieau, Benjamin and Elmanto, Elden , TITLE =. Surveys on recent developments in algebraic geometry , SERIES =. 2017 , MRCLASS =

    2017

  121. [130]

    Duke Math

    Asok, Aravind and Hoyois, Marc and Wendt, Matthias , TITLE =. Duke Math. J. , FJOURNAL =. 2017 , NUMBER =. doi:10.1215/00127094-0000014X , URL =

    2017 doi

  122. [131]

    Ayoub, Joseph , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2010 , NUMBER =. doi:10.1017/S1474748009000127 , URL =

    2010 doi

  123. [132]

    Simpson, Carlos and Teleman, Constantin , Note =. de

  124. [133]

    Cohesion and global homotopy theory , Year =

    Rezk, Charles , Month =. Cohesion and global homotopy theory , Year =

  125. [134]

    Differential cohomology , Year =

    Bunke, Ulrich , Note =. Differential cohomology , Year =

  126. [135]

    Differential function spectra, the differential

    Bunke, Ulrich and Gepner, David , Month =. Differential function spectra, the differential

  127. [136]

    Sheaves and homotopy theory , Year =

    Dugger, Daniel , Month =. Sheaves and homotopy theory , Year =

  128. [137]

    Dugger, Daniel , TITLE =. Adv. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1006/aima.2001.2014 , URL =

    2001 doi

  129. [138]

    1994 , PAGES =

    Adámek, Jiří and Rosický, Jiří , TITLE =. 1994 , PAGES =. doi:10.1017/CBO9780511600579 , URL =

    1994 doi

  130. [139]

    Answer to Math

    Hoyois, Marc , Month =. Answer to Math

  131. [140]

    Chromatic fracture cubes , Year =

    Barthel, Tobias and Antolín-Camarena, Omar , Month =. Chromatic fracture cubes , Year =

  132. [141]

    Answer to Math

    Nardin, Denis , Month =. Answer to Math

  133. [142]

    A note on stable reollements , Year =

    Barwick, Clark and Glasman, Saul , Month =. A note on stable reollements , Year =

  134. [143]

    and Guillemin, V

    Golubitsky, M. and Guillemin, V. , TITLE =. 1973 , PAGES =

    1973

  135. [144]

    2007 , PAGES =

    Voisin, Claire , TITLE =. 2007 , PAGES =

    2007

  136. [145]

    Losik, M. V. , TITLE =. Cahiers Topologie Géom. Différentielle Catég. , FJOURNAL =. 1994 , NUMBER =

    1994

  137. [146]

    Losik, M. V. , TITLE =. Izv. Vyssh. Uchebn. Zaved. Mat. , FJOURNAL =. 1992 , NUMBER =

    1992

  138. [147]

    , TITLE =

    Hain, Richard M. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/2042717 , URL =

    1979 doi

  139. [148]

    Waldorf, Konrad , TITLE =. Cah. Topol. Géom. Différ. Catég. , FJOURNAL =. 2012 , NUMBER =

    2012

  140. [149]

    Differential characters , SERIES =

    B\". Differential characters , SERIES =. 2014 , PAGES =. doi:10.1007/978-3-319-07034-6 , URL =

    2014 doi

  141. [150]

    Geometry and topology (

    Cheeger, Jeff and Simons, James , TITLE =. Geometry and topology (. 1985 , MRCLASS =. doi:10.1007/BFb0075216 , URL =

    1985 doi

  142. [151]

    Simons, James and Sullivan, Dennis , TITLE =. J. Topol. , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/jtopol/jtm006 , URL =

    2008 doi

  143. [152]

    , TITLE =

    Bott, R. , TITLE =. Advances in Math. , FJOURNAL =. 1973 , PAGES =. doi:10.1016/0001-8708(73)90012-1 , URL =

    1973 doi

  144. [153]

    De Marco, Giuseppe and Orsatti, Adalberto , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1971 , PAGES =. doi:10.2307/2037716 , URL =

    1971 doi

  145. [154]

    Metric and differential geometry , SERIES =

    Simons, James and Sullivan, Dennis , TITLE =. Metric and differential geometry , SERIES =. 2012 , MRCLASS =. doi:10.1007/978-3-0348-0257-4_12 , URL =

    2012 doi

  146. [155]

    Quanta of maths , SERIES =

    Simons, James and Sullivan, Dennis , TITLE =. Quanta of maths , SERIES =. 2010 , MRCLASS =

    2010

  147. [156]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S0273-0979-2013-01415-0 , URL =

    2013 doi

  148. [157]

    , TITLE =

    Harpaz, Yonatan and Schlank, Tomer M. , TITLE =. Torsors, étale homotopy and applications to rational points , SERIES =. 2013 , MRCLASS =. doi:10.1017/CBO9781139525350.010 , URL =

    2013 doi

  149. [158]

    Hopkins, M. J. and Singer, I. M. , TITLE =. J. Differential Geom. , FJOURNAL =. 2005 , NUMBER =

    2005

  150. [159]

    Gwilliam, Owen and Grady, Ryan , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/agt.2014.14.2299 , URL =

    2014 doi

  151. [160]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin , TITLE =. J. Topol. , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/jtopol/jtm001 , URL =

    2008 doi

  152. [161]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.4007/annals.2011.174.2.5 , URL =

    2011 doi

  153. [162]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S0894-0347-2013-00761-4 , URL =

    2013 doi

  154. [163]

    and Hopkins, Michael J

    Freed, Daniel S. and Hopkins, Michael J. and Teleman, Constantin , TITLE =. J. Topol. , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/jtopol/jtr019 , URL =

    2011 doi

  155. [164]

    Differential cohomology theories as sheaves of spectra , JOURNAL =

    Bunke, Ulrich and Nikolaus, Thomas and V\". Differential cohomology theories as sheaves of spectra , JOURNAL =. 2016 , NUMBER =. doi:10.1007/s40062-014-0092-5 , URL =

    2016 doi

  156. [165]

    1998 , PAGES =

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

    1998

  157. [166]

    , TITLE =

    Thom, R. , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1969 , PAGES =. doi:10.1090/S0002-9904-1969-12138-5 , URL =

    1969 doi

  158. [167]

    Mather, John , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0273-0979-2012-01383-6 , URL =

    2012 doi

  159. [168]

    Ayala, David and Francis, John and Tanaka, Hiro Lee , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2016.11.032 , URL =

    2017 doi

  160. [169]

    , TITLE =

    Fritsch, Rudolf and Piccinini, Renzo A. , TITLE =. 1990 , PAGES =. doi:10.1017/CBO9780511983948 , URL =

    1990 doi

  161. [170]

    Woolf, Jon , TITLE =. J. Homotopy Relat. Struct. , FJOURNAL =. 2009 , NUMBER =

    2009

  162. [171]

    This is the (co)end, my only (co)friend , Year =

    Loregian, Fosco , Month =. This is the (co)end, my only (co)friend , Year =

  163. [172]

    geometric realization , howpublished =

  164. [173]

    and Zisman, M

    Gabriel, P. and Zisman, M. , TITLE =. 1967 , PAGES =

    1967

  165. [174]

    Lifting accessible model structures , Year =

    Garner, Richard and Kędziorek, Magdalena and Riehl, Emily , Month =. Lifting accessible model structures , Year =

  166. [175]

    Étude homotopique des espaces stratifiés , Year =

    Douteau, Sylvain , Note =. Étude homotopique des espaces stratifiés , Year =

  167. [176]

    Kyushu J

    Shimakawa, Kazuhisa and Yoshida, Kohei and Haraguchi, Tadayuki , TITLE =. Kyushu J. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.2206/kyushujm.72.239 , URL =

    2018 doi

  168. [177]

    Spectral

    Barwick, Clark , Doi =. Spectral. Adv. Math. , Mrclass =

  169. [178]

    A Simplicial Approach to Stratified Homotopy Theory , Year =

    Nand-Lal, Stephen , School =. A Simplicial Approach to Stratified Homotopy Theory , Year =

  170. [179]

    , Date-Added =

    Miller, David A. , Date-Added =. Strongly stratified homotopy theory , Url =. Trans. Amer. Math. Soc. , Mrclass =. 2013 , Bdsk-Url-1 =. doi:10.1090/S0002-9947-2013-05795-9 , Fjournal =

    2013 doi

  171. [180]

    , Date-Added =

    Miller, David A. , Date-Added =. Popaths and holinks , Volume =. J. Homotopy Relat. Struct. , Mrclass =

  172. [181]

    Quinn, Frank , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1988 , NUMBER =. doi:10.2307/1990924 , URL =

    1988 doi

  173. [182]

    2012 , PAGES =

    Morishita, Masanori , TITLE =. 2012 , PAGES =. doi:10.1007/978-1-4471-2158-9 , URL =

    2012 doi

  174. [183]

    Sugaku Expositions , FJOURNAL =

    Morishita, Masanori , TITLE =. Sugaku Expositions , FJOURNAL =. 2010 , NUMBER =

    2010

  175. [184]

    Remarks on the

    Mazur, Barry , Note =. Remarks on the

  176. [185]

    and Mazur, Barry , TITLE =

    Artin, M. and Mazur, Barry , TITLE =. Proc. 1967 , MRCLASS =

    1967

  177. [186]

    Beĭlinson, A. A. and Bernstein, J. and Deligne, P. and Gabber, Ofer , TITLE =. Analysis and topology on singular spaces,. 1982 , MRCLASS =. doi:10.24033/ast.1042 , Note =

    1982 doi

  178. [187]

    and Migliorini, Luca , TITLE =

    de Cataldo, Mark Andrea A. and Migliorini, Luca , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0273-0979-09-01260-9 , URL =

    2009 doi

  179. [188]

    Goresky, Mark and MacPherson, Robert , TITLE =. Invent. Math. , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF01389130 , URL =

    1983 doi

  180. [189]

    Topology , FJOURNAL =

    Goresky, Mark and MacPherson, Robert , TITLE =. Topology , FJOURNAL =. 1980 , NUMBER =. doi:10.1016/0040-9383(80)90003-8 , URL =

    1980 doi

  181. [190]

    Dynamical systems for arithmetic schemes , Year =

    Deninger, Christopher , Month =. Dynamical systems for arithmetic schemes , Year =

  182. [191]

    Jahresber

    Deninger, Christopher , TITLE =. Jahresber. Deutsch. Math.-Verein. , FJOURNAL =. 2001 , NUMBER =

    2001

  183. [192]

    Deninger, Christopher , TITLE =. Doc. Math. , FJOURNAL =. 1998 , NUMBER =

    1998

  184. [193]

    Regulators in analysis, geometry and number theory , SERIES =

    Deninger, Christopher , TITLE =. Regulators in analysis, geometry and number theory , SERIES =. 2000 , MRCLASS =

    2000

  185. [194]

    Algebraic number theory and algebraic geometry , SERIES =

    Deninger, Christopher , TITLE =. Algebraic number theory and algebraic geometry , SERIES =. 2002 , MRCLASS =. doi:10.1090/conm/300/05144 , URL =

    2002 doi

  186. [195]

    Étale cohomology of algebraic number fields , Year =

    Conrad, Brian and Masullo, Alessandro Maria , Month =. Étale cohomology of algebraic number fields , Year =

  187. [196]

    Cohomological descent , author=

  188. [197]

    and Verdier, J.-L , TITLE =

    Artin, M. and Verdier, J.-L , TITLE =. Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, Massachusetts, July 6--July 31, 1964. , PUBLISHER =

    1964

  189. [198]

    Mazur, Barry , TITLE =. Ann. Sci. École Norm. Sup. (4) , FJOURNAL =. 1973 , PAGES =

    1973

  190. [199]

    Weil's Conjecture for Function Fields , Year =

    Gaitsgory, Dennis and Lurie, Jacob , Month =. Weil's Conjecture for Function Fields , Year =

  191. [200]

    Morel, Sophie , Month =. Mixed

  192. [201]

    Compositio Math

    Huber, Annette , TITLE =. Compositio Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1023/A:1000273606373 , URL =

    1997 doi

  193. [202]

    Abelian arithmetic

    Chung, Hee-Joong and Kim, Dohyeong and Kim, Minhyong and Pappas, George and Park, Jeehoon and Yoo, Hwajong , Month =. Abelian arithmetic

  194. [203]

    Arithmetic

    Chung, Hee-Joong and Kim, Dohyeong and Kim, Minhyong and Park, Jeehoon and Yoo, Hwajong , Month =. Arithmetic

  195. [204]

    Arithmetic

    Kim, Minhyong , Month =. Arithmetic

  196. [205]

    Modern Phys

    Kim, Minhyong , TITLE =. Modern Phys. Lett. A , FJOURNAL =. 2018 , NUMBER =. doi:10.1142/S0217732318300124 , URL =

    2018 doi

  197. [206]

    Toën, Bertrand , TITLE =. Cah. Topol. Géom. Différ. Catég. , FJOURNAL =. 2002 , NUMBER =

    2002

  198. [207]

    Clausen, Dustin , Month =. A

  199. [208]

    2013 , PAGES =

    Clausen, Dustin , TITLE =. 2013 , PAGES =

    2013

  200. [209]

    2022 , NOTE =

    Clausen, Dustin and Scholze, Peter , TITLE =. 2022 , NOTE =

    2022

  201. [210]

    Densities and stability via factorization homology , Year =

    Hồ, Quốc Phú , Month =. Densities and stability via factorization homology , Year =

  202. [211]

    2018 , NOTE =

    Carchedi, David and Elmanto, Elden , TITLE =. 2018 , NOTE =

    2018

  203. [212]

    2017 , PAGES =

    Hồ, Quốc Phú , TITLE =. 2017 , PAGES =

    2017

  204. [213]

    Selecta Math

    Hồ, Quốc Phú , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00029-017-0339-1 , URL =

    2017 doi

  205. [214]

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

    2017

  206. [215]

    2019 , PAGES =

    Gaitsgory, Dennis and Lurie, Jacob , TITLE =. 2019 , PAGES =

    2019

  207. [216]

    , Month =

    Barwick, Clark and Haine, Peter J. , Month =. Pyknotic objects,

  208. [217]

    Selecta Math

    Ayala, David and Francis, John and Tanaka, Hiro Lee , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00029-016-0242-1 , URL =

    2017 doi

  209. [218]

    Ayala, David and Francis, John , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00220-019-03311-z , URL =

    2019 doi

  210. [219]

    Intersection cohomology (

    Goresky, Mark and MacPherson, Robert , TITLE =. Intersection cohomology (. 1984 , MRCLASS =. doi:10.1007/978-0-8176-4765-0_9 , URL =

    1984 doi

  211. [220]

    , Month =

    Barwick, Clark and Haine, Peter J. , Month =. Exodromy for stacks , Year =

  212. [221]

    Morel, Fabien and Voevodsky, Vladimir , TITLE =. Inst. Hautes Études Sci. Publ. Math. , FJOURNAL =. 1999 , PAGES =

    1999

  213. [222]

    Stix, Jakob , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 2006 , NUMBER =

    2006

  214. [223]

    Bunge, Marta and Lack, Stephen , TITLE =. Adv. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0001-8708(03)00010-0 , URL =

    2003 doi

  215. [224]

    Zoonekynd, Vincent , TITLE =. Ann. Math. Blaise Pascal , FJOURNAL =. 2002 , NUMBER =

    2002

  216. [225]

    Theory Appl

    Hoyois, Marc , TITLE =. Theory Appl. Categ. , FJOURNAL =. 2019 , PAGES =

    2019

  217. [226]

    , TITLE =

    Misamore, Michael D. , TITLE =. Canad. J. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.4153/CJM-2011-030-x , URL =

    2011 doi

  218. [227]

    , TITLE =

    Milne, James S. , TITLE =. 1980 , PAGES =

    1980

  219. [228]

    , TITLE =

    Isaksen, Daniel C. , TITLE =. Adv. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/S0001-8708(03)00094-X , URL =

    2004 doi

  220. [229]

    , Month =

    Schlank, Tomer M. , Month =. Answer to Math

  221. [230]

    Milne, J. S. , TITLE =. 2006 , PAGES =

    2006

  222. [231]

    , TITLE =

    Hoobler, Raymond T. , TITLE =. Pacific J. Math. , FJOURNAL =. 1972 , PAGES =

    1972

  223. [232]

    Henriques, André , Note =

  224. [233]

    Orbispaces , Year =

    Henriques, André , School =. Orbispaces , Year =

  225. [234]

    Algebra Universalis , FJOURNAL =

    Bunge, Marta , TITLE =. Algebra Universalis , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01181872 , URL =

    1995 doi

  226. [235]

    2006 , PAGES =

    Bunge, Marta and Funk, Jonathon , TITLE =. 2006 , PAGES =

    2006

  227. [236]

    Categories in algebra, geometry and mathematical physics , SERIES =

    Bunge, Marta and Funk, Jonathon , TITLE =. Categories in algebra, geometry and mathematical physics , SERIES =. 2007 , MRCLASS =. doi:10.1090/conm/431/08269 , URL =

    2007 doi

  228. [237]

    Gabber, Ofer and Kelly, Shane , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2015 , NUMBER =. doi:10.1016/j.jpaa.2015.03.001 , URL =

    2015 doi

  229. [238]

    2016 , PAGES =

    Abbes, Ahmed and Gros, Michel and Tsuji, Takeshi , TITLE =. 2016 , PAGES =. doi:10.1515/9781400881239 , URL =

    2016 doi

  230. [239]

    Grothendieck topologies from unique factorisation systems , Year =

    Anel, Mathieu , Month =. Grothendieck topologies from unique factorisation systems , Year =

  231. [240]

    Compositio Math

    Moerdijk, Ieke , TITLE =. Compositio Math. , FJOURNAL =. 1986 , NUMBER =

    1986

  232. [241]

    , TITLE =

    Dubuc, Eduardo J. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/S0022-4049(03)00141-5 , URL =

    2004 doi

  233. [242]

    Rydh, David , TITLE =. J. Algebra , FJOURNAL =. 2011 , PAGES =. doi:10.1016/j.jalgebra.2011.01.006 , URL =

    2011 doi

  234. [243]

    , Date-Added =

    Hopkins, Michael J. , Date-Added =. Complex oriented Cohomology Theories and the Language of Stacks , Year =

  235. [244]

    General topology

    Bourbaki, Nicolas , Isbn =. General topology

  236. [245]

    Lecture 30: Localizations and the Adams--Novikov Spectral Sequence , Year =

    Lurie, Jacob , Date-Added =. Lecture 30: Localizations and the Adams--Novikov Spectral Sequence , Year =

  237. [246]

    Nilpotence and descent in equivariant stable homotopy theory , Url =

    Mathew, Akhil and Naumann, Niko and Noel, Justin , Date-Added =. Nilpotence and descent in equivariant stable homotopy theory , Url =. Adv. Math. , Mrclass =. 2017 , Bdsk-Url-1 =. doi:10.1016/j.aim.2016.09.027 , Fjournal =

    2017 doi

  238. [247]

    A relative

    Mazel-Gee, Aaron and Peterson, Eric and Stapleton, Nathaniel , Date-Added =. A relative. Algebr. Geom. Topol. , Mrclass =. 2015 , Bdsk-Url-1 =. doi:10.2140/agt.2015.15.2239 , Fjournal =

    2015 doi

  239. [248]

    Existence theorem for higher local fields , Url =

    Kato, Kazuya , Booktitle =. Existence theorem for higher local fields , Url =. 2000 , Bdsk-Url-1 =. doi:10.2140/gtm.2000.3.165 , Mrclass =

    2000 doi

  240. [249]

    , Date-Added =

    Waterhouse, William C. , Date-Added =. Basically bounded functors and flat sheaves , Url =. Pacific J. Math. , Mrclass =. 1975 , Bdsk-Url-1 =

    1975

  241. [250]

    Constructing the determinant sphere using a

    Barthel, Tobias and Beaudry, Agn. Constructing the determinant sphere using a

  242. [251]

    Condensed mathematics , Year =

    Scholze, Peter , Date-Added =. Condensed mathematics , Year =

  243. [252]

    Lectures on Analytic Geometry , Year =

    Scholze, Peter , Month =. Lectures on Analytic Geometry , Year =

  244. [253]

    Lectures on Condensed Mathematics , Year =

    Scholze, Peter , Month =. Lectures on Condensed Mathematics , Year =

  245. [254]

    Condensed mathematics , Url =

    Scholze, Peter , Date-Added =. Condensed mathematics , Url =. 2019 , Bdsk-Url-1 =

    2019

  246. [255]

    2019 , Bdsk-Url-1 =

    Chough, Chang-Yeon , Date-Added =. 2019 , Bdsk-Url-1 =. doi:10.1093/imrn/rnz065 , Eprint =

    2019 doi

  247. [256]

    A trivial remark on the

    Hoyois, Marc , Date-Added =. A trivial remark on the

  248. [257]

    , Date-Added =

    Haine, Peter J. , Date-Added =. On coherent topoi & coherent

  249. [258]

    Homological algebra with locally compact abelian groups , Url =

    Hoffmann, Norbert and Spitzweck, Markus , Date-Added =. Homological algebra with locally compact abelian groups , Url =. Adv. Math. , Mrclass =. 2007 , Bdsk-Url-1 =. doi:10.1016/j.aim.2006.09.019 , Fjournal =

    2007 doi

  250. [259]

    Kerodon , Year =

    Lurie, Jacob , Howpublished =. Kerodon , Year =

  251. [260]

    Lecture 11: Coherent topoi , Year =

    Lurie, Jacob , Month =. Lecture 11: Coherent topoi , Year =

  252. [261]

    , Booktitle =

    Nisnevich, Yevsey A. , Booktitle =. The completely decomposed topology on schemes and associated descent spectral sequences in algebraic

  253. [262]

    Notes on the

    Hoyois, Marc , Date-Added =. Notes on the

  254. [263]

    Chough, Chang-Yeon , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2022 , NUMBER =. doi:10.4171/jems/1263 , URL =

    2022 doi

  255. [264]

    Lecture 30

    Lurie, Jacob , Date-Added =. Lecture 30

  256. [265]

    , TITLE =

    Gleason, Andrew M. , TITLE =. Illinois J. Math. , FJOURNAL =. 1958 , PAGES =

    1958

  257. [266]

    Scholze, Peter , Date-Added =

  258. [267]

    and Yanovski, Lior , Date-Added =

    Schlank, Tomer M. and Yanovski, Lior , Date-Added =. On

  259. [268]

    1970 , PAGES =

    Raynaud, Michel , TITLE =. 1970 , PAGES =

    1970

  260. [269]

    Ultracategories , Year =

    Lurie, Jacob , Date-Added =. Ultracategories , Year =

  261. [270]

    Algorithmic canonical stratifications of simplicial complexes , Year =

    Asai, Ryo and Shah, Jay , Date-Added =. Algorithmic canonical stratifications of simplicial complexes , Year =

  262. [271]

    Lecture 13: Elimination of Imaginaries , Year =

    Lurie, Jacob , Date-Added =. Lecture 13: Elimination of Imaginaries , Year =

  263. [272]

    Lecture 7: Pretopoi , Year =

    Lurie, Jacob , Date-Added =. Lecture 7: Pretopoi , Year =

  264. [273]

    Hochster duality in derived categories and point-free reconstruction of schemes , Url =

    Kock, Joachim and Pitsch, Wolfgang , Date-Added =. Hochster duality in derived categories and point-free reconstruction of schemes , Url =. Trans. Amer. Math. Soc. , Mrclass =. 2017 , Bdsk-Url-1 =. doi:10.1090/tran/6773 , Fjournal =

    2017 doi

  265. [274]

    Norms in motivic homotopy theory , Year =

    Bachmann, Tom and Hoyois, Marc , Date-Added =. Norms in motivic homotopy theory , Year =

  266. [275]

    , Date-Added =

    Barwick, Clark and Haine, Peter J. , Date-Added =. Exodromy for

  267. [276]

    2016 , PAGES =

    Cho, Chang-Yeon , TITLE =. 2016 , PAGES =

    2016

  268. [277]

    , TITLE =

    Barnea, Ilan and Schlank, Tomer M. , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2015.11.014 , URL =

    2016 doi

  269. [278]

    , Fjournal =

    Cox, David A. , Fjournal =. Homotopy theory of simplicial schemes , Url =. Compositio Math. , Mrclass =. 1979 , Bdsk-Url-1 =

    1979

  270. [279]

    A Simplicial Approach to Stratified Homotopy Theory , Year =

    Douteau, Sylvain , Date-Added =. A Simplicial Approach to Stratified Homotopy Theory , Year =

  271. [280]

    Differential cohomology in a cohesive

    Schreiber, Urs , Date-Added =. Differential cohomology in a cohesive

  272. [281]

    , Date-Added =

    Haine, Peter J. , Date-Added =. A model for the

  273. [282]

    , Month =

    Haine, Peter J. , Month =. Extended étale homotopy groups from profinite

  274. [283]

    McCarthy, John , Date-Added =. A ``. Trans. Amer. Math. Soc. , Mrclass =. 1985 , Bdsk-Url-1 =. doi:10.2307/2000100 , Fjournal =

    1985 doi

  275. [284]

    , Date-Added =

    Ivanov, Nikolai V. , Date-Added =. Subgroups of

  276. [285]

    Genetics of homotopy theory and the

    Sullivan, Dennis , Date-Added =. Genetics of homotopy theory and the. Ann. of Math. (2) , Mrclass =. 1974 , Bdsk-Url-1 =. doi:10.2307/1970841 , Fjournal =

    1974 doi

  277. [286]

    and Mislin, Guido , Date-Added =

    Friedlander, Eric M. and Mislin, Guido , Date-Added =. Cohomology of classifying spaces of complex. Comment. Math. Helv. , Mrclass =. 1984 , Bdsk-Url-1 =. doi:10.1007/BF02566356 , Fjournal =

    1984 doi

  278. [287]

    and Parshall, Brian , Booktitle =

    Friedlander, Eric M. and Parshall, Brian , Booktitle =. Étale cohomology of reductive groups , Volume =

  279. [288]

    , Date-Added =

    Knudson, Kevin P. , Date-Added =. Homology of linear groups , Url =. 2001 , Bdsk-Url-1 =. doi:10.1007/978-3-0348-8338-2 , Isbn =

    2001 doi

  280. [289]

    Morel, Fabien , Booktitle =. On the

  281. [290]

    Galois cohomology , Year =

    Serre, Jean-Pierre , Date-Added =. Galois cohomology , Year =

  282. [291]

    Oda, Takayuki , Booktitle =

  283. [292]

    , Booktitle =

    Isaksen, Daniel C. , Booktitle =. Strict model structures for pro-categories , Volume =

  284. [293]

    , Date-Added =

    Isaksen, Daniel C. , Date-Added =. A model structure on the category of pro-simplicial sets , Url =. Trans. Amer. Math. Soc. , Mrclass =. 2001 , Bdsk-Url-1 =. doi:10.1090/S0002-9947-01-02722-2 , Fjournal =

    2001 doi

  285. [294]

    The étale symmetric

    Hoyois, Marc , Month =. The étale symmetric

  286. [295]

    Barwick, Clark , Date-Added =. On

  287. [296]

    , Date-Added =

    Johnstone, Peter T. , Date-Added =. Stone spaces , Volume =

  288. [297]

    , Date-Added =

    Cox, David A. , Date-Added =. Algebraic tubular neighborhoods. Math. Scand. , Mrclass =. 1978 , Bdsk-Url-1 =. doi:10.7146/math.scand.a-11750 , Fjournal =

    1978 doi

  289. [298]

    , Date-Added =

    Cox, David A. , Date-Added =. Algebraic tubular neighborhoods. Math. Scand. , Mrclass =. 1978 , Bdsk-Url-1 =. doi:10.7146/math.scand.a-11749 , Fjournal =

    1978 doi

  290. [299]

    Theory of tubular neighborhood in étale topology , Url =

    Fujiwara, Kazuhiro , Date-Added =. Theory of tubular neighborhood in étale topology , Url =. Duke Math. J. , Mrclass =. 1995 , Bdsk-Url-1 =. doi:10.1215/S0012-7094-95-08002-8 , Fjournal =

    1995 doi

  291. [300]

    Algebra Number Theory , FJOURNAL =

    Holschbach, Armin and Schmidt, Johannes and Stix, Jakob , TITLE =. Algebra Number Theory , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/ant.2014.8.1037 , URL =

    2014 doi

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