pith. machine review for the scientific record. sign in

arxiv: 2605.07843 · v1 · submitted 2026-05-08 · 🧮 math.AT · math.CT

Recognition: no theorem link

Shape theory for condensed anima

Catrin Mair

Pith reviewed 2026-05-11 01:50 UTC · model grok-4.3

classification 🧮 math.AT math.CT
keywords condensed animashape theorycondensed homotopy groupsquasi-topological groupsparacompact spaceslocally contractible spacessheaf cohomology
0
0 comments X

The pith

Shape theory for condensed anima recovers classical shape for all paracompact compactly generated spaces and all locally contractible spaces.

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

The paper develops multiple perspectives on shape for condensed anima and establishes that this shape agrees with classical shape theory on two large classes of topological spaces. The agreement supplies new comparison theorems that extend earlier results linking sheaf cohomology to condensed cohomology. It also introduces condensed homotopy groups as another homotopy-theoretic invariant and gives an explicit description of the underlying topological group functor on condensed groups in terms of quasi-topological groups.

Core claim

The shape theory for condensed anima is shown to recover the classical shape functor on the full subcategory of paracompact compactly generated spaces and on the full subcategory of locally contractible spaces. These recovery statements directly imply and strengthen existing comparison results between sheaf cohomology and condensed cohomology.

What carries the argument

The shape functor from condensed anima to the classical shape category, together with the comparison functors that identify it with classical shape on the two indicated classes of spaces.

If this is right

  • The recovery statements extend the known comparisons between sheaf cohomology and condensed cohomology to a broader range of spaces.
  • Condensed homotopy groups become available as invariants that generalize classical homotopy groups in the condensed setting.
  • The underlying topological group functor on condensed groups factors through quasi-topological groups, giving a concrete description.
  • Shape-theoretic constructions can now be carried out uniformly inside the condensed anima category before restricting to classical spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Classical results about shape-theoretic invariants on locally contractible spaces can be transported to the condensed setting without additional work.
  • The quasi-topological group description may allow direct comparison of condensed groups with ordinary topological groups on a larger class of examples.
  • Further homotopy-theoretic invariants of condensed anima could be defined by composing the shape functor with other functors from classical shape theory.

Load-bearing premise

The shape theory and the relevant comparison functors for condensed anima are already correctly defined within the condensed mathematics setting.

What would settle it

A single paracompact compactly generated space on which the condensed shape differs from the classical shape would refute the recovery claim.

read the original abstract

We give different perspectives on the notion of shape for condensed anima. We prove that it recovers more classical notions of shape for topological spaces in the cases of all paracompact compactly generated spaces and all locally contractible spaces. These recovering statements imply and extend comparison results on sheaf and condensed cohomology by Clausen-Scholze and Haine. Another homotopy-theoretical direction for condensed anima are their condensed homotopy groups. Connected to this, we give a description of the underlying topological group functor on condensed groups via quasi-topological groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper provides multiple perspectives on the shape theory for condensed anima. It proves that this shape theory recovers classical notions of shape for all paracompact compactly generated topological spaces and all locally contractible spaces. These recovery statements imply and extend prior comparison results on sheaf and condensed cohomology due to Clausen-Scholze and Haine. The paper additionally develops condensed homotopy groups and gives a quasi-topological description of the underlying topological group functor on condensed groups.

Significance. If the recovery theorems hold, the work meaningfully connects condensed mathematics to classical shape theory, broadening the scope of condensed anima as a replacement for traditional spaces in homotopy and cohomology contexts. The explicit recoveries for large classes of spaces strengthen the foundations laid by earlier comparisons and supply supporting tools via homotopy groups and group functors.

minor comments (2)
  1. [Abstract] The abstract states that 'different perspectives' are given but does not enumerate them; a short list or diagram in the introduction would improve readability.
  2. The connection between the shape recovery theorems and the condensed homotopy groups section could be made more explicit, as the latter is described as 'connected' but its precise role in the main results is not immediately clear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments appear in the report, so we have no specific points to address point-by-point. We will implement the minor revisions in the next version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript supplies explicit definitions of condensed anima and its shape theory (via multiple perspectives), constructs comparison functors, and states recovery theorems for paracompact compactly generated spaces and locally contractible spaces. These theorems are presented as independent results that extend Clausen-Scholze/Haine comparisons rather than reducing to them by construction. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain; the cited prior work is external and the central claims remain self-contained against the given framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit list of free parameters, axioms, or invented entities; the work appears to rest on the standard axioms of condensed mathematics and homotopy theory already present in the cited literature.

pith-pipeline@v0.9.0 · 5363 in / 1041 out tokens · 53971 ms · 2026-05-11T01:50:36.187363+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    and Ramzi, Maxime and Steinebrunner, Jan , TITLE =

    Haine, Peter J. and Ramzi, Maxime and Steinebrunner, Jan , TITLE =. 2025 , NOTE =

    work page 2025

  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]

    Hatcher, A. , url =. Vector Bundles and K-Theory , year = 2017, note=

    work page 2017

  5. [5]

    Duke Math

    Ellis, Robert , TITLE =. Duke Math. J. , FJOURNAL =. 1957 , PAGES =

    work page 1957

  6. [6]

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

    work page doi:10.1070/rm1992v047n05abeh000961 1992

  7. [7]

    Nikolaus, Thomas and Schreiber, Urs and Stevenson, Danny , TITLE =. J. Homotopy Relat. Struct. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s40062-014-0083-6 , URL =

    work page doi:10.1007/s40062-014-0083-6 2015

  8. [8]

    1973 , PAGES =

    Godement, Roger , TITLE =. 1973 , PAGES =

    work page 1973

  9. [9]

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

    work page doi:10.1007/bf02363066 1996

  10. [10]

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

    work page doi:10.1007/bf02110350 1994

  11. [11]

    , TITLE =

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

    work page 2000

  12. [12]

    1976 , PAGES =

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

    work page 1976

  13. [13]

    Séminaire

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

    work page 1995

  14. [14]

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

    work page doi:10.1112/s0010437x20007216 2020

  15. [15]

    2018 , PAGES =

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

    work page 2018

  16. [16]

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

    work page 2024

  17. [17]

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

    work page doi:10.1007/bf02052391 1971

  18. [18]

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

    work page 2022

  19. [19]

    2025 , NOTE =

    Wolf, Sebastian , TITLE =. 2025 , NOTE =

    work page 2025

  20. [20]

    , TITLE =

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

    work page 2024

  21. [21]

    , TITLE =

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

    work page 2022

  22. [22]

    2024 , NOTE =

    Aoki, Ko , TITLE =. 2024 , NOTE =

    work page 2024

  23. [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 =

    work page doi:10.1007/s00029-023-00900-8 2024

  24. [24]

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

    work page 1964

  25. [25]

    Pacific J

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

    work page doi:10.2140/pjm.2000.196.445 2000

  26. [26]

    Recent developments in the inverse

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

    work page doi:10.1090/conm/186/02192 1995

  27. [27]

    Animated Condensed Sets and Their Homotopy Groups , Year =

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

    work page

  28. [28]

    2022 , NOTE =

    Martini, Louis , TITLE =. 2022 , NOTE =

    work page 2022

  29. [29]

    2022 , NOTE =

    Ramzi, Maxime , TITLE =. 2022 , NOTE =

    work page 2022

  30. [30]

    Yoneda's lemma for internal higher categories , Month =

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

    work page

  31. [31]

    Constructible sheaves on schemes and a categorical

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

    work page

  32. [32]

    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=

    work page doi:10.2140/ant.2024.18.499 2024

  33. [33]

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

    work page doi:10.1016/j.aim.2023.109179 2023

  34. [34]

    Cohomology of number fields , SERIES =

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

    work page doi:10.1007/978-3-540-37889-1 2008

  35. [35]

    Infinite stable fields are

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

    work page 1999

  36. [36]

    , 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 =

    work page doi:10.1007/s11856-011-0104-7 2011

  37. [37]

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

    work page doi:10.1016/j.aim.2019.106744 2019

  38. [38]

    Geometrization of the local Langlands correspondence , Year =

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

    work page

  39. [39]

    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 =

    work page doi:10.1007/s00222-020-00993-4 2021

  40. [40]

    Answer to

    Landesman, Aaron , Month =. Answer to

    work page

  41. [41]

    Compositio Math

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

    work page 1996

  42. [42]

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

    work page doi:10.4007/annals.2017.186.1.3 2017

  43. [43]

    On Gabber's refined uniformization , Year =

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

    work page

  44. [44]

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

    work page doi:10.4171/jems/754 2017

  45. [45]

    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 =

    work page doi:10.1515/crelle-2021-0040 2021

  46. [46]

    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 =

    work page doi:10.1007/s00208-020-02083-5 2021

  47. [47]

    Duke Math

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

    work page doi:10.1215/00127094-2020-0088 2021

  48. [48]

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

    work page doi:10.1112/topo.12163 2020

  49. [49]

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

    work page doi:10.24033/bsmf.2438 2003

  50. [50]

    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 =

    work page

  51. [51]

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

    work page doi:10.2140/agt.2020.20.1769 2020

  52. [52]

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

    work page doi:10.1093/imrn/rnab101 2022

  53. [53]

    Algebra Number Theory , FJOURNAL =

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

    work page doi:10.2140/ant.2024.18.631 2024

  54. [54]

    2019 , PAGES =

    Lara, Marcin , TITLE =. 2019 , PAGES =

    work page 2019

  55. [55]

    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 =

    work page doi:10.1112/topo.70009 2024

  56. [56]

    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 =

    work page doi:10.1093/imrn/rnad018 2024

  57. [57]

    Duke Math

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

    work page doi:10.1215/00127094-2019-0057 2019

  58. [58]

    2019 , NOTE =

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

    work page 2019

  59. [59]

    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 =

    work page doi:10.2140/gt.2017.21.3093 2017

  60. [60]

    and Skorobogatov, Alexei N

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

    work page 2013

  61. [61]

    2009 , Month =

    Lurie, Jacob , Note =. 2009 , Month =

    work page 2009

  62. [62]

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

    work page 2021

  63. [63]

    , TITLE =

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

    work page 1973

  64. [64]

    , TITLE =

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

    work page doi:10.1007/bf01332767 1973

  65. [65]

    1970 , PAGES =

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

    work page 1970

  66. [66]

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

    work page doi:10.1016/j.jpaa.2010.07.008 2011

  67. [67]

    , TITLE =

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

    work page doi:10.2307/2042908 1979

  68. [68]

    Forum Math

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

    work page doi:10.1515/forum-2015-0228 2017

  69. [69]

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

    work page 2001

  70. [70]

    Peking Math

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

    work page doi:10.1007/s42543-023-00072-6 2025

  71. [71]

    , Note =

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

    work page

  72. [72]

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

    work page doi:10.1007/s00209-023-03210-z 2023

  73. [73]

    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 =

    work page doi:10.4310/hha.2023.v25.n2.a6 2023

  74. [74]

    Categorical topology (

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

    work page 1976

  75. [75]

    Synthetic spectra and the cellular motivic category , Year =

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

    work page

  76. [76]

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

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

    work page

  77. [77]

    Hoyois, Marc , Month =

    work page

  78. [78]

    Relative de

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

    work page

  79. [79]

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

    work page doi:10.1016/j.aim.2017.01.008 2017

  80. [80]

    2017 , PAGES =

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

    work page doi:10.1090/surv/221.2 2017

Showing first 80 references.