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arxiv: 2312.07906 · v3 · submitted 2023-12-13 · 🧮 math.AT · math.CT

Mapping spaces between operads in relation to bimodules

Hoang Truong This is my paper

Pith reviewed 2026-05-24 05:05 UTC · model grok-4.3

classification 🧮 math.AT math.CT
keywords operadsbimodulesmapping spacesfiber sequencessimplicial operadsdg operadsDold-Kan correspondenceenriched categories
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The pith

Mapping spaces between enriched operads relate to those between operadic bimodules through fiber sequences.

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

This paper shows that the mapping space from one enriched operad to another stands in a fiber sequence with mapping spaces that involve operadic bimodules. The fiber sequences give a direct relation that lets one pass between the two kinds of mapping spaces. The statements are proved first for simplicial operads and then for operads enriched in simplicial modules over a commutative ring. They are carried over to connective dg operads by means of an operadic Dold-Kan correspondence. A sympathetic reader would care because the relation supplies a concrete tool for comparing homotopy data in different models of operads.

Core claim

The paper claims that convenient fiber sequences exist relating the mapping spaces between enriched operads to the mapping spaces between operadic bimodules, with the statements holding for simplicial operads, operads enriched in simplicial modules over a commutative ring, and connective dg operads via an operadic version of the Dold-Kan correspondence.

What carries the argument

Fiber sequences relating the mapping space between two operads to mapping spaces between their associated bimodules.

If this is right

  • The fiber sequences hold for simplicial operads.
  • The fiber sequences extend to operads enriched in simplicial modules over a commutative ring.
  • The fiber sequences apply to connective dg operads once the operadic Dold-Kan correspondence is used.

Where Pith is reading between the lines

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

  • The relation may reduce the computation of certain operad automorphism spaces to bimodule data.
  • Bimodule mapping spaces may encode the essential homotopy information needed to compare operads.
  • Analogous fiber sequences could be sought in other monoidal model categories that admit an operad theory.

Load-bearing premise

An operadic version of the Dold-Kan correspondence exists and applies to connective dg operads, allowing the fiber-sequence statements to be translated from the simplicial setting.

What would settle it

A concrete pair of simplicial operads for which the claimed fiber sequence fails to relate the operad mapping space to the bimodule mapping spaces would disprove the main statements.

read the original abstract

This paper investigates mapping spaces between enriched operads and relates these spaces to those between operadic bimodules via convenient fiber sequences. The main statements hold for simplicial operads, operads enriched in simplicial modules over a commutative ring, and for connective dg operads, with the last case relying on an operadic version of the Dold-Kan correspondence.

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

1 major / 0 minor

Summary. The paper investigates mapping spaces between enriched operads and relates these spaces to those between operadic bimodules via convenient fiber sequences. The main statements hold for simplicial operads, operads enriched in simplicial modules over a commutative ring, and for connective dg operads, with the last case relying on an operadic version of the Dold-Kan correspondence.

Significance. If the fiber sequences are established correctly, the work supplies a useful relation between mapping spaces of operads and bimodules that may simplify certain computations in enriched operad homotopy theory. The explicit routing of the dg case through an operadic Dold-Kan equivalence is a positive feature when the correspondence preserves the relevant mapping-space structures.

major comments (1)
  1. Abstract: the abstract asserts that the main statements hold but supplies no derivation details, error handling, or verification steps; soundness cannot be assessed from the given information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the abstract asserts that the main statements hold but supplies no derivation details, error handling, or verification steps; soundness cannot be assessed from the given information.

    Authors: Abstracts are conventionally concise overviews and are not intended to contain full derivations or verification steps; those appear in the body of the paper. The fiber sequences relating mapping spaces of enriched operads to those of bimodules are derived in detail for the simplicial case, the case of operads enriched in simplicial modules, and the connective dg case (via the operadic Dold-Kan correspondence) in the subsequent sections, together with the necessary technical verifications. Soundness is therefore to be judged from the complete manuscript rather than the abstract alone. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from given material

full rationale

The provided material consists solely of the abstract and a high-level summary with no equations, derivations, or explicit self-citations visible. The central claim (fiber sequences relating mapping spaces of enriched operads to bimodules, holding across simplicial, simplicial-module, and connective dg settings via an operadic Dold-Kan correspondence) is stated as a proven result without any reduction to fitted inputs, self-definitional loops, or load-bearing self-citations that collapse the argument. No load-bearing step can be quoted or exhibited as equivalent to its inputs by construction. This is the expected honest non-finding when the derivation chain is not reproduced in the input.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no access to free parameters, axioms, or invented entities; the mention of an operadic Dold-Kan correspondence suggests reliance on a prior or constructed equivalence whose status cannot be checked.

pith-pipeline@v0.9.0 · 5565 in / 1055 out tokens · 24479 ms · 2026-05-24T05:05:45.055343+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    Dold, Homology of symmetric products and other functor s of complexes, Annals of Math- ematics Second Series

    A. Dold, Homology of symmetric products and other functor s of complexes, Annals of Math- ematics Second Series. 68(1), p. 54-80 (1958)

    work page 1958

  2. [2]

    Dold and D

    A. Dold and D. Puppe, Homologie nicht-additiver Funktore n. Anwendungen, Annales de l’institut Fourier. 11, p. 201-312 (1961)

    work page 1961

  3. [3]

    Schwede and B

    S. Schwede and B. Shipley, Equivalences of monoidal model categories , Algebraic and Geo- metric Topology. 3, p. 287-334 (2003)

    work page 2003

  4. [4]

    Mandell, Topological Andr´ e-Quillen cohomology andE∞ Andr´ e-Quillen cohomology, Ad- vances in Mathematics

    M. Mandell, Topological Andr´ e-Quillen cohomology andE∞ Andr´ e-Quillen cohomology, Ad- vances in Mathematics. 177(2), p. 227-279 (2003)

    work page 2003

  5. [5]

    White, D

    D. White, D. Yau, Homotopical adjoint lifting theorem , Applied Categorical Structures. 27, p. 385-426 (2019)

    work page 2019

  6. [6]

    Berger and I

    C. Berger and I. Moerdijk, Axiomatic homotopy theory for operads , Comment. Math. Helv. 78, p. 805-831 (2003)

    work page 2003

  7. [7]

    Pavlov and J

    D. Pavlov and J. Scholbach, Admissibility and rectification of colored symmetric opera ds, Journal of Topology. 11 (3), p. 559-601 (2018)

    work page 2018

  8. [8]

    Operads, Algebras and Modules in General Model Categories

    M. Spitzweck, Operads, algebras and modules in general model categories , preprint, arXiv:math/0101102, (2001)

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  9. [9]

    Tabuada, Differential graded versus simplicial categories , Topology Appl

    G. Tabuada, Differential graded versus simplicial categories , Topology Appl. 157, no. 3, p. 563-593 (2010)

    work page 2010

  10. [10]

    Dwyer and D.M

    W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra , Journal of Topology 19(4), p. 427-440 (1980)

    work page 1980

  11. [11]

    Hinich, Dwyer-Kan localization revisited , Homology Homotopy Appl

    V. Hinich, Dwyer-Kan localization revisited , Homology Homotopy Appl. 18, p. 27-48 (2016)

    work page 2016

  12. [12]

    W. G. Dwyer and K. Hess, Long knots and maps between operads , Geometric and Topology. 16, p. 919-955 (2012)

    work page 2012

  13. [13]

    Ducoulombier, Delooping derived mapping spaces of bimodules over an opera d, Journal of Homotopy and relative structure

    J. Ducoulombier, Delooping derived mapping spaces of bimodules over an opera d, Journal of Homotopy and relative structure. 14, p. 411-453 (2019)

    work page 2019

  14. [14]

    Hoang, Quillen cohomology of enriched operads, arXiv :2005.01198 (2020)

    T. Hoang, Quillen cohomology of enriched operads, arXiv :2005.01198 (2020)

    work page arXiv 2005

  15. [15]

    Loday and B

    J.-L. Loday and B. Vallette, Algebraic Operads, Springer, (2012)

    work page 2012

  16. [16]

    Berger and I

    C. Berger and I. Moerdijk, On the derived category of an algebra over an operad , Georgian Mathematical Journal. 16(1), p. 13-28 (2009)

    work page 2009

  17. [17]

    Fresse, Modules over operads and functors , Springer, (2009)

    B. Fresse, Modules over operads and functors , Springer, (2009)

    work page 2009

  18. [18]

    J. J. Guti´ errez and R. M. Vogt, A model structure for coloured operads in symmetric spectra , Mathematische Zeitschrift. 270, p. 223-239, (2012)

    work page 2012

  19. [19]

    Caviglia, A model structure for enriched coloured operads , available at author’s homepage https://www.math.ru.nl/~gcaviglia/ (2015)

    G. Caviglia, A model structure for enriched coloured operads , available at author’s homepage https://www.math.ru.nl/~gcaviglia/ (2015)

    work page 2015

  20. [20]

    Lurie, Higher topos theory , Annals of Mathematics Studies

    J. Lurie, Higher topos theory , Annals of Mathematics Studies. 170, Princeton University Press (2009)

    work page 2009

  21. [21]

    Muro, Dwyer-Kan homotopy theory of enriched categories , Journal of Topology

    F. Muro, Dwyer-Kan homotopy theory of enriched categories , Journal of Topology. 8, p. 377- 413 (2015)

    work page 2015

  22. [22]

    J. E. Bergner, A model category structure on the category of simplicial cat egories, Trans. Amer. Math. Soc. 359, p.2043-2058 (2007)

    work page 2043

  23. [23]

    Berger and I

    C. Berger and I. Moerdijk, On the homotopy theory of enriched categories , Quarterly Journal of Mathematics. 64, p. 805-846, (2013)

    work page 2013

  24. [24]

    W eibel, An introduction to homological algebra , Cambridge Studies in Advanced Mathe- matics

    C. W eibel, An introduction to homological algebra , Cambridge Studies in Advanced Mathe- matics. 38, Cambridge University Press, Cambridge (1994)

    work page 1994

  25. [25]

    Goerss and R

    P. Goerss and R. Jardine, Simplicial homotopy theory , Springer. 174 (2009)

    work page 2009

  26. [26]

    Rezk, Spaces of algebra structures and cohomology of operads , PhD Thesis, Massachusetts Institute of Technology, (1996)

    C. Rezk, Spaces of algebra structures and cohomology of operads , PhD Thesis, Massachusetts Institute of Technology, (1996)

    work page 1996

  27. [27]

    Lurie, Higher algebra , preprint, available at author’s homepage http://www.math.harvard.edu/~lurie/ (2011)

    J. Lurie, Higher algebra , preprint, available at author’s homepage http://www.math.harvard.edu/~lurie/ (2011)

    work page 2011

  28. [28]

    Dwyer and D.M

    W.G. Dwyer and D.M. Kan, A classification theorem for diagrams of simplicial sets , Journal of Topology. 23, p. 139-155 (1984). 36 HOANG TRUONG DEPARTMENT OF MATHEMATICS, FPT UNIVERSITY, HA NOI, VIET NAM. Email address : truonghm@fe.edu.vn

    work page 1984