Pith. sign in

REVIEW 3 major objections 4 minor 71 references

Models for cyclic infinity operads

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The homotopy theory of cyclic infinity-operads has three equivalent presentations.

desk verdict Strong, new model-theoretic results for cyclic infinity-operads, with detailed main proofs; the unproved Exercise 4.19 leaves a load-bearing but likely repairable combinatorial gap. read the letter →

arxiv 2506.15622 v2 pith:CSUQKWG6 submitted 2025-06-18 math.AT math.CT

Brandon Doherty , Philip Hackney This is my paper · ORCID
classification math.ATmath.CT MSC 18M8518N4055U3518N70
keywords cyclicoperadinfinity-operaddendroidalsetmodelcategoryQuillenequivalenceSegalspacehomotopycoherentnerve
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 seeks to establish that cyclic infinity-operads admit a well-behaved homotopy theory in three equivalent guises: cyclic dendroidal sets, cyclic dendroidal spaces, and simplicial cyclic operads. This matters because cyclic operads encode algebraic structures with no distinguished output, as in cyclic homology and topological field theory, and an infinity-categorical model lets homotopy-invariant versions of those structures be studied. The authors prove the required model structures exist and are Quillen equivalent, resolving a conjecture in the literature and completing the model-categorical foundation for connecting cyclic 2-Segal spaces to planar cyclic infinity-operads.

What carries the argument

The load-bearing object is root-elision, the functor from rooted trees to unrooted trees that forgets the chosen root. The target category has as objects unrooted trees whose boundary is a set of arcs, and presheaves on it are cyclic dendroidal sets. The paper shows root-elision is a discrete fibration, meaning every tree map with a root in the codomain lifts uniquely; moreover, its restriction to active maps, those bijecting boundaries, is a discrete opfibration, and active maps are exactly those that do not factor through outer cofaces. These combinatorial facts produce the pushout decompositions of Lemma 6.2, which yield the right-induced model structure on cyclic dendroidal sets, the Quillen property of the associated adjoint string, and the Beck-Chevalley induction in Proposition 7.13 that upgrades the known rooted equivalences to the cyclic ones. The natural isomorphism between the cyclic and rooted rigidification functors provides the adjunction compatibility that this induction uses.

What would settle it

Search the category of unrooted trees for an active tree map that does not admit a unique lift after a root is chosen, or an active map that factors through an outer coface; the paper's propositions assert neither happens, and the pushout decomposition in Lemma 6.2, which drives the rest of the argument, would fail if either did.

Watch

Extended reading notes

Core claim

The paper's central claim is that cyclic infinity-operads can be presented homotopically in three equivalent ways. It constructs a model structure on cyclic dendroidal sets whose fibrant objects are cyclic quasi-operads, meaning presheaves on the unrooted-tree category with fillers for all inner horns, and shows the homotopy coherent nerve and rigidification adjunction between this category and simplicial cyclic operads is a Quillen equivalence. It also constructs the cyclic dendroidal Rezk model structure on simplicial-valued presheaves whose fibrant objects are complete cyclic dendroidal Segal spaces, and proves the inclusion of ordinary cyclic dendroidal sets into these spaces is a left Quillen equivalence. The planar versions of all statements follow by slicing over the associative cyclic operad. Together these results affirmatively answer the question of whether a model structure for cyclic infinity-operads exists and is comparable to the known Dwyer-Kan model structure on simplicial cyclic operads.

Load-bearing premise

The chain of model structures and equivalences rests on root-elision: every map of unrooted trees with a chosen boundary element must lift uniquely to a map of rooted trees, and active maps, those that biject boundaries, must lift uniquely while never factoring through outer faces.

Editorial extensions

If this is right

  • The homotopy coherent nerve from simplicial cyclic operads to cyclic dendroidal sets becomes a right Quillen equivalence, so any infinity-categorical construction in one model transfers to the others.
  • Complete cyclic dendroidal Segal spaces give a second presentation of cyclic infinity-operads via Segal conditions plus Rezk completeness, and the inclusion of discrete objects is a Quillen equivalence.
  • Planar cyclic quasi-operads, planar simplicial cyclic operads, and planar cyclic dendroidal Rezk spaces are all Quillen equivalent, completing the bridge between cyclic 2-Segal spaces and invertible planar cyclic infinity-operads.
  • The model structure for cyclic quasi-operads is both right- and left-induced from the rooted dendroidal model structure, giving explicit generating cofibrations and a concrete description of cofibrant objects.

Reading between the lines

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

  • A general template emerges: whenever a root-forgetting functor between tree categories is a discrete fibration and a discrete opfibration on active maps, the rooted model structures and equivalences should lift to the unrooted side; modular shape categories are a natural test case.
  • The planar results make it likely that cyclic 2-Segal spaces are equivalent, as infinity-categories, to invertible planar cyclic infinity-operads, since the restriction functor to Connes' cyclic category plus the new planar model structures supply the missing equivalence.
  • The Beck-Chevalley induction on skeleta suggests a reusable proof strategy for lifting Quillen equivalences along discrete fibrations, which could be tested on other families of generalized operads such as profinite or enriched variants.
  • If a cyclic Boardman-Vogt tensor product were constructed, one could check for a reverse Quillen equivalence from complete cyclic dendroidal spaces back to cyclic quasi-operads; the paper explicitly leaves this direction open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs model structures on cyclic dendroidal sets and cyclic dendroidal spaces: Theorem A gives a model structure on bΥ whose fibrant objects are the cyclic quasi-operads, Theorem C gives the cyclic dendroidal Rezk model structure on sbΥ, and Theorems B and D show that these are Quillen equivalent to the existing model structure on simplicial cyclic operads. Theorem E extends the statement to planar cyclic and planar non-cyclic operads, with the planar cyclic case intended to complete Walde's comparison with cyclic 2-Segal spaces. The main technical device is the root-elision functor f: Ω → Υ, used to right-induce model structures from the known dendroidal ones and to lift the Cisinski--Moerdijk Quillen equivalence via a Beck--Chevalley condition.

Significance. If the arguments are correct, the paper resolves the Drummond-Cole--Hackney conjecture and establishes model-independence across the cyclic quasi-operad, simplicial cyclic operad, and cyclic dendroidal Rezk settings. The paper contains substantial and mostly detailed proofs: Lemma 6.2 is a careful pushout computation, Proposition 7.13 is an induction over skeleta, and Theorem 8.2 is a nontrivial application of the localization calculus in Appendix A. The planar consequences are a genuine extra contribution. The main weakness is that a load-bearing combinatorial assertion, Exercise 4.19 on the skeletal/EZ structure of Υ, is left unproved; because Proposition 4.25, the boundary/skeleton identifications, and the induction in Proposition 7.13 all depend on it, the central scaffolding is not yet fully verified. The gap appears fillable, but it must be closed in the manuscript.

major comments (3)
  1. [Section 4, Exercise 4.19] Exercise 4.19 asserts that the four properties of Ω transfer to Υ and that Υ is consequently a catégorie squelettique and a Berger--Moerdijk EZ category, but no proof is supplied. This is not a harmless exercise: Proposition 4.25 invokes Cisinski 8.1.35 to identify normal monomorphisms as the saturation of boundary inclusions, and Definition 4.22 uses [BM11, Corollary 6.10] for the equality ∂ΥT = sk_{n-1}ΥT, both of which require the skeletal/EZ structure. The transfer of property (4), absolute pushouts, is not a formal consequence of Propositions 4.12 and 4.16 alone: one must either construct the absolute pushout in Υ explicitly or prove that a pair of negative maps in Υ lifts to a common domain in Ω and that the image of the absolute Ω-pushout is a pushout in Υ. This argument, or a precise reference to it, must be included.
  2. [Section 6, Theorem 6.5 and Proposition 4.25] The characterization of cofibrations in the cyclic quasi-operad model structure as the normal monomorphisms depends on Proposition 4.25, which is stated without proof and whose cited source, Cisinski 8.1.35, applies only after Υ is known to be a catégorie squelettique. Since that skeletal structure is exactly what Exercise 4.19 is supposed to establish, Theorem A is currently contingent on an unverified combinatorial fact. The proof should either promote Exercise 4.19 to a theorem with a complete proof or replace Proposition 4.25 with a directly verified statement for Υ.
  3. [Section 7, Proposition 7.13] The induction proving the Beck--Chevalley isomorphism uses that ∂ΥT is the (n-1)-skeleton of ΥT for T of degree n, citing [BM11, Corollary 6.10]. That corollary applies in the EZ-categorical setting, and the EZ axioms are precisely the content of Exercise 4.19. Thus the proof of Proposition 7.13, and with it Theorem 7.14, is not fully justified until the missing skeletal/EZ argument is supplied. This is the same underlying gap as in the previous comments, but it is load-bearing at the exact point where the main Quillen equivalence is concluded.
minor comments (4)
  1. [Section 7, before Definition 7.2] The sentence 'we next introduct an analogue' contains a typo; it should read 'we next introduce an analogue'.
  2. [Author affiliation] The affiliation of the second author is given as 'University of Louisiana at Laf ayette'; this should be corrected to 'Lafayette'.
  3. [Section 4, Definition 4.22] The sentence 'by [BM11, Corollary 6.10] we have ∂ΥT = sk_{n-1}ΥT for T ∈ Υ of degree n' is stated as if the applicability of the cited corollary were already established; given the dependence on Exercise 4.19, this should be flagged explicitly at this point or moved after the skeletal structure is proved.
  4. [Section 7, Lemma 7.7 and Corollary 7.8] In Corollary 7.8, the map γ∂ΩT,t is referenced without having been explicitly named beforehand; a short phrase identifying it as the component of γ at the boundary ∂ΩT,t would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central Quillen equivalences are proven via independent model-categorical inputs and a proven Beck–Chevalley isomorphism, not by construction or by self-citation chains.

full rationale

The paper's central claims are not circular. Theorem A is obtained by right-inducing the Cisinski–Moerdijk quasi-operad model structure along root-elision f*: bΥ -> bΩ, and the key adjunction f* f! is shown Quillen via the genuinely combinatorial Lemma 6.2, whose proof reduces to Proposition 4.16 and Lemma 4.21; the existence of the right-induced structure comes from the general lifting Lemma 2.3. Theorem B lifts the known Cisinski–Moerdijk Quillen equivalence C: bΩ ⇄ sOp : N along the adjoint string F: sCyc ⇄ sOp, and the load-bearing Beck–Chevalley condition is proved in Proposition 7.13 by induction on skeleta, using the pushout Lemma 7.12 and the fact that β is an isomorphism at boundaries; it is not assumed or defined into existence. Theorem D follows from the relevant Rezk localization results of Cisinski–Moerdijk and the general lifting Theorem A.2, while Theorem E is an application of slicing results (Li, Gagna, Moerdijk) to the already-established cyclic and non-cyclic equivalences. The paper does cite the authors' own prior work: the model structure on sCyc from DCH21, the right-induced model structure criterion from DCH19, and the discrete fibration property of root-elision from Hac24a. These citations are load-bearing in the sense that the paper builds on them, but they are not equivalent to the new theorems: they are prior, externally published results whose statements do not include the conjecture being answered. No parameter is fitted to a subset of data and then reported as a prediction, and no known result is merely renamed. The one flagged gap is Exercise 4.19, which asks the reader to transfer four skeletal/EZ properties from Ω to Υ and concludes that both are catégories squelettiques and Berger–Moerdijk EZ categories. No proof is supplied, and the paper later relies on these properties for Proposition 4.25, the identification ∂ΥT = sk_{n-1}ΥT, and the induction in Proposition 7.13. This is an omitted proof and a correctness risk, not a circularity: the properties are asserted as transferable consequences of Propositions 4.12 and 4.16 rather than being defined in terms of, or derived from, the model structures being constructed. Honest non-finding is therefore appropriate: the derivation chain is self-contained against external benchmarks, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data; the paper is purely combinatorial and categorical. The model structures are constructed on previously defined categories, and the planar results are obtained by slicing over the associative cyclic operad. No new speculative entities are introduced.

assumptions (5)
  • domain assumption Model structure for quasi-operads on dendroidal sets bΩ and its Quillen equivalence with simplicial operads sOp (Cisinski-Moerdijk).
    Used as the base for right-inducing the bΥ model structure (Thm 6.5) and for lifting the Quillen equivalence to sCyc (Thm 7.14).
  • domain assumption Model structure on simplicial cyclic operads sCyc from DCH21 (Theorem 6.3).
    The target model category in Theorem B; established by the second author and Drummond-Cole prior to this work.
  • domain assumption Root-elision f: Ω -> Υ is a discrete fibration, and its restriction Ωact -> Υact is a discrete opfibration.
    Background about the cyclic dendroidal category, cited to Hac24a, underpinning Lemmas 6.2 and 8.1.
  • domain assumption Dendroidal Rezk model structure on sbΩ and the Quillen equivalence bΩ -> sbΩRezk (CM13a).
    Used to construct sbΥRezk in Theorem 8.2 and its Quillen equivalence in Theorem 8.4.
  • domain assumption Slice equivalences plOp ≃ Op/Ass and plCyc ≃ Cyc/Ass.
    Used in Section 9 to transfer the main results to planar cyclic operads.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Models for cyclic infinity operads." pith.science (2026). https://pith.science/paper/CSUQKWG6

@misc{pith2026250615622,
  author       = {Pith},
  title        = {Pith review of: Models for cyclic infinity operads},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSUQKWG6}},
  note         = {Machine review of arXiv:2506.15622}
}
abstract

We construct model structures on cyclic dendroidal sets and cyclic dendroidal spaces for cyclic quasi-operads and complete cyclic dendroidal Segal spaces, respectively. We show these models are Quillen equivalent to the model structure for simplicial cyclic operads. This answers in the affirmative a question of the second author and Drummond-Cole concerning model structures for cyclic $\infty$-operads. We infer similar statements for planar cyclic $\infty$-operads, providing the model-categorical foundation needed to complete Walde's program on the relationship between cyclic 2-Segal spaces and planar cyclic $\infty$-operads.

Figures

Figures reproduced from arXiv: 2506.15622 by the authors.

Figure 1
Figure 1. Ternary composition in a cyclic operad the cyclic dendroidal category Υ introduced by the second author, Robertson, and Yau [HRY20a]. Theorem A. There is a Quillen model structure on the category of cyclic dendroidal sets whose fibrant objects are the cyclic quasi-operads and whose cofibrations are the normal monomorphisms. A morphism is a weak equivalence just when the underlying map of dendroidal sets is a weak op… view at source ↗
Figure 2
Figure 2. An edge, a star, and a linear graph 4. Rooted and unrooted trees In this section we establish background we will need concerning tree categories. We will use the following definition of graph with loose ends from [JK11, §3], which is based on arcs, which are edges equipped with on of the two possible orientations. Definition 4.1. An undirected graph G is a diagram of finite sets † A D V t with † a fixpoint-free invo… view at source ↗
Figure 3
Figure 3. Objects in ∆ versus their counterparts in Ω There is a fully faithful functor ∆ → Ω sending [n] = {0, 1, . . . , n} to the rooted linear tree (⟨n⟩, n). We will simply regard ∆ as a full subcategory of Ω, ignoring the distinction between the objects [n] and (⟨n⟩, n). See [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The isomorphism f ∗N ∼= NF Υb Ωb sOp Υb sCyc sOp f ∗ id wε C f! wκ L id wη C F κ L f! β := F C C f ∗ F C C f ∗ [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The natural transformation β : Cf ∗ ⇒ FC κ C L f! β = = α C N F f ∗ N N F f ∗ N N F f ∗ N [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Proof of Lemma 7.6 Lemma 7.6. The following mate of α is equal to β. Υb sCyc sOp Υb Ωb sOp C id tη F N tα N id tε f ∗ C Proof. Figure [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Proof of Lemma 7.11 Here each vertical map acts on the component corresponding to t ∈ bd T as the relevant component of γ. Proof. If T ∼= ⟨0⟩ is an edge, then ∂ΥT is ∅. As all functors in the diagram are left adjoints, the above square is a pushout since the left verti…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 65 canonical work pages

  1. [1]

    12 (2010), no

    Clark Barwick, On left and right model categories and left and right B ousfield localizations , Homology Homotopy Appl. 12 (2010), no. 2, 245--320. 2771591

    work page 2010

  2. [2]

    , From operator categories to higher operads, Geom. Topol. 22 (2018), no. 4, 1893--1959. 3784514

    work page 2018

  3. [3]

    M. A. Batanin and C. Berger, The lattice path operad and H ochschild cochains , Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 23--52. 2581904

    work page 2009

  4. [4]

    Bergner, A model category structure on the category of simplicial categories, Trans

    Julia E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043--2058. 2276611

    work page 2007

  5. [5]

    Clemens Berger, Moment categories and operads, Theory Appl. Categ. 38 (2022), Paper No. 39, 1485--1537. 4541944

    work page 2022

  6. [6]

    Thomas Blom and Ieke Moerdijk, Profinite completions of topological operads, preprint arXiv:2312.12567 [math.AT]

    work page arXiv

  7. [7]

    Clemens Berger and Ieke Moerdijk, On an extension of the notion of R eedy category , Math. Z. 269 (2011), no. 3-4, 977--1004. 2860274

    work page 2011

  8. [8]

    Thomas Blom and Ieke Moerdijk, Profinite -operads , Adv. Math. 408 (2022), Paper No. 108601, 50. 4462939

    work page 2022

Show all 71 references
  1. [9]

    Pereira, Rigidification of dendroidal infinity-operads, Homology Homotopy Appl

    Peter Bonventre and Lu\' s A. Pereira, Rigidification of dendroidal infinity-operads, Homology Homotopy Appl. 23 (2021), no. 2, 349--372. 4317574

    2021

  2. [10]

    Luciana Basualdo Bonatto and Marcy Robertson, Galois actions on surfaces and a higher genus G rothendieck-- T eichmüller group , in preparation

  3. [11]

    Shaul Barkan and Jan Steinebrunner, Cyclic -operads, in preparation

  4. [12]

    K-Theory 13 (2014), no

    Eugenia Cheng, Nick Gurski, and Emily Riehl, Cyclic multicategories, multivariable adjunctions and mates, J. K-Theory 13 (2014), no. 2, 337--396. 3189430

    2014

  5. [13]

    Hongyi Chu and Rune Haugseng, Homotopy-coherent algebra via S egal conditions , Adv. Math. 385 (2021), Paper No. 107733, 95. 4256131

    2021

  6. [14]

    Hongyi Chu, Rune Haugseng, and Gijs Heuts, Two models for the homotopy theory of -operads , J. Topol. 11 (2018), no. 4, 857--873. 3847208

    2018

  7. [15]

    308, xxiv+390

    Denis-Charles Cisinski, Les pr\'efaisceaux comme mod\`eles des types d'homotopie, Ast\'erisque (2006), no. 308, xxiv+390. 2294028 (2007k:55002)

    2006

  8. [16]

    Denis-Charles Cisinski and Ieke Moerdijk, Dendroidal sets as models for homotopy operads, J. Topol. 4 (2011), no. 2, 257--299. 2805991

    2011

  9. [17]

    , Dendroidal S egal spaces and -operads , J. Topol. 6 (2013), no. 3, 675--704. 3100887

    2013

  10. [18]

    , Dendroidal sets and simplicial operads, J. Topol. 6 (2013), no. 3, 705--756. 3100888

    2013

  11. [19]

    Alain Connes, Cohomologie cyclique et foncteurs Ext ^n , C. R. Acad. Sci. Paris S\' e r. I Math. 296 (1983), no. 23, 953--958. 777584

    1983

  12. [20]

    Drummond-Cole and Philip Hackney, A criterion for existence of right-induced model structures, Bull

    Gabriel C. Drummond-Cole and Philip Hackney, A criterion for existence of right-induced model structures, Bull. Lond. Math. Soc. 51 (2019), no. 2, 309--326. 3937590

    2019

  13. [21]

    , Dwyer- K an homotopy theory for cyclic operads , Proc. Edinb. Math. Soc. (2) 64 (2021), no. 1, 29--58. 4249838

    2021

  14. [22]

    Tobias Dyckerhoff and Mikhail Kapranov, Triangulated surfaces in triangulated categories, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 6, 1473--1524. 3801819

    2018

  15. [23]

    2244, Springer, Cham, 2019

    , Higher S egal spaces , Lecture Notes in Mathematics, vol. 2244, Springer, Cham, 2019. 3970975

    2019

  16. [24]

    thesis, University of Melbourne, 2023

    Patrick Elliott, Homotopy coherent cyclic operads, Ph.D. thesis, University of Melbourne, 2023

    2023

  17. [25]

    Zbigniew Fiedorowicz and Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57--87. 998125

    1991

  18. [26]

    Andrea Gagna, The C isinski-- M oerdijk model structure on planar dendroidal sets , Master's thesis, Universiteit Leiden, 2015

    2015

  19. [27]

    Getzler and M

    E. Getzler and M. M. Kapranov, Cyclic operads and cyclic homology, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 167--201. 1358617

    1995

  20. [28]

    110 (1998), no

    , Modular operads, Compositio Math. 110 (1998), no. 1, 65--126. 1601666

    1998

  21. [29]

    Richard Garner, Magdalena Kędziorek, and Emily Riehl, Lifting accessible model structures, J. Topol. 13 (2020), no. 1, 59--76. 3999672

    2020

  22. [30]

    Philip Hackney, Categories of graphs for operadic structures, Math. Proc. Cambridge Philos. Soc. 176 (2024), no. 1, 155--212. 4680484

    2024

  23. [31]

    Math., vol

    , Segal conditions for generalized operads, Higher S tructures in T opology, G eometry, and P hysics, Contemp. Math., vol. 802, Amer. Math. Soc., Providence, RI, 2024, pp. 161--194. 4773880

    2024

  24. [32]

    Gijs Heuts, Vladimir Hinich, and Ieke Moerdijk, On the equivalence between L urie's model and the dendroidal model for infinity-operads , Adv. Math. 302 (2016), 869--1043. 3545944

    2016

  25. [33]

    Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol

    Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. 1944041

    2003

  26. [34]

    Homotopy Relat

    , Overcategories and undercategories of cofibrantly generated model categories, J. Homotopy Relat. Struct. 16 (2021), no. 4, 753--768. 4343079

    2021

  27. [35]

    Kathryn Hess, Magdalena Kędziorek, Emily Riehl, and Brooke Shipley, A necessary and sufficient condition for induced model structures, J. Topol. 10 (2017), no. 2, 324--369. 3653314

    2017

  28. [36]

    Gijs Heuts and Ieke Moerdijk, Simplicial and dendroidal homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 75, Springer, Cham, 2022. 4485749

    2022

  29. [37]

    Ralf Hinze and Dan Marsden, Introducing string diagrams: the art of category theory, Cambridge University Press, New York, 2023. 4739685

    2023

  30. [38]

    Philip Hackney and Martina Rovelli, Induced model structures for higher categories, Proc. Amer. Math. Soc. 150 (2022), no. 11, 4629--4644. 4489301

    2022

  31. [39]

    Philip Hackney, Marcy Robertson, and Donald Yau, Higher cyclic operads, Algebr. Geom. Topol. 19 (2019), no. 2, 863--940. 3924179

    2019

  32. [40]

    , A graphical category for higher modular operads, Adv. Math. 365 (2020), 107044, 61. 4064770

    2020

  33. [41]

    , Modular operads and the nerve theorem, Adv. Math. 370 (2020), 107206, 39. 4099828

    2020

  34. [42]

    Notes Theor

    Andr\'e Joyal and Joachim Kock, Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract), Electron. Notes Theor. Comput. Sci. 270 (2011), no. 2, 105--113

    2011

  35. [43]

    45, CRM, Barcelona, 2008, https://mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern45-2.pdf mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern45-2.pdf

    Andr \'e Joyal, The theory of quasi-categories and its applications, Quaderns, no. 45, CRM, Barcelona, 2008, https://mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern45-2.pdf mat.uab.cat/ kock/crm/hocat/advanced-course/Quadern45-2.pdf

    2008

  36. [44]

    Andr\'e Joyal and Ross Street, The geometry of tensor calculus. I , Adv. Math. 88 (1991), no. 1, 55--112. 1113284

    1991

  37. [45]

    Math., vol

    Andr\'e Joyal and Myles Tierney, Quasi-categories vs S egal spaces , Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 277--326. 2342834

    2007

  38. [46]

    Joachim Kock, Polynomial functors and trees, Int. Math. Res. Not. IMRN (2011), no. 3, 609--673. 2764874

    2011

  39. [47]

    G. M. Kelly and Ross Street, Review of the elements of 2 -categories , Category S eminar ( P roc. S em., S ydney, 1972/1973), Lecture Notes in Math., vol. Vol. 420, Springer, Berlin-New York, 1974, pp. 75--103. 357542

    1972

  40. [48]

    Lauda, Frobenius algebras and ambidextrous adjunctions, Theory Appl

    Aaron D. Lauda, Frobenius algebras and ambidextrous adjunctions, Theory Appl. Categ. 16 (2006), No. 4, 84--122. 2210667

    2006

  41. [49]

    Zhiwei Li, A note on model (co)slice categories, Chinese Ann. Math. Ser. B 37 (2016), no. 1, 95--102. 3436599

    2016

  42. [50]

    thesis, Universiteit Utrecht, 2010

    Andor Luk\'acs, Cyclic O perads, D endroidal S tructures, H igher C ategories , Ph.D. thesis, Universiteit Utrecht, 2010

    2010

  43. [51]

    170, Princeton University Press, Princeton, NJ, 2009

    Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. 2522659

    2009

  44. [52]

    Martin Markl, Operads and PROP s , Handbook of algebra. V ol. 5, Handb. Algebr., vol. 5, Elsevier/North-Holland, Amsterdam, 2008, pp. 87--140. 2523450

    2008

  45. [53]

    Noncommut

    , Modular envelopes, OSFT and nonsymmetric (non- ) modular operads , J. Noncommut. Geom. 10 (2016), no. 2, 775--809. 3519052

    2016

  46. [54]

    J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, vol. Vol. 271, Springer-Verlag, Berlin-New York, 1972. 420610

    1972

  47. [55]

    Ieke Moerdijk and Joost Nuiten, Minimal fibrations of dendroidal sets, Algebr. Geom. Topol. 16 (2016), no. 6, 3581--3614. 3584268

    2016

  48. [56]

    Courses Math

    Ieke Moerdijk, Lectures on dendroidal sets, Simplicial methods for operads and algebraic geometry, Adv. Courses Math. CRM Barcelona, Birkh\" a user/Springer Basel AG, Basel, 2010, Notes written by Javier J. Guti\' e rrez, pp. 1--118. 2778589

    2010

  49. [57]

    J. P. May and K. Ponto, More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012. 2884233

    2012

  50. [58]

    Ieke Moerdijk and Ittay Weiss, Dendroidal sets, Algebr. Geom. Topol. 7 (2007), 1441--1470. 2366165

    2007

  51. [59]

    Moerdijk and I

    I. Moerdijk and I. Weiss, On inner K an complexes in the category of dendroidal sets , Adv. Math. 221 (2009), no. 2, 343--389. 2508925

    2009

  52. [60]

    Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol

    Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. No. 43, Springer-Verlag, Berlin-New York, 1967. 223432

    1967

  53. [61]

    Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973--1007. 1804411

    2001

  54. [62]

    Michael Shulman, All ( ,1) -toposes have strict univalent universes, preprint arXiv:1904.07004 [math.AT]

    1904 arXiv

  55. [63]

    Michael Shulman, The 2- C hu- D ialectica construction and the polycategory of multivariable adjunctions , Theory Appl. Categ. 35 (2020), Paper No. 4, 89--136. 4063570

    2020

  56. [64]

    Jan Steinebrunner, 2-dimensional TFT s via modular -operads , preprint arXiv:2506.22104 [math.CT]

    arXiv

  57. [65]

    Michelle Strumila, Quasi modular operads, preprint arXiv:2504.06522 [math.CT]

    arXiv

  58. [66]

    Tashi Walde, 2- S egal spaces as invertible infinity-operads , Algebr. Geom. Topol. 21 (2021), no. 1, 211--246. 4224740

    2021

  59. [67]

    Mark Weber, Familial 2-functors and parametric right adjoints, Theory Appl. Categ. 18 (2007), No. 22, 665--732. 2369114

    2007

  60. [68]

    thesis, Universiteit Utrecht, 2007

    Ittay Weiss, Dendroidal sets, Ph.D. thesis, Universiteit Utrecht, 2007

    2007

  61. [69]

    , From operads to dendroidal sets, Mathematical foundations of quantum field theory and perturbative string theory, Proc. Sympos. Pure Math., vol. 83, Amer. Math. Soc., Providence, RI, 2011, pp. 31--70. 2742425

    2011

  62. [70]

    170, American Mathematical Society, Providence, RI, 2016

    Donald Yau, Colored operads, Graduate Studies in Mathematics, vol. 170, American Mathematical Society, Providence, RI, 2016. 3444662

    2016

  63. [71]

    Johnson, Boardman-- V ogt resolutions of generalized props , draft available at u.osu.edu/yau.22/main https://u.osu.edu/yau.22/main/

    Donald Yau and Mark W. Johnson, Boardman-- V ogt resolutions of generalized props , draft available at u.osu.edu/yau.22/main https://u.osu.edu/yau.22/main/

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.