Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture

Boris Shoikhet; Michael Batanin

REVIEW 4 major objections 4 minor 33 references

Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For any dg algebra $A$, the paper exhibits a model of the chain Swiss Cheese operad for which the category of actions on $A$ is equivalent to maps into the Hochschild complex $\mathrm{Hoch}(A)$, making $(\mathrm{Hoch}(A),A)$ a final…

desk verdict A clean chain-level Swiss Cheese theorem for a specific model, but the bridge to the actual Swiss Cheese operad is a one-sentence 'straightforward' identification that still needs a real proof. read the letter →

arxiv 2412.03239 v2 pith:DMINJ7ZQ submitted 2024-12-04 math.CT math.KTmath.QA

Michael Batanin , Boris Shoikhet This is my paper · ORCID

classification math.CTmath.KTmath.QA MSC 18M7018D2016E40

keywords SwissCheeseconjecturetwistedtensorproductsmalldgcategoriesbraceoperadHochschildcomplexskew-monoidalcategorycoloredEckmann-Hiltonargument

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 proves a strict, chain-level version of the Swiss Cheese conjecture for a specially chosen model of the chain Swiss Cheese operad. For any dg algebra $A$, the category of actions of a brace/$E_2$-type algebra on $A$ is equivalent to the category of such algebras equipped with a map into the Hochschild complex $\mathrm{Hoch}(A)$. Because the identity map on $\mathrm{Hoch}(A)$ is a final object in that comma category, the pair $(\mathrm{Hoch}(A), A)$ is a final object in the category of actions before any localisation. This gives the conjectured universal action an explicit chain-level representative.

What carries the argument

The load-bearing object is the twisted tensor product $\widetilde{\otimes}$ on the category of small dg categories: a skew-monoidal structure whose right adjoint is the coherent internal Hom $[D,E]_{\mathrm{coh}}$, with morphism complexes given by reduced Hochschild complexes. The adjunction $\mathrm{Hom}(C\widetilde{\otimes}D,E)\simeq \mathrm{Hom}(C,[D,E]_{\mathrm{coh}})$ and the one-sided associator formula, which is literally the brace relation, convert an action of $B$ on $A$ into a monoid map $B\to [A,A]_{\mathrm{coh}}(\mathrm{id},\mathrm{id})=\mathrm{Hoch}(A)$. Lemma 1.2 is the hinge: a dg algebra is a $\widetilde{\otimes}$-monoid sharing its unit exactly when it is an algebra over the brace operad, and the brace operad is weakly equivalent to $E_2$. This is the relaxed Eckmann-Hilton setting in which the strict center $Z(A)$ is replaced by the full Hochschild complex.

What would settle it

Take a small dg algebra $A$, for instance $A=k[x]/(x^2)$, and a free brace algebra $B$ on one generator; compute the set of $\mathrm{O}_0$-actions of $B$ on $A$ and compare it with the set of brace-algebra maps $B\to\mathrm{Hoch}(A)$: a mismatch would falsify Corollary 2.7 and Theorem 1.1, and a direct comparison of a defining relation of $\mathrm{O}_0$ with the simplicial condensation would test the Section 3.2 identification.

Watch

Extended reading notes

Core claim

Theorem 1.1 states that there is a colored dg operad $\mathrm{O}_0$ with two colors, weakly equivalent to the chain Swiss Cheese operad $\mathrm{SC}_1$, such that for any dg algebra $A$ one has an equivalence of categories $\mathrm{Alg}(\mathrm{O}_0)_A \simeq \mathrm{Alg}(E_2^{\mathrm{O}_0})/\mathrm{Hoch}(A)$. Here $E_2^{\mathrm{O}_0}$ is the restriction of $\mathrm{O}_0$ to the color for the 2-algebra argument, and the comma category consists of maps $B\to\mathrm{Hoch}(A)$ compatible with the monoid structures. In particular, the homotopy category $\mathrm{Ho}(\mathrm{Alg}(\mathrm{O}_0)_A)$ has a final object represented by $(\mathrm{Hoch}(A),A)$. The operad $\mathrm{O}_0$ is identified with the reduced $k$-linear condensation of the simplicial Swiss Cheese operad of natural operations, and an algebra over it is a pair $(B,A)$ in which $B$ is a monoid for the twisted tensor product of dg categories acting on $A$; by Lemma 1.2 such monoids are exactly brace algebras.

Load-bearing premise

The whole argument rests on the identification, asserted in Section 3.2 with details left to the reader, between the operad $\mathrm{O}_0$ built from the twisted tensor product and the normalized $k$-linear condensation of the simplicial Swiss Cheese operad of [DTT]; if those two operads are not isomorphic, Theorem 1.1 concerns an operad unrelated to Swiss Cheese.

Editorial extensions

If this is right

For the model $\mathrm{O}_0$, $\mathrm{Ho}(\mathrm{Alg}(\mathrm{O}_0)_A)$ has final object $(\mathrm{Hoch}(A),A)$, so the homotopy-categorical content of the Swiss Cheese conjecture holds without localising.
Every action of a brace algebra $B$ on $A$ determines a canonical map $B\to\mathrm{Hoch}(A)$, giving a universal solution to the Deligne conjecture for $A$ in this model.
The strict statement replaces the center $Z(A)$ in the baby Eckmann-Hilton argument by the full Hochschild complex, which is exactly the relaxed behaviour expected for $E_2$ acting on $E_1$.
The same formal argument suggests Conjecture 1.3: for any closed multitensor associated to a 2-operad $\mathcal{P}$, actions on a $P_1$-algebra $A$ are equivalent to maps into $[A,A]_{\mathrm{coh}}(\mathrm{id},\mathrm{id})$.

Reading between the lines

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

The same two-step scheme should prove Swiss-Cheese-type final-object theorems for any contractible 2-operad whose associated multitensor is closed; the main obstruction is closure, which already fails for the Gray tensor product of 2-categories.
The chain-level equivalence makes the universal action $B\to\mathrm{Hoch}(A)$ computable in examples via brace operations, for instance for $A=k[x]/(x^2)$, without passing through cofibrant resolutions.
Writing out the Section 3.2 comparison in detail, and checking whether $\mathrm{O}_0$ is $\Sigma$-cofibrant, would be natural first steps toward the announced transfer of Theorem 1.1 to an arbitrary $\Sigma$-cofibrant model of $\mathrm{SC}_1$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main thing to know: this paper has one genuinely new result and one large unproven identification. The new result is Theorem 1.1: for a specific two-colored dg operad O0, the category of actions on a fixed dg algebra A is equivalent on the nose to Alg(E2^O)/Hoch(A), giving a chain-level final object (Hoch(A), A) without passing to the homotopy category. That is stronger than the homotopy-categorical Swiss Cheese statement for this model, and the formal argument in Section 3.3 is short and convincing: it goes through the adjunction for the twisted tensor product, Lemma 1.2 identifying ∼⊗-monoids with brace algebras, and Corollary 2.7. Lemma 1.2 is the key new observation, and the computation in Section 3.1 is plausible; the signs are outsourced to [Sh2], which is a minor annoyance.

The soft spot is exactly the one the stress-test note names. Section 3.2 defines O0 by a universal property and then says it is 'straightforwardly' isomorphic to the normalized condensation of the simplicial Swiss Cheese operad of [DTT], leaving details to the reader. That one sentence is the bridge between the whole paper and Kontsevich's conjecture. [DTT, Th. 2.1] proves weak equivalence to SC1 for their operad, not for O0 unless the isomorphism is established. If a low-arity component or symmetric group action of O0 differs from the condensation, Theorem 1.1 is a chain-level final object for an operad that may have nothing to do with Swiss Cheese. I expect the identification is true—both sides have brace algebras in one color and the construction is natural—but as written it is a conjecture, not a proof. This is the first thing a referee should demand.

Corollary 2.7 is also only sketched, but that one is standard and I am not worried. The paper is otherwise clean; I found no fitting, circularity, or invented entities. The citations are appropriate, and self-citation is not a problem when the cited results are prior and independently published.

Who is this for? People working on Swiss Cheese, Deligne conjecture, and higher operads. The framework via skew-monoidal categories and 2-operads is useful, and Conjecture 1.3 is a nice programmatic generalization. It deserves a serious referee; the theorem is important enough and the proof is likely repairable. I would not desk reject it.

Referee Report

4 major / 4 minor

Summary. The paper claims a chain-level version of Kontsevich's Swiss Cheese conjecture for a specific model of the Swiss Cheese operad. For a dg algebra A, the authors define a two-colored dg operad O0 via the twisted tensor product ∼⊗ on small dg categories, and state that O0 is weakly equivalent to the chain Swiss Cheese operad SC1. They prove that the category Alg(O0)_A of actions on A is equivalent to the comma category Alg(E2^{O0})/Hoch(A), so that the pair (Hoch(A), A) is a final object in Alg(O0)_A without passing to the homotopy category. The proof proceeds by identifying ∼⊗-monoids with brace algebras (Lemma 1.2), using the closed skew-monoidal structure to identify actions with maps into an internal Hom, and then invoking the identification of O0 with the normalized dg condensation of the simplicial Swiss Cheese operad from [DTT].

Significance. If the main theorem is fully justified, the paper makes a significant conceptual contribution: it gives a clean, non-geometric route to a strict version of the Swiss Cheese conjecture, and it highlights the role of the twisted tensor product of dg categories as a 'relaxed' Eckmann-Hilton context. The idea of using a closed skew-monoidal structure to reduce Swiss Cheese-type statements to comma categories is elegant and potentially generalizable, as the concluding conjecture indicates. The paper is honest that passage from O0 to arbitrary weakly equivalent Σ-cofibrant models is deferred to a sequel. However, the current manuscript is very short and several load-bearing identifications and proofs are left to the reader, most importantly the isomorphism between O0 and the [DTT] operad. The strength of the paper therefore depends on details that are not yet written down.

major comments (4)

[3.2] The statement immediately after the definition of O0 — 'This identiﬁcation is straightforward, so we leave details to the reader' — is load-bearing for the whole paper. Theorem 1.1 is a statement about a Swiss Cheese-type operad only if O0 is the normalized k-linear condensation |sØ|_norm of the simplicial Swiss Cheese operad of [DTT] (or is otherwise shown weakly equivalent to SC1). The manuscript does not prove the isomorphism of colored dg operads, and it does not give a precise reference to a statement that establishes it. In particular, the low-arity components, the differential on the ε-generators, and the symmetric group actions of O0 must be matched with those of |sØ|_norm. I request a full proof or a precise citation to a theorem that contains this isomorphism.
[3.3 / Corollary 2.7] The proof of Theorem 1.1 is one paragraph and rests entirely on Corollary 2.7, whose proof is left to the reader. The supporting Lemma 2.6 says 'It is straightforward, anyway, and the details are left to the reader.' The correspondence between module structures over a skew-monoidal monoid and monoid maps into [A,A](id,id) is not automatic: the one-sided associator and the non-invertible unit maps require checking that the two equaliser diagrams in the proof of Lemma 2.6 correspond under the adjunction. Since the final-object statement follows from this equivalence, this is not a cosmetic omission. Please include the missing proof or a direct reference to a published full proof.
[3.2, last paragraph] The weak equivalence O0 ≃ SC1 is asserted by combining the unproved isomorphism with [DTT, Th. 2.1] and 'arguments similar to [BBM]'. The paper does not explain which theorem in [BBM] is being used or how the isomorphism transfers the statement. Because Theorem 1.1's title claim ('Kontsevich's Swiss Cheese conjecture') depends on this weak equivalence, the inference needs to be written out: state precisely what [DTT, Th. 2.1] proves and why it applies to O0.
[3.1, proof of Lemma 1.2] The proof of Lemma 1.2 asserts that equations (3.1)–(3.4) follow from (2.4), (2.6), and (2.10), but no sign check is included. The signs in the associator α of Theorem 2.2(iv) are explicitly deferred to [Sh2], and the signs in the brace differential (3.4) are not matched with the differential of ε in (2.6). Since Lemma 1.2 is the bridge between the twisted tensor product formalism and the brace operad, the sign verification should be given in full or the specific statements in [Sh2] should be quoted.

minor comments (4)

[Abstract, §1.3, §3.2] The notation for the two colors is inconsistent: the abstract uses a and b, while §3.2 uses c and a; the restriction operad is called E2^O in the abstract and E^{O0}_2 in Theorem 1.1. Please unify.
[Throughout] There are several typographical errors: 'the the category' (Abstract), 'the fact the the pair' (§1.2), 'possess the naturality' (§2.1), and 'the reason considering ... is that' (§2.1).
[2.2, relation (R1)] The relation (R1) is written with '*' but composition order is not defined; also the formula '(f1 ⊗ idd) ∗ (f2 ⊗ idd) = (f1f2) ∗ idd' has a typesetting issue that should be corrected.
[2.1] The term 'reduced Hochschild complex' is used for [D,E](F,G), but the relationship to the usual (non-reduced) Hochschild cochain complex of a dg algebra is not spelled out; a brief remark would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the category equivalence in Theorem 1.1 is a formal adjunction consequence, and the Swiss Cheese identification, while asserted rather than proved, is a non-circular gap.

full rationale

The main equivalence Alg(O0)_A ≃ Alg(E2^O)/Hoch(A) is derived in Section 3.3 from Corollary 2.7 and Lemma 2.6: in the closed skew-monoidal category (Cat_dg(k), ∼⊗), a ∼⊗-monoid action of B on A is bijective with monoid maps B → [A,A](id,id), and [A,A](id,id) is by definition Hoch(A). The identity map on [A,A](id,id) is then the final object in the comma category. Lemma 1.2, proved in Section 3.1, identifies ∼⊗-monoids with brace algebras; this is an internal computation, not an assumption of the theorem. The weak equivalence O0 ≃ SC1 is not proved in the paper: Section 3.2 asserts an isomorphism O0 ≅ |sØ|_norm as 'straightforward' without details and then cites [DTT, Th. 2.1]. This is a load-bearing gap and a correctness risk, but it is not circular—the [DTT] theorem does not assume Theorem 1.1, and the paper's formal result would remain a valid statement about O0 even if the comparison failed. Self-citations to [Sh1], [Sh2], and [BBM] supply prior published constructions (twisted tensor product, skew-monoidal structure, brace-operad identification); these do not assume the theorem's conclusion, so they do not make the derivation circular. Accordingly, no specific circular step is exhibited; the minor self-citation presence justifies a low nonzero score without a circular reduction.

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

No free parameters or invented entities. The paper's derivation is categorical, not numerical. The main external dependencies are [Sh2]'s adjunction and [DTT]'s weak equivalence, plus [T4]. These are prior literature, not circular inputs; however the identification with [DTT] is asserted without proof.

assumptions (5)

domain assumption Hom(C∼⊗D,E) ≃ Hom(C,[D,E]coh) holds for the twisted tensor product (Eq. 2.2).
Cited from [Sh2, Prop. 2.4]; the whole action-monoid correspondence depends on this adjunction.
domain assumption A dg algebra B is a monoid for ∼⊗ with the same unit if and only if it is a brace algebra (Lemma 1.2).
Proven in the paper by computation, but signs are delegated to [Sh2].
domain assumption O0 is isomorphic to the normalized condensation of the simplicial Swiss Cheese operad of [DTT], hence weakly equivalent to SC1.
Stated in Section 3.2 with details left to the reader, citing [DTT, Th. 2.1] and [BBM].
domain assumption The brace operad Br is weakly equivalent to the chain little disks operad E2.
Used to identify Alg(E2^O) with brace algebras; cited from Tamarkin [T4].
domain assumption Monoid actions correspond to monoid maps B to [A,A] in a closed skew-monoidal category (Corollary 2.7).
The proof is sketched in Lemma 2.6 with details left to the reader; this is the heart of Theorem 1.1.

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Pith. "Pith review of Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture." pith.science (2026). https://pith.science/paper/DMINJ7ZQ

@misc{pith2026241203239,
  author       = {Pith},
  title        = {Pith review of: Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMINJ7ZQ}},
  note         = {Machine review of arXiv:2412.03239}
}

abstract

Let $A$ be a $1$-algebra. The Kontsevich Swiss Cheese conjecture [K2] states that the homotopy category $\mathrm{Ho}(\mathrm{Act}(A))$ of actions of $2$-algebras on $A$ has a final object and that this object is weakly equivalent to the pair $(\mathrm{Hoch}(A), A),$ where $\mathrm{Hoch}(A)$ is the Hochschild complex of $A$. Here the category of actions is the category whose objects are pairs $(B,A)$ which are algebras of the chain Swiss Cheese operad such that the induced action of the little interval operad on the component $A$ coincides with the $1$-structure on $A$. We prove that there is a colored dg operad ${O}$ with 2 colors, weakly equivalent to the chain Swiss Cheese operad for which the following ``stricter" version of the Kontsevich Swiss Cheese conjecture holds. Denote the two colors of ${O}$ by $a$ (for the 1-algebra argument) and $b$ (for the 2-algebra argument), denote by $E_1^{{O}}$ the restriction of ${O}$ to the color $a$, and by $E_2^{{O}}$ the restriction of ${O}$ to the color $b$. Let $\mathrm{Alg}({O})$ be the category of dg algebras over ${O}$. For a fixed $1$-algebra $A$ we also have the the category of action $\mathrm{Alg}({O})_A$ (equal to $\mathrm{Act}(A)$ in case of the Swiss Cheese operad). We prove that there is an equivalence of categories $$\mathrm{Alg}({O})_A \cong \mathrm{Alg}(E_2^{{O}})/\mathrm{Hoch}(A)$$ We stress that for this particular model of Swiss Cheese operad the statement holds on the chain level, without passage to the homotopy category.

Figures

Figures reproduced from arXiv: 2412.03239 by the authors.

**Figure 1.** The Swiss Cheese operad component SC1(b n , am; a). One has natural operad embeddings i: E1 → SC1 and j : E2 → SC1 which send the only color of the source operad to the color a and the color b, correspondingly. An algebra over SC1 is a pair A, B of topological spaces (so that A corresponds to the color a and B corresponds to the color b) equipped with maps SC1(b n ; b) × B ×n → B, SC1(b n , am; a) × B ×n × A ×m → A … view at source ↗

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Works this paper leans on

33 extracted references · 32 canonical work pages

[1]

Batanin, The symmetrisation of n -operads and compactification of real configuration spaces, Advances in Math

M.A. Batanin, The symmetrisation of n -operads and compactification of real configuration spaces, Advances in Math. 211 (2007), 684-725

work page 2007
[2]

M.Batanin, From group action to Kontsevich Swiss-Cheese conjecture, slides for the talk at University of Adelaide, June 3, 2011

work page 2011
[3]

Batanin, C

M.A. Batanin, C. Berger, Homotopy theory for algebras over polynomial monads, Theory and Applications of Categories, Vol. 32 , 2017, No. 6, pp 148-253

work page 2017
[4]

Batanin, C

M.A. Batanin, C. Berger, M. Markl, Operads of natural operations, I: lattice paths, braces and Hochschild cochains, in Loday, Jean-Louis (ed.) et al., Operads 2009 , Proceedings of the school and conference, Luminy, France, April 20–30, 2009, Société Mathématique de France, Séminaires et Congrès 26, 1-33 (2011)

work page 2011
[5]

Batanin, D.-C

M. Batanin, D.-C. Cisinski, M. Weber, Multitensor lifting and strictly unital higher category theory, Theory Appl. Categ. , 28 , 804-856 (2013)

work page 2013
[6]

J.Bourke, S.Lack, Free skew monoidal categories, Journal of Pure and Applied Algebra 222 (10) (2018), 3255-3281

work page 2018
[7]

Berger, I

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

work page 2003
[8]

Berger, I

C. Berger, I. Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, Contemp. Math. 431 (2007), 31-58

work page 2007

Show all 33 references

[9]

Baues, The double bar and cobar constructions, Compositio Mathematica , 43 (3)(1981), 331-341

H.J. Baues, The double bar and cobar constructions, Compositio Mathematica , 43 (3)(1981), 331-341

1981
[10]

Crans, A tensor product for Gray-categories , Theory and Applications of Categories, Vol

S. Crans, A tensor product for Gray-categories , Theory and Applications of Categories, Vol. 5 , 1999, No. 2, pp 12-69

1999
[11]

Dolgushev, D

V. Dolgushev, D. Tamarkin, B. Tsygan, Int. Math. Res. Not. IMRN 20 2011, 4666-4746

2011
[12]

Noncommut

G.Faonte, A_ -functors and homotopy theory of dg-categories, J. Noncommut. Geom. 11 (3) (2017), 957-1000

2017
[13]

Getzler, J.D.S

E. Getzler, J.D.S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, archive preprint hep-th/9403055

arXiv
[14]

Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10)

V. Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)

1997
[15]

Hirschhorn, Model Categories and their Localisations , AMS Math

Ph. Hirschhorn, Model Categories and their Localisations , AMS Math. Surveys and Monographs, vol. 99, 2002

2002
[16]

Hovey, Model Categories , AMS Math

M. Hovey, Model Categories , AMS Math. Surveys and Monographs, vol. 63, 1998

1998
[17]

Hyeon Tai Jung, Gray Tensor product and Kontsevich's Swiss-Cheese conjecture, Macquarie University Master Thesis
[18]

Kontsevich, Deformation quantization of Poisson manifolds, I, Lett

M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (3) (2003), 157–216

2003
[19]

Kontsevich, Operads and motives in deformation quantisation, Lett

M. Kontsevich, Operads and motives in deformation quantisation, Lett. Math. Phys. 48 (1) (1999), 35–72

1999
[20]

Kontsevich, Y

M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and Deligne's conjecture, in Conference Moshe Flato 1999, Quantization, Deformations, and Symmetries , vol. I, Ed. G. Dito and D. Sternheimer, Kluwer Academic Publishers, 2000, 255-307

1999
[21]

Kadeishvili, On the cobar construction of a bialgebra, Homology, Homotopy and Appl

T. Kadeishvili, On the cobar construction of a bialgebra, Homology, Homotopy and Appl. , 7 (2) , 2005, 109-122

2005
[22]

Keller, Derived invariance of higher structures on the Hochschild complex, available at https://webusers.imj-prg.fr/ bernhard.keller/publ/

B. Keller, Derived invariance of higher structures on the Hochschild complex, available at https://webusers.imj-prg.fr/ bernhard.keller/publ/
[23]

Lef\` e vre-Hasegawa, Sur les A_ -cat\' e gories, Thesis, available at https://webusers.imj-prg.fr/ bernhard.keller/lefevre/publ.html

K. Lef\` e vre-Hasegawa, Sur les A_ -cat\' e gories, Thesis, available at https://webusers.imj-prg.fr/ bernhard.keller/lefevre/publ.html
[24]

McClure, J

J. McClure, J. Smith, Multivariable cochain operations and little n-cubes, J. Amer. Math. Soc. , 16 (3)(2003), 681-704

2003
[25]

Street, Skew-closed categories, Journal of Pure and Advanced Algebra , 217 (2013) 973-988

R. Street, Skew-closed categories, Journal of Pure and Advanced Algebra , 217 (2013) 973-988

2013
[26]

Shoikhet, On the twisted tensor product of small dg categories, J

B. Shoikhet, On the twisted tensor product of small dg categories, J. of Noncommutative geometry , 14 (2)(2020), 789-820

2020
[27]

Shoikhet, The twisted tensor product of dg categories and a contractible 2-operad, Advances in Math

B. Shoikhet, The twisted tensor product of dg categories and a contractible 2-operad, Advances in Math. 434 (2023), 42 pages

2023
[28]

Tamarkin, What do dg categories form? Compos

D. Tamarkin, What do dg categories form? Compos. Math. 143 (5), (2007), 1335-1358

2007
[29]

Tamarkin, Quantization of Lie bialgebras via the formality of the operad of little discs, Geom

D. Tamarkin, Quantization of Lie bialgebras via the formality of the operad of little discs, Geom. Funct. Anal. 17 (2007), no. 2, 537-604

2007
[30]

Tamarkin, Formality of chain operad of little discs, Lett

D. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (1-2) (2003), pp. 65–72

2003
[31]

Tamarkin, Another proof of M.Kontsevich formality theorem for R ^n , preprint math.QA.9803025

D. Tamarkin, Another proof of M.Kontsevich formality theorem for R ^n , preprint math.QA.9803025
[32]

Thomas, Kontsevich's Swiss cheese conjecture, Geom

J. Thomas, Kontsevich's Swiss cheese conjecture, Geom. Topol. 20(1) : 1-48 (2016). DOI: 10.2140/gt.2016.20.1

2016 doi
[33]

Voronov, The Swiss-cheese operad, In: Homotopy invariant algebraic structures (Baltimore, MD, 1998) , Contemp

A. Voronov, The Swiss-cheese operad, In: Homotopy invariant algebraic structures (Baltimore, MD, 1998) , Contemp. Math. 239 1999, pp. 365–373

1998

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Batanin, The symmetrisation of n -operads and compactification of real configuration spaces, Advances in Math

M.A. Batanin, The symmetrisation of n -operads and compactification of real configuration spaces, Advances in Math. 211 (2007), 684-725

work page 2007

[2] [2]

M.Batanin, From group action to Kontsevich Swiss-Cheese conjecture, slides for the talk at University of Adelaide, June 3, 2011

work page 2011

[3] [3]

Batanin, C

M.A. Batanin, C. Berger, Homotopy theory for algebras over polynomial monads, Theory and Applications of Categories, Vol. 32 , 2017, No. 6, pp 148-253

work page 2017

[4] [4]

Batanin, C

M.A. Batanin, C. Berger, M. Markl, Operads of natural operations, I: lattice paths, braces and Hochschild cochains, in Loday, Jean-Louis (ed.) et al., Operads 2009 , Proceedings of the school and conference, Luminy, France, April 20–30, 2009, Société Mathématique de France, Séminaires et Congrès 26, 1-33 (2011)

work page 2011

[5] [5]

Batanin, D.-C

M. Batanin, D.-C. Cisinski, M. Weber, Multitensor lifting and strictly unital higher category theory, Theory Appl. Categ. , 28 , 804-856 (2013)

work page 2013

[6] [6]

J.Bourke, S.Lack, Free skew monoidal categories, Journal of Pure and Applied Algebra 222 (10) (2018), 3255-3281

work page 2018

[7] [7]

Berger, I

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

work page 2003

[8] [8]

Berger, I

C. Berger, I. Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, Contemp. Math. 431 (2007), 31-58

work page 2007