N∞-operads and associahedra

Balchin, Scott; Barnes, David; Roitzheim, Constanze

doi:10.2140/pjm.2021.315.285

Cited by 13 publications

(25 citation statements)

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“…We can put a partial order on the set of rooted binary trees with n leaves by saying that one tree is larger than another if it can be obtained from the latter by rotating a branch to the right. Balchin-Barnes-Roitzheim found that, indeed, N ∞ -diagrams for G = C p n and rooted binary trees with n + 2 leaves are isomorphic as posets [BBR19]. However, we do not wish to elaborate on this result here.…”

Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 88%

“…Rubin [Rub17], Gutierrez-White [GW18] and Bonventre-Pereira [BP17] independently showed that for every such indexing system one can construct a corresponding operad. Barnes-Balchin-Roitzheim [BBR19] showed that indexing systems are equivalent to the transfer systems given in the above version of this theorem.…”

Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 89%

“…In particular, there can be only finitely many N ∞ -operads for a finite group G, and as such, it makes sense to count them. We will denote by N ∞ (G) the set of all N ∞ -operads on G. For G = C p n , the number of N ∞ -operads plus some additional structure has been determined in [BBR19].…”

Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 99%

“…It then becomes an intriguing question to see how many different types of equivariant homotopy commutativity are possible for a finite group G, and how these are related. For a cyclic group of order p n−1 , an answer was given in [BBR19], namely, the number of N ∞ -diagrams for C p n−1 is the n th Catalan number. Moreover, the set of N ∞ -diagrams for a fixed group is a lattice, which in the case of G = C p n−1 is isomorphic to the n-Tamari lattice (the vertex set of the n-associahedron).…”

Section: Introductionmentioning

confidence: 99%

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Equivariant homotopy commutativity for $G=C_{pqr}$

Balchin¹,

Bearup²,

Pech³

et al. 2020

Preprint

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Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 88%

Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 89%

Section: N ∞ -Operads and N ∞ -Diagramsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Equivariant homotopy commutativity for $G=C_{pqr}$

Balchin¹,

Bearup²,

Pech³

et al. 2020

Preprint

Self Cite

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“…Completely computing the poset I G of G-transfer systems is in general a very hard task, and it has only been done for some concrete finite groups (see [BBR21], [Rub21b], [BBPR20]). However the subposet I G of G-transfer systems closed under arbitrary homomorphisms is much easier to compute, as we will show through the following examples.…”

Section: Relating Global Transfer Systems To G-transfer Systemsmentioning

confidence: 99%

Global $N_\infty$-operads

Barrero¹

2022

Preprint

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We define N∞-operads in the globally equivariant setting and completely classify them. These global N∞-operads model intermediate levels of equivariant commutativity in the global world, i. e. in the setting where objects have compatible actions by all compact Lie groups. We classify global N∞-operads by giving an equivalence between the homotopy category of global N∞-operads and the partially ordered set of global transfer systems, which are much simpler, algebraic objects. We also explore the relation between global N∞-operads and N∞-operads for a single group, recently introduced by Blumberg and Hill. One interesting consequence of our results is the fact that not all equivariant N∞-operads can appear as restrictions of global N∞-operads. Contents 1. Introduction 1 2. Background on global homotopy theory 3 3. N ∞ -symmetric sequences in Spc 4 4. Global N ∞ -operads and global transfer systems 7 5. The structure of the homotopy category of global N ∞ -operads 8 6. Constructing global N ∞ -operads 11 7. Relating global transfer systems to G-transfer systems 15 Appendix A. A quick note on definitions of operads 17 References 18 MIGUEL BARREROThe homotopy groups of algebras over equivariant N ∞ -operads inherit additional structure. Let G be a finite group, and let X be a G-space. For each K H G there is a restriction morphism res H K (X) : π H * (X) → π K * (X) on equivariant homotopy groups. Each equivariant N ∞ -operad encodes the existence of certain transfer morphisms tr H K (X) : π K * (X) → π H * (X) on its algebras X, for some K H G. If this is the case we say that the abstract transfer from K to H is admissible for the given N ∞ -operad. These transfer and restriction morphisms assemble into an incomplete Mackey functor. If X is a G-spectrum and an algebra over an equivariant N ∞ -operad we can instead construct norm maps that assemble into an incomplete Tambara functor.The collection of admissible abstract transfers for an equivariant N ∞ -operad forms a relation on the set of subgroups of G. This relation has to be a partial order, and furthermore it has to be closed under conjugation and restriction to subgroups, due to the structure of the equivariant N ∞ -operads. The relations on the subgroups of G that satisfy these conditions are called G-transfer systems. Blumberg and Hill conjectured that the associated G-transfer systems completely classify the equivalence types of equivariant N ∞ -operads. The conjecture was proven separately by Gutiérrez and White in [GW18], by Rubin in [Rub21a], and by Bonventre and Pereira in [BP21].In this paper we are interested in analogous operads encoding different levels of commutativity in the setting of global homotopy theory. This is the homotopy theory of spaces which have compatible actions by all compact Lie groups. A globally equivariant space X has an associated G-space X G for each compact Lie group G. These G-spaces are compatible for varying G in the following strong sense. For any continuous homomorphism α : K → H of compact Lie groups, there is a natural ...

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Model structures on finite total orders

et al. 2023

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We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order [n], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [n].

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N∞-operads and associahedra

Cited by 13 publications

References 13 publications

Equivariant homotopy commutativity for $G=C_{pqr}$

Equivariant homotopy commutativity for $G=C_{pqr}$

Global $N_\infty$-operads

Model structures on finite total orders

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