Equivariant Trees and G-Dendroidal Sets

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Abner Kory O’Brien’
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1 Equivariant Trees and -Dendroidal Sets 31st Summer Conference in Topology and Applications: Algebraic Topology Special Session Leicester, UK August 5, 2016 Joint with Luis Pereira

2 Motivating Questions : finite group -operads: operads enriched in -spaces (or -sets) Question: What is the homotopy theory of -operads? Question: Is there an additivity theorem for N -operads? Question: What controls the combinatorics for -operads? Question: Is there a combinatorical model for - -operads?

3 Trees and Combinatorics, Non-Equivariantly Operad Combinatorics, Non-Equivariantly Question: What controls the combinatorics for operads? Answer: Trees! [Boardman-Vogt] Example (Trees encode composition diagrams via grafting:) e 1 T e 2 e 1 e 2 S 1 S 2 e 1 S e 2

4 Trees and Combinatorics, Non-Equivariantly Evaluation of Operads on Trees Example e 1 T e 2 e 1 e 2 S 1 S 2 e 1 S e 2 NO(T ) = O(3), NO(S 1 ) = O(4), NO(S 2 ) = O(2) NO(S) = O(3) O(4) O(2) O(7)

5 Trees and Combinatorics, Non-Equivariantly Dendroidal Sets This story was categorified and explored in the past decade by Weiss, Moerdijk, and Cisinski (and others) Ω = dendroidal category of the Ω(T ), free operads generated by trees Set Ωop = dendroidal sets N d : Op Set Ωop given by NO(T ) = Op(Ω(T ), O) Proposition (Weiss, 07) N d is a fully-faithful embedding, with image those dendroidal sets which are strict inner Kan complexes.

6 Trees and Combinatorics, Non-Equivariantly Infinity Operads Theorem (Cisinski-Moerdijk, 13) The category of dendroidal sets has a Quillen model structure, with fibrant objects inner Kan complexes, which is Quillen equivalent to the category of simplicially enriched operads. Theorem (Chu-Haugseng-Heuts, Cisinski-Moerdijk, Barwick) The homotopy theory of Lurie s -operads is equivalent to that of dendroidal sets.

7 Trees and Combinatorics, Non-Equivariantly Equivariantly Question: What objects describe the equivariant combinatorics of -operads? Some hints: -E -operads: model genuine commutative -ring spectra O(n) Γ for all graph subgroups Γ = Γ(ϕ), all other fixed points empty {Γ(ϕ)} {finite H-sets, H } N -operads: Not all finite H-sets, but some subsystem thereof [Blumberg-Hill]. Need: encode all H-sets into our trees

8 First uess First uess Non-equivariantly: the collection of all trees with leaves any finite set [n]. Equivariantly: the collection of all trees with leaves any finite H-set X, H. Remark Ignoring nullary vertices, an action on the leaves induces a unique action on the tree.

9 First uess Example ( = H = C 6, L = /2) b a T b+1 T /2 / Example ( = C 6, H = L = 2C 6 ) γ γ + 2 γ β S S 2/2

10 First uess Example ( = H = L = C 6 ) c c + 2 c + 4 c + 1 c + 3 c + 5 b b + 1 /2 a / R R

11 First uess Remark An arbitrary tree cannot have an arbitrary H-set as its leaves, for often the tree doesn t have the required symmetries. Example (Non-Example for = H = L = C 4 ) c c + 2 b c + 1 a c + 3

12 First uess rafting? Example (T b,β S?) b a T b+1 γ β S γ + 2 γ + 4 γ γ + 2 γ + 4 b = β a T b=β S b+1

13 First uess rafting Orbital representation: Example /2 /2 T / S /2 / T /2 S = R Question: What is the expanded representation for S?

14 First uess What s Missing? Example c b c+2 c+4 c+1 H S = S b+1 c+3 c+5 S /2 Definition A (genuine) -tree is a based, indecomposable -forest. rafting can be done whenever the orbital representations allow it.

15 enuine -Trees Evaluation of Operads on -Trees NO(T ) := Op (Ω(T ), O) Example b a T b+1 T /2 / NO(T ) = O(2) Γ /2

16 enuine -Trees Evaluation of Operads on -Trees Example c b c+2 c+4 c+1 H S = S b+1 c+3 c+5 S /2 NO(S ) = (O(3) Γ 2 O(3) Γ 2 ) Γ /2 = O( 2 ) Γ 2

17 enuine -Trees Evaluation of Operads on enuine -Corollas Example ( arbitrary, H, L = A) /K 1 /H/K n C A A = i H/K i Set H, NO(C A ) = O( A ) Γ A

18 enuine -Trees Evaluation of Operads on Free -Trees T 0 Example ( = C 2, T 0 = C 2 Ω) y z y + 1 z + 1 x x + 1 T 0 T 0 NO( T 0 ) = ( O(2)) Γ / / = O(2)

19 enuine -Trees Observations Remark T = H T H T H Ω H All tree components of T have a same underlying shape Remark T = T 0 /N T 0 Ω N Aut(T 0 ) graph subgroup

20 Morphisms Categories and Morphisms Definition The dendroidal category Ω is the full subcategory of the category of multicategories (i.e. symmetric colored operads) spanned by the free operads generated by trees. Definition The -dendroidal category Ω is the full subcategory of the category of -multicategories spanned by the free -operads generated by genuine -trees.

21 Morphisms Morphisms: Non-Equivariantly Proposition (Moerdijk-Weiss) Every map f in Ω has a unique decomposition of the form f = ϕπσ, where ϕ is a composition of face maps, π is an isomorphism, and σ is a composition of degeneracies.

22 Morphisms Morphisms: Non-Equivariantly Degeneracies: d e b d e b a c b a c Faces: Outer: Inner: b a c b a c e a d c e d c

23 Morphisms Morphisms: Equivariantly (On Orbital Representation) Degeneracies: /L 1 /L 2 /K 1 /L 1 /L 2 /K 1/H /K 2 /K 1/H /K 1 Faces: Outer: Inner: /K 1 /H /L 1 /L 2 /L 3 /K 2 /K 1 /K 2 /H /L 1 /H /L 2 /K 2 /L 1 /L 2 /K 2 /K 1

24 Morphisms Quotients: K i K T H H T H c c + 2 c c + 2 c + 1 c + 3 c + 1 c + 3 γ γ + 2 γ + 1 γ + 3 b a b b + 1 a + 1 b + 1 β α β + 1 /2 /2 /2 /2 /

25 Morphisms Problem: Quotient maps lead to non-unique factorizations: Example ( = C 2 ) a a + 1 a a + 1 b / a a b Consequence: Defining boundaries and horns requires some care

26 Morphisms -Dendroidal Sets Definition The category of -dendroidal sets is the presheaf category Set Ωop. Example (Representable Sheaves) iven a -tree T = T 0 /N, we have two representable functors: Ω [T ] := Ω (, T ) Ω [ T 0 ]/N := Ω (, T 0 )/N. These agree on free -trees, but the latter object is often better behaved.

27 Morphisms Free Face Maps Definition A free face map is a map ϕ : Ω [ R 0 ] Ω [ T 0 ]/N such that the associated map ϕ : R 0 UT 0 is a face map in Ω. Remark The set of face maps into Ω [ T 0 ]/N is a -set

28 Morphisms Example ( = C 4 ) c a c + 2 c + 1 b + 1 c + 3 b c c + 2 b a c + 1 b + 1 c + 3 W 0 T 0 /N c + 2 c c + 1 a c + 3 b c c + 2 b a c + 1 b + 1 c + 3 W 0 b /2 / W 0 /M / T 0 /N

29 Inner Kan Complexes Boundaries and Horns iven T = T 0 /N, where N = Γ(ϕ), ϕ : H Aut(T 0 ). Definition The boundary of T is given by Ω [T ] := Ω [T 0 ]/N. Definition iven an inner face map β of degree 1, the β-horn of Ω [ T 0 ]/N is given by Λ β [T ] := Λ β [T 0 ]/N.

30 Inner Kan Complexes Inner Kan Complexes Definition iven X Set (Ω ) op, we call X an inner Kan complex if X has lifts against all inner horn inclusions: Λ H β [T ] X Ω[ T 0 ]/N

31 Inner Kan Complexes Inner Kan Complexes II Ho : Kan(Set Ωop ) genop genuine -operads are indexed over all genuine -corollas We have a diagram of fully-faithful functors Op genop N d Set Ωop N d

32 Inner Kan Complexes Theorem (B.-Pereira) Let X Set Ωop. Then X is a strict inner Kan complex iff X is isomorphic to the nerve of some genuine -operad.

33 Inner Kan Complexes Next Steps Algebras over inner Kan complexes Homotopy Coherent Nerve Model structure on category of -dendroidal sets

34 Thank You Questions? References: [1] H. Chu, R. Haugseng, and. Heuts. Two models for the homotopy theory of -operads. arxiv Preprint: , [2] D. Cisinski and I. Moerdijk. Dendroidal sets and simplicial operads. Journal of Topology, 6, [3] I. Moerdijk and I. Weiss. On inner kan complexes in the category of dendroidal sets. Advances in Mathematics, 221, [4] I. Weiss. Dendroidal Sets. PhD thesis, University of Utrecht, 2007.

Higher Prop(erad)s. Philip Hackney, Marcy Robertson*, and Donald Yau UCLA. July 1, 2014

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