Homotopy theory of Segal cyclic operads Philip Hackney, Marcy Robertson, Donald Yau

Cyclic Operads

Cyclic operads • Operad P . Σ n = Aut{1,…,n} acts on P(n)

Cyclic operads • Operad P . Σ n = Aut{1,…,n} acts on P(n) • Extend the Σ n action to Σ n+1 = Aut{0,1,…,n} action

Cyclic operads • Operad P . Σ n = Aut{1,…,n} acts on P(n) • Extend the Σ n action to Σ n+1 = Aut{0,1,…,n} action • Example: V a finite dimensional vector space with an inner product. End V (n) = hom(V ⊗ n ,V) = hom(V ⊗ n ,V*) = hom(V ⊗ (n+1) ,k)

Cyclic operads • Operad P . Σ n = Aut{1,…,n} acts on P(n) • Extend the Σ n action to Σ n+1 = Aut{0,1,…,n} action • Example: V a finite dimensional vector space with an inner product. End V (n) = hom(V ⊗ n ,V) = hom(V ⊗ n ,V*) = hom(V ⊗ (n+1) ,k) • Compatibility with operadic structure? Let τ = τ n+1 = (012…n) of order n+1

Cyclic operads

Cyclic operads Apply τ to a composition:

Cyclic operads An operad P together with an extension of the Σ n = Aut{1,…,n} action on P(n) to a Σ n+1 = Aut{0,1,…,n} action, with τ n+1 = (01…n) satisfying 1 ∙τ 2 = 1 and ( ( g · τ ) � i − 1 f i > 1 ( g � i f ) · τ = ( f · τ ) � n ( g · τ ) i = 0

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov)

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov) • fD n (Budney)

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov) • fD n (Budney) • P(n) = ∅ for n ≠ 1: monoid with involution (xy)* = y*x* and (x*)* = x

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov) • fD n (Budney) • P(n) = ∅ for n ≠ 1: monoid with involution (xy)* = y*x* and (x*)* = x Example: groups

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov) • fD n (Budney) • P(n) = ∅ for n ≠ 1: monoid with involution (xy)* = y*x* and (x*)* = x Example: groups • Example: M = Z /2 × Z /2 = {00, 10, 01, 11}. Abelian, so look at order 1 & 2 elements in Aut(M) ≅ Σ 3

Examples • Ass, Com, Lie, BV, A ∞ ,… (Getzler-Kapranov) • fD n (Budney) • P(n) = ∅ for n ≠ 1: monoid with involution (xy)* = y*x* and (x*)* = x Example: groups • Example: M = Z /2 × Z /2 = {00, 10, 01, 11}. Abelian, so look at order 1 & 2 elements in Aut(M) ≅ Σ 3 Example 1: m* = m -1 = m Example 2: 10* = 01, 11* = 11

Colored Objects • Colored cyclic operads: τ : P(c 1 , …, c n ; c 0 ) → P(c 2 , …, c n , c 0 ; c 1 ) • If P(c 1 , … c n ; c 0 ) = ∅ for n ≠ 1: dagger categories Given f : c → d, get f † : d → c… (gf) † = f † g † , (f † ) † , id † = id • Example: groupoids

Unrooted Tree Category

Trees • Graphs with loose ends (legs), contractible, with at least one leg. • V=V(R) set of vertices, E=E(R) set of edges legs(R) set of legs, nb(v) = edges adjacent to v

Important Examples • ☆ n , n ≥ 1. One vertex, n edges. • L n , n ≥ 0. n bivalent vertices, n+1 edges. • L 0 . One edge, no vertex.

(Special) Subtrees • A naive subtree S ⊆ R is a pair of subsets E(S) ⊆ E(R) and V(S) ⊆ V(R) with incidence relation inherited from R, so that S is connected.

(Special) Subtrees • A naive subtree S ⊆ R is a pair of subsets E(S) ⊆ E(R) and V(S) ⊆ V(R) with incidence relation inherited from R, so that S is connected. • A special subtree S ⊆ R is naive subtree so that if v ∊ V(S) then every edge of R incident to v is in E(S). nb(v) ⊆ E(S)

(Special) Subtrees • A naive subtree S ⊆ R is a pair of subsets E(S) ⊆ E(R) and V(S) ⊆ V(R) with incidence relation inherited from R, so that S is connected. • A special subtree S ⊆ R is naive subtree so that if v ∊ V(S) then every edge of R incident to v is in E(S). nb(v) ⊆ E(S) • Sb(R) = set of (special) subtrees of R

(Special) Subtrees • A naive subtree S ⊆ R is a pair of subsets E(S) ⊆ E(R) and V(S) ⊆ V(R) with incidence relation inherited from R, so that S is connected. • A special subtree S ⊆ R is naive subtree so that if v ∊ V(S) then every edge of R incident to v is in E(S). nb(v) ⊆ E(S) • Sb(R) = set of (special) subtrees of R • V(R) and E(R) can be regarded as subsets of Sb(R) - ☆ v , | e

Morphisms • A morphism φ : R → S consists of two functions: φ 0 : E(R) → E(S) φ 1 : V(R) → Sb(S)

Morphisms • A morphism φ : R → S consists of two functions: φ 0 : E(R) → E(S) φ 1 : V(R) → Sb(S) • The following axioms are satisfied:

Morphisms • A morphism φ : R → S consists of two functions: φ 0 : E(R) → E(S) φ 1 : V(R) → Sb(S) • The following axioms are satisfied: 1. If φ 0 | nb(v) is not injective then v is bivalent

Morphisms • A morphism φ : R → S consists of two functions: φ 0 : E(R) → E(S) φ 1 : V(R) → Sb(S) • The following axioms are satisfied: 1. If φ 0 | nb(v) is not injective then v is bivalent 2. If φ 0 | nb(v) is injective, then φ 0 (nb(v)) = legs( φ 1 (v)) If φ 0 | nb(v) is not injective, then φ 1 (v) = | φ (e) = | φ (e’)

Morphisms • A morphism φ : R → S consists of two functions: φ 0 : E(R) → E(S) φ 1 : V(R) → Sb(S) • The following axioms are satisfied: 1. If φ 0 | nb(v) is not injective then v is bivalent 2. If φ 0 | nb(v) is injective, then φ 0 (nb(v)) = legs( φ 1 (v)) If φ 0 | nb(v) is not injective, then φ 1 (v) = | φ (e) = | φ (e’) 3. V( φ 1 (v)) ∩ V( φ 1 (w)) = ∅

Generating morphisms • Inner coface ∂ e

Generating morphisms • Inner coface ∂ e • Outer coface ∂ v

Generating morphisms • Inner coface ∂ e • Outer coface ∂ v • Codegeneracy σ v

Relationship to Cyc • A tree R determines a free object in the category Cyc| E(R) of cyclic operads with set of colors E(R). The generating set for this cyclic operad is V(R). • Functor i : Ξ → Cyc.

nonfullness of i : Ξ → Cyc • Ξ (L 1 , L 1 ) ⊆ Set({0,1}, {0,1}) = 4 • Cyc( iL 1 , iL 1 ) infinite, since iL 1 (0,0) is infinite

Nerve Functor Set Ξ op � Cyc : N • i: Ξ → Cyc gives rise to an adjunction with N(P) R = Cyc(iR, P) • For each presheaf X, there is a “Segal map” v → e when X R → E ( R ) ,V ( R ) X S lim e ∊ nb(v) • Theorem: N is fully faithful. The essential image consists of those X with the Segal map an isomorphism.

Relationship to Ω Ω = Moerdijk-Weiss category of rooted trees: • Suppose r, s are legs of R, S. A morphism φ : R → S is oriented (wrt r & s) if dist( φ 0 (r),s) ≤ dist( φ 0 (e),s) for every edge e.

Relationship to Ω Ω = Moerdijk-Weiss category of rooted trees: • Suppose r, s are legs of R, S. A morphism φ : R → S is oriented (wrt r & s) if dist( φ 0 (r),s) ≤ dist( φ 0 (e),s) for every edge e. • Ω (R r , S s ) ⊆ Ξ (R, S) the set of oriented maps

Relationship to Ω Ω = Moerdijk-Weiss category of rooted trees: • Suppose r, s are legs of R, S. A morphism φ : R → S is oriented (wrt r & s) if dist( φ 0 (r),s) ≤ dist( φ 0 (e),s) for every edge e. • Ω (R r , S s ) ⊆ Ξ (R, S) the set of oriented maps • For each s ∊ legs(S), the map ∐ legs(R) Ω (R r , S s ) → Ξ (R, S) is a bijection away from constants.

Segal Cyclic Operads

Reduced presheaves Reduced simplicial presheaves: those X which are a point when evaluated at L 0 . sSet Ξ op ∗ Think: if P is a one-colored cyclic operad, then X = N(P) satisfies this.

Model structure sSet Ξ op Theorem : There is a Quillen model structure on ∗ with fibrant objects those X which satisfy 1. X is Reedy fibrant, and Y 2. The Segal map X R → X F v v ∈ V ( R ) is a weak equivalence for all R

Sketch Lift model structure from generalized Reedy model structure sSet Ξ op � sSet Ξ op ∗

Sketch Lift model structure from generalized Reedy model structure sSet Ξ op � sSet Ξ op ∗ Localize with respect to reduction of [ Ξ [ F v ] , → Ξ [ R ] v ∈ R

Similar model structure for reduced dendroidal presheaves sSet Ω op ∗ sSet Ω op 6' sSet Ξ op ∗ ∗ sSet Ω op (Bergner & H., Cisinski & Moerdijk) is Quillen ∗ equivalent to the model structure on one-colored simplicial operads sSet Ξ op Conjecture: simplicial cyclic operads is equivalent to ∗

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