Homotopy theory of Segal cyclic operads Philip Hackney, Marcy - - PowerPoint PPT Presentation

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Homotopy theory of Segal cyclic operads Philip Hackney, Marcy - - PowerPoint PPT Presentation

Feb 23, 2023 242 likes •723 views

Homotopy theory of Segal cyclic operads Philip Hackney, Marcy Robertson, Donald Yau Cyclic Operads Cyclic operads Operad P . n = Aut{1,,n} acts on P(n) Cyclic operads Operad P . n = Aut{1,,n} acts on P(n) Extend the n

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Homotopy theory of Segal cyclic operads

Philip Hackney, Marcy Robertson, Donald Yau

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Cyclic Operads

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Cyclic operads

  • Operad P

. Σn = Aut{1,…,n} acts on P(n)

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Cyclic operads

  • Operad P

. Σn = Aut{1,…,n} acts on P(n)

  • Extend the Σn action to Σn+1= Aut{0,1,…,n} action
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Cyclic operads

  • Operad P

. Σn = Aut{1,…,n} acts on P(n)

  • Extend the Σn action to Σn+1= Aut{0,1,…,n} action
  • Example:


V a finite dimensional vector space with an inner product.
 EndV(n) = hom(V⊗n,V) = hom(V⊗n,V*) = hom(V⊗(n+1),k)

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Cyclic operads

  • Operad P

. Σn = Aut{1,…,n} acts on P(n)

  • Extend the Σn action to Σn+1= Aut{0,1,…,n} action
  • Example:


V a finite dimensional vector space with an inner product.
 EndV(n) = hom(V⊗n,V) = hom(V⊗n,V*) = hom(V⊗(n+1),k)

  • Compatibility with operadic structure?


Let τ = τn+1 = (012…n) of order n+1

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Cyclic operads

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Cyclic operads

Apply τ to a composition:

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Cyclic operads

An operad P together with an extension of the 
 Σn = Aut{1,…,n} action on P(n) to a Σn+1= Aut{0,1,…,n}
 action, with τn+1 = (01…n) satisfying 1∙τ2 = 1 and

(g i f) · τ = ( (g · τ) i−1 f i > 1 (f · τ) n (g · τ) i = 0

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Examples

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

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

  • Ass, Com, Lie, BV, A∞,… (Getzler-Kapranov)
  • fDn (Budney)
  • P(n) = ∅ for n ≠ 1: monoid with involution 


(xy)* = y*x* and (x*)* = x

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Examples

  • Ass, Com, Lie, BV, A∞,… (Getzler-Kapranov)
  • fDn (Budney)
  • P(n) = ∅ for n ≠ 1: monoid with involution 


(xy)* = y*x* and (x*)* = x Example: groups

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Examples

  • Ass, Com, Lie, BV, A∞,… (Getzler-Kapranov)
  • fDn (Budney)
  • P(n) = ∅ for n ≠ 1: monoid with involution 


(xy)* = y*x* and (x*)* = x

  • Example: M = Z/2 × Z/2 = {00, 10, 01, 11}. 


Abelian, so look at order 1 & 2 elements in Aut(M) ≅ Σ3 Example: groups

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Examples

  • Ass, Com, Lie, BV, A∞,… (Getzler-Kapranov)
  • fDn (Budney)
  • P(n) = ∅ for n ≠ 1: monoid with involution 


(xy)* = y*x* and (x*)* = x

  • Example: M = Z/2 × Z/2 = {00, 10, 01, 11}. 


Abelian, so look at order 1 & 2 elements in Aut(M) ≅ Σ3 Example 1: m* = m-1 = m
 Example 2: 10* = 01, 11* = 11 Example: groups

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Colored Objects

  • Colored cyclic operads: 


τ : P(c1, …, cn; c0) → P(c2, …, cn, c0; c1)

  • If P(c1, … cn; c0) = ∅ for n ≠ 1: dagger categories


Given f : c → d, get f† : d → c… (gf)† = f†g† , (f†)† , id† = id

  • Example: groupoids
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Unrooted Tree Category

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Trees

  • Graphs with loose ends (legs), contractible, with at least
  • ne leg.
  • V=V(R) set of vertices, E=E(R) set of edges


legs(R) set of legs, nb(v) = edges adjacent to v

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Important Examples

  • ☆n, n ≥ 1. One vertex, n edges.
  • Ln, n ≥ 0. n bivalent vertices, n+1 edges.
  • L0. One edge, no vertex.
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(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.

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(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)

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(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
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(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
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Morphisms

  • A morphism φ : R → S consists of two functions:


φ0 : E(R) → E(S)
 φ1 : V(R) → Sb(S)

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Morphisms

  • A morphism φ : R → S consists of two functions:


φ0 : E(R) → E(S)
 φ1 : V(R) → Sb(S)

  • The following axioms are satisfied:
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Morphisms

  • A morphism φ : R → S consists of two functions:


φ0 : E(R) → E(S)
 φ1 : V(R) → Sb(S)

  • The following axioms are satisfied:
  • 1. If φ0|nb(v) is not injective then v is bivalent
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Morphisms

  • A morphism φ : R → S consists of two functions:


φ0 : E(R) → E(S)
 φ1 : V(R) → Sb(S)

  • The following axioms are satisfied:
  • 1. If φ0|nb(v) is not injective then v is bivalent
  • 2. If φ0|nb(v) is injective, then φ0(nb(v)) = legs(φ1(v))


If φ0|nb(v) is not injective, then φ1(v) = |φ(e) = |φ(e’)

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Morphisms

  • A morphism φ : R → S consists of two functions:


φ0 : E(R) → E(S)
 φ1 : V(R) → Sb(S)

  • The following axioms are satisfied:
  • 1. If φ0|nb(v) is not injective then v is bivalent
  • 2. If φ0|nb(v) is injective, then φ0(nb(v)) = legs(φ1(v))


If φ0|nb(v) is not injective, then φ1(v) = |φ(e) = |φ(e’)

  • 3. V(φ1(v)) ∩ V(φ1(w)) = ∅
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Generating morphisms

  • Inner coface ∂e
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Generating morphisms

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

  • Inner coface ∂e
  • Outer coface ∂v
  • Codegeneracy σv
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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.
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nonfullness of i : Ξ → Cyc

  • Ξ(L1, L1) ⊆ Set({0,1}, {0,1}) = 4
  • Cyc( iL1 , iL1 ) infinite, since iL1(0,0) is infinite
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Nerve Functor

  • i: Ξ → Cyc gives rise to an adjunction


with N(P)R = Cyc(iR, P)

  • For each presheaf X, there is a “Segal map”
  • Theorem: N is fully faithful. The essential image consists
  • f those X with the Segal map an isomorphism.

SetΞop Cyc : N XR → lim

E(R),V (R) XS

v → e when e ∊ nb(v)

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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(φ0(r),s) ≤ dist(φ0(e),s) for every edge e.

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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(φ0(r),s) ≤ dist(φ0(e),s) for every edge e.

  • Ω(Rr, Ss) ⊆ Ξ(R, S) the set of oriented maps
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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(φ0(r),s) ≤ dist(φ0(e),s) for every edge e.

  • Ω(Rr, Ss) ⊆ Ξ(R, S) the set of oriented maps
  • For each s ∊ legs(S), the map


∐legs(R) Ω(Rr, Ss) → Ξ(R, S)
 is a bijection away from constants.

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Segal Cyclic Operads

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Reduced presheaves

Reduced simplicial presheaves: those X which are a point when evaluated at L0. Think: if P is a one-colored cyclic operad, then X = N(P) satisfies this.

sSetΞop

∗

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Model structure

Theorem: There is a Quillen model structure on with fibrant objects those X which satisfy

  • 1. X is Reedy fibrant, and
  • 2. The Segal map 



 
 is a weak equivalence for all R


sSetΞop

∗

XR → Y

v∈V (R)

XFv

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Sketch

Lift model structure from generalized Reedy model structure

sSetΞop sSetΞop

∗

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Sketch

Lift model structure from generalized Reedy model structure Localize with respect to reduction of

sSetΞop sSetΞop

∗

[

v∈R

Ξ[Fv] , → Ξ[R]

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Similar model structure for reduced dendroidal presheaves (Bergner & H., Cisinski & Moerdijk) is Quillen equivalent to the model structure on one-colored simplicial

  • perads

Conjecture: simplicial cyclic operads is equivalent to

sSetΩop

∗

sSetΩop

∗

6' sSetΞop

∗

sSetΩop

∗

sSetΞop

∗