Todd Trimble
Cyclic operads

Graft product of species

Let FBFB be the monoidal groupoid of finite sets and bijections, where the monoidal product is obtained by restricting the coproduct on the category of finite sets. Let F,G:FBVF, G: FB \to V be species valued in a bicomplete symmetric monoidal closed category VV. The graft product F*GF \ast G is defined by the formula

(F*G)[S]= TSF[S/T]G[T](F \ast G)[S] = \sum_{T \subseteq S} F[S/T] \otimes G[T]

where S/TS/T denotes the pushout of T1T \to 1 along the inclusion TST \subseteq S. This includes the degenerate case where T=0T = 0; in this case S/0S/0 is the result of freely adjoining a point to SS.

The set S/TS/T has T/TT/T as distinguished basepoint, and if UU is complementary to TT, we have a natural bijection S/TU+{T/T}S/T \cong U + \{T/T\}. It follows that

(F*G)[S] U+T=SF[U+*]G[T] U+T=SF[U]G[T] (FG)[S]\array{ (F \ast G)[S] & \cong & \sum_{U + T = S} F[U + *] \otimes G[T] \\ & \cong & \sum_{U + T = S} F'[U] \otimes G[T] \\ & \cong & (F' \otimes G)[S] }

where FGF' \otimes G refers to the usual Day convolution product, induced from the tensor product on FBFB. Since differentiation of species,

V FB+*V FB,V^{FB} \stackrel{- + *}{\to} V^{FB},

is cocontinuous, and since Day convolution is cocontinuous in each of its arguments, we see that

F*GFGF \ast G \cong F' \otimes G

is separately cocontinuous in each of its arguments FF, GG. We remark below on the fundamental implications of this observation.

For V=SetV = Set, a structure of species F*GF \ast G is given by three data:

Lax monoidal structure of graft product

We note that the graft poduct is not associative up to isomorphism, but there is a lax associativity. Specifically, we calculate

(F*G)*H (FG)H FGH+FGH FGH+F*(G*H)\array{ (F \ast G) \ast H & \cong & (F' \otimes G)' \otimes H \\ & \cong & F'' \otimes G \otimes H + F' \otimes G' \otimes H \\ & \cong & F'' \otimes G \otimes H + F \ast (G \ast H) }

so that there is a noninvertible associativity (an inclusion)

α FGH:F*(G*H)(F*G)*H\alpha_{F G H}: F \ast (G \ast H) \to (F \ast G) \ast H

which is natural in each of its arguments FF, GG, and HH, and which satisfies an evident pentagon coherence condition. Additionally, there is a lax unit, defined as the species XX for which X[S]X[S] is the monoidal unit of VV if card(S)=1card(S) = 1, else X[S]X[S] is initial, for which we have evident natural maps

λ F:X*FF,ρ F:F*XF\lambda_F: X \ast F \to F, \qquad \rho_F: F \ast X \to F

The first map λ F\lambda_F is invertible, but the second is not: its component at a finite set SS is the codiagonal

:SF[S]F[S]\nabla: S \cdot F[S] \to F[S]

However, the standard unit coherence conditions for monoidal categories hold. We may call such a structure, relaxing the condition of invertibility of α\alpha and ρ\rho but retaining the naturality and coherence conditions, a lax monoidal category.

In the sequel, F *nF^{\ast n} will denote the iterated graft product defined recursively by

F *0=X,F *(n+1)=F *n*FF^{\ast 0} = X, \qquad F^{\ast (n+1)} = F^{\ast n} \ast F

so that all parentheses in F *nF^{\ast n} are to the left. We have a partial coherence theorem:

Proposition

Any two maps

F *m*F *nF *(m+n)F^{\ast m} \ast F^{\ast n} \to F^{\ast (m+n)}

definable in the language of lax monoidal categories are equal. This map by α mn\alpha_{m n}.

For FF a SetSet-valued species, an element of F *nF^{\ast n} is described by data as follows:

  1. A structure of tree constructed from nn sprouts in a prescribed grafting order;

  2. An element of F[σ]F[\sigma] for each sprout σ\sigma used in the construction of the tree in 1.

Thus we expect some connection between the “geometric series” nF *n\sum_n F^{\ast n} and the free operad on FF. The only difference between the two is that the standard construction of free operads via trees is in no way dependent on grafting order. Thus, to obtain the free operad, we have to mod out by isomorphisms on F *nF^{\ast n} induced by “inessential” differences in grafting order. This will be described in detail in the section on trees and hierarchies.

Trees and hierarchies

It is interesting to resolve F *nF^{\ast n} into a sum of monomials, each of which is a tensor product of derivatives of FF. For example, we have

F*F FF (F*F)*F FF 2+FFF F *4 FF 3+2FFF+FFFF+FFF 2+(F) 3F\array{ F \ast F & \cong & F' \otimes F \\ (F \ast F) \ast F & \cong & F'' \otimes F^{\otimes 2} + F' \otimes F' \otimes F \\ F^{\ast 4} & \cong & F''' \otimes F^{\otimes 3} + 2 F'' \otimes F' \otimes F + F'' \otimes F \otimes F' \otimes F + F' \otimes F'' \otimes F^{\otimes 2} + (F')^{\otimes 3} \otimes F }

There are in fact (n1)!(n-1)! monomial terms, each a tensor product of nn derivative terms, and each corresponding to a function f:[2,n][1,n]f: [2, n] \to [1, n] such that f(k)<kf(k) \lt k for all k[2,n]k \in [2, n]. We call such an ff a hierarchy. In detail, if the S thS^{th} derivative of a species FF is defined by F (S)[T]=F[S+T]F^{(S)}[T] = F[S + T], we have

F *n= fHier(n)F (f 1(1))F (f 1(2))F (f 1(n1))FF^{\ast n} = \sum_{f \in Hier(n)} F^{(f^{-1}(1))} \otimes F^{(f^{-1}(2))} \otimes \ldots \otimes F^{(f^{-1}(n-1))} \otimes F

as may be proved by an easy induction.

Each fHier(n)f \in Hier(n) endows the set [1,n][1, n] with a rooted tree structure. A rooted tree structure on a set SS is precisely a function f:SSf: S \to S with a fixed point rr (the root) such that for every sSs \in S, some finite iterate of ff takes ss to rr, i.e., f (k)(s)=rf^{(k)}(s) = r for some kk. (The function ff simply moves a vertex one step closer to the root along the unique path to the root.) Given a hierarchy f:[2,n][1,n]f: [2, n] \to [1, n], the corresponding rooted tree structure on [1,n][1, n] is the extension of ff which fixes the point 11. The well-ordering of the vertices 1<<n1 \lt \ldots \lt n serves to prescribe a grafting order on sprouts, starting from the sprout at the root 11, grafting the sprout rooted at 22 to a leaf of the initial sprout, and so on.

Somewhat more generally, given any finite well-ordered set S={s 1<<s n}S = \{s_1 \lt \ldots \lt s_n\}, a hierarchy on SS is a function f:S{s 1}Sf: S - \{s_1\} \to S such that f(s i)<s if(s_i) \lt s_i for 1<in1 \lt i \leq n, and this also gives rise to a rooted tree structure. A rooted tree whose vertices are equipped with a well-ordering and whose tree structure arises from a hierarchy function will be called a hierarchical tree. The root rr is of course the initial element s 1s_1.

Now by removing the root of a finite rooted tree (S,f,r)(S, f, r), we obtain a forest of finite rooted trees S/sS/s whose roots ss range over f 1(r)f^{-1}(r). An isomorphism Φ\Phi of rooted trees (S,f,r)(S,f,r)(S, f, r) \to (S', f', r') may be described recursively as consisting of

  1. A bijection ϕ:f 1(r)(f) 1(r)\phi: f^{-1}(r) \to (f')^{-1}(r');

  2. Isomorphisms of rooted trees Φ s:S/sS/ϕ(s)\Phi_s: S/s \to S'/\phi(s) where sf 1(r)s \in f^{-1}(r).

If (S,f)(S, f) is hierarchical, then the subtrees S/sS/s inherit hierarchical tree structures from SS by restricting both the well-ordering and f:SSf: S \to S to a function f|:S/sS/sf|: S/s \to S/s. Thus isomorphisms of hierarchical trees SS (defined to be isomorphisms of the underlying rooted trees) can also be described recursively.

If (S={s 1<<s n},f)(S = \{s_1 \lt \ldots \lt s_n\}, f) is a hierarchical tree and FF is a species, we define a species

F S,f= k=1 nF (f 1(s k))F_{S, f} = \bigotimes_{k=1}^{n} F^{(f^{-1}(s_k))}

Observe that there is a well-defined isomorphism

F S,fF (f 1(r))( sf 1(r)F S/s,f|)F_{S, f} \cong F^{(f^{-1}(r))} \otimes (\bigotimes_{s \in f^{-1}(r)} F_{S/s, f|})

by permuting tensor factors. (The order of tensor factors on either side is fixed by the well-ordering of vertices.)

Now if Φ:(S,f)(S,f)\Phi: (S, f) \to (S', f') is an isomorphism of hierarchical trees, we define by recursion a species isomorphism to be the composite of

F (f 1(s 1)) sf 1(s 1)F S/s,f|F (ϕ)( sf 1(s 1)F Φ s)F ((f) 1(r))( s(f) 1(r)F S/s,f|)F^{(f^{-1}(s_1))} \otimes \bigotimes_{s \in f^{-1}(s_1)} F_{S/s, f|} \stackrel{F^{(\phi)} \otimes (\bigotimes_{s \in f^{-1}(s_1)} F_{\Phi_s})}{\to} F^{((f')^{-1}(r'))} \otimes (\bigotimes_{s' \in (f')^{-1}(r')} F_{S'/s', f'|})

with factor-permuting isomorphisms

F S,fF (f 1(r))( sf 1(r)F S/s,f|)F ((f) 1(r))( s(f) 1(r)F S/s,f|)F S,fF_{S, f} \cong F^{(f^{-1}(r))} \otimes (\bigotimes_{s \in f^{-1}(r)} F_{S/s, f|}) \qquad F^{((f')^{-1}(r'))} \otimes (\bigotimes_{s' \in (f')^{-1}(r')} F_{S'/s', f'|}) \cong F_{S', f'}

on either side. The result is an isomorphism F Φ:F S,fF S,fF_\Phi: F_{S, f} \to F_{S', f'}.

There is a groupoid consisting of hierarchical trees and isomorphisms between them. Let Hier nHier_n be the full subgroupoid whose objects are hierarchical tree structures on [1,n][1, n]. (The isomorphisms in Hier nHier_n can be thought of as permutations on [1,n][1, n] that relabel vertices to give a new grafting order.) Given a VV-species FF, there is a functor

Hier nV FB:(Φ:([1,n],f)([1,n],g))F [1,n],fF [1,n],gHier_n \to V^{FB}: (\Phi: ([1, n], f) \to ([1, n], g)) \mapsto F_{[1, n], f} \to F_{[1, n], g}

which defines a diagram in the category of species. The colimit of this diagram is a certain quotient of

F *n= fHier(n)F [1,n],fF^{\ast n} = \sum_{f \in Hier(n)} F_{[1, n], f}

which we will denote as F *n/Hier nF^{\ast n}/Hier_n. By taking this quotient, we eliminate dependence on a specified hierarchy or grafting order.

The constructions are made in view of the following result.

Theorem

The species

𝒪(F)= n0F *n/Hier n\mathcal{O}(F) = \sum_{n \geq 0} F^{\ast n}/Hier_n

is the underlying species of the free operad on FF.

This will be explored further in the next section.

Operads

The notion of (permutative) VV-valued operad may be formulated in terms of the graft product. Specifically, if MM is an operad, there is a natural map

ψ:M*MMM\psi: M \ast M \to M \circ M

and whose unit is again u:XMu: X \to M. The structure of an operad MM may be described entirely in terms of m:M*MMm: M \ast M \to M, u:XMu: X \to M, and the notion of operad may be recast as appropriate *\ast-monoids (with a slight extra twist).

The map ψ\psi is not hard to describe: the graft product F*GF \ast G may be alternatively described by the formula

(F*G)[S]= k 1kk(F[k]G^ j)[S](F \ast G)[S] = \sum_k \sum_{1 \leq k \leq k} (F[k] \otimes \widehat{G}_j)[S]

where G^ j\widehat{G}_j is a tensor product of kk factors

XGX,X \otimes \ldots \otimes G \otimes \ldots \otimes X,

all factors being XX except for a single instance of GG as the j thj^{th} tensor factor. If there is in addition a map i:XGi: X \to G (for example, a unit map u:XMu: X \to M of an operad), there is an induced map

i j=i1 Gi:G^ jG ki_j = i \otimes \ldots \otimes 1_G \otimes \ldots \otimes i: \widehat{G}_j \to G^{\otimes k}

and under these circumstances there is an induced map

F*G kF[k]G k kF[k] S kG k=FGF \ast G \to \sum_k F[k] \otimes G^{\otimes k} \to \sum_k F[k] \otimes_{S_k} G^{\otimes k} = F \circ G

These must satisfy the following axioms:

  1. F*(F*F) 1*m F*F α m (F*F)*F m*1F*F m F\array{ F \ast (F \ast F) & \stackrel{1 \ast m}{\to} & F \ast F & \\ \alpha \downarrow & & & \searrow m \\ (F \ast F) \ast F & \underset{m \ast 1}{\to} F \ast F & \underset{m}{\to} & F }

    commutes;

  2. The two composites FF 2FF'' \otimes F^{\otimes 2} \to F named in

    F(FF)1σF(FF)i(F*F)*Fm(m*1)FF'' \otimes (F \otimes F) \stackrel{\overset{\sigma}{\to}}{\underset{1}{\to}} F'' \otimes (F \otimes F) \stackrel{i}{\to} (F \ast F) \ast F \stackrel{m(m \ast 1)}{\to} F

    commute. Here ii denotes the inclusion complementary to α:F*(F*F)(F*F)*F\alpha: F \ast (F \ast F) \to (F \ast F) \ast F, and σ\sigma is the involution

    σ 1σ 2:FF 2FF 2\sigma_1 \otimes \sigma_2: F'' \otimes F^{\otimes 2} \to F'' \otimes F^{\otimes 2}

    where σ:FF\sigma: F'' \to F'' is the natural involution on the second derivative

    ()=(+2)*:V FBV FB(-)'' = (- + 2)*: V^{FB} \to V^{FB}

    induced by the nonidentity involution 222 \to 2, and σ 2\sigma_2 is the symmetry isomorphism F 2F 2F^{\otimes 2} \to F^{\otimes 2}.

Cyclic operads

Definition

A cyclic operad is a species YY with Y[0]=0Y[0] = 0 and equipped with a morphism

π:YYY\pi: Y' \otimes Y' \to Y

satisfying two axioms:

  1. Symmetry: πσ=π\pi \circ \sigma = \pi where σ=σ Y,Y:YYYY\sigma = \sigma_{Y', Y'}: Y' \otimes Y' \to Y \otimes Y' is the symmetry;

  2. Operad axiom: The composite

    YYYY+YY(YY)πYY'' \otimes Y' \hookrightarrow Y'' \otimes Y' + Y' \otimes Y'' \cong (Y' \otimes Y')' \stackrel{\pi'}{\to} Y'

    is a structure of operad m:Y*YYm: Y' \ast Y' \to Y' on YY'.

We observe that by the symmetry axiom, the composite

μ=(YYYY+YY(YY)πY)\mu = (Y' \otimes Y'' \hookrightarrow Y'' \otimes Y' + Y' \otimes Y'' \cong (Y' \otimes Y')' \stackrel{\pi'}{\to} Y')

is equal to

ν=(YYσ Y,YYYmY)\nu = (Y' \otimes Y'' \stackrel{\sigma_{Y', Y''}}{\to} Y'' \otimes Y' \stackrel{m}{\to} Y')

so that π\pi' (and hence π\pi) is uniquely determined by mm.

Because π\pi is uniquely determined from mm under the μ=ν\mu = \nu condition, one could define a cyclic operad to be an operad MM together with a suitable identification ϕ:YM\phi: Y' \cong M, so that it is the derivative MM (and not YY) that is the “operad” per se. In essence, this is what is usually done in the literature. But definition 1 seems simpler.

Example

Let YY be the species of necklaces (or cyclic orders), where sY[S]s \in Y[S] is a permutation s:SSs: S \to S of order |S|{|S|}. We define a map

π:YYY\pi: Y' \otimes Y' \to Y

where π\pi takes a pair of necklaces (α,β)Y[S+*]×Y[T+*](\alpha, \beta) \in Y[S + \ast] \times Y[T + \ast] to the necklace γ\gamma on S+TS + T such that

  1. For sSs \in S, γ(s)=β(*)\gamma(s) = \beta(\ast) if α(s)=*\alpha(s) = \ast, else γ(s)=α(s)\gamma(s) = \alpha(s);

  2. For tTt \in T, γ(t)=α(*)\gamma(t) = \alpha(\ast) if β(t)=*\beta(t) = \ast, else γ(t)=β(t)\gamma(t) = \beta(t).

It is clear from the definition that π\pi is symmetric. Moreover, the derivative of YY is AssocAssoc, the operad for associative algebras, for which Y[S]Y'[S] is isomorphic to the species of linear orders on SS (if α\alpha is a necklace on S+*S + \ast, then the corresponding linear order on SS has first element α(*)\alpha(\ast) and last element α 1(*)\alpha^{-1}(\ast), and the successor function on the linear order given by α\alpha itself).

Example

There is of course a unique cyclic operad structure on Y=exp(X)Y = \exp(X), since Y=YY' = Y.

Revised on April 5, 2013 at 21:24:41 by Todd Trimble