non-unital operad


A non-unital operad is a structure like an operad but without a unit or operadic identity. However, in contrast to operads where the multiplication operation substitutes mm operations simultaneously into a single mm-ary operation, the primary operations for a non-unital operad are ones which substitute only one operation at a time.

It is possible to package these individual substitution operations into a single operation, using not the plethystic or substitution monoidal product (usually denoted \circ) but a certain lax monoidal product introduced below, called the graft product. At all events, it must be understood that the desired notion is not simply a species equipped FF equipped with an associative product FFFF \circ F \to F, because that is not expressive enough.


There are various possible notions of non-unital operad; here we focus on the permutative and non-permutative cases.

Graft product of (permutative) 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 (Joyal) 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 in the category of finite sets. 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. In the theory of species, there is a fundamental differentiation functor FFF \mapsto F' defined by the formula F[S]=F[S/0]F'[S] = F[S/0].

The set S/TS/T has T/TT/T as distinguished basepoint, and if UU is complementary to TT, we have natural bijections S/TU+{T/T}U/0S/T \cong U + \{T/T\} \cong U/0. 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 \otimes refers to the usual Day convolution product on species, 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. (Contrast with the plethystic or substitution product on species FGF \circ G, which is cocontinuous in the first argument FF but not in the second GG.)

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

  • A tree obtained by grafting the root of a sprout (aka corolla) with leaf set τ\tau to a leaf of another sprout with leaf set σ\sigma;

  • An element of F[σ]F[\sigma];

  • An element of G[τ]G[\tau].

Lax monoidal structure of graft product

The graft poduct is not associative up to isomorphism, but there is a lax associativity. Specifically, we calculate (using a categorified Leibniz rule)

(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 (although we won’t really need this here), there is a lax monoidal 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 usual 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 state without proof the following partial coherence theorem:


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. We denote this map by α mn\alpha_{m n}.

Permutative non-unital operads


A (permutative) non-unital operad in a cosmos VV is a species F:FB opVF: FB^{op} \to V equipped with a multiplication m:F*FFm: F \ast F \to F, satisfying the following two associativity 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 }


  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}.

Of course the notion can be reformulated perfectly well in a general symmetric monoidal category; the use of coproducts just makes for a more efficient packaging.

Graft product of (non-permutative) species

A non-permutative species in a monoidally cocomplete category VV is simply an \mathbb{N}-graded object in VV, which we may think of as equivalent to a functor V\mathbb{N} \to V from the monoidal groupoid of finite linear orders, with the monoidal product given by restriction of the ordinal sum to the core groupoid. If SS is a finite linear order and T=[a,b]ST = [a, b] \subseteq S is a subinterval? of SS, then we may again define S/TS/T via a pushout construction in the category of finite linear orders, and define a corresponding graft product by the formula

(F*G)[S]= subintervalsTF[S/T]G[T].(F \ast G)[S] = \sum_{subintervals T} F[S/T] \otimes G[T].

This graft product shares certain formal properties with the permutative graft product defined above, such as the fact that it is a lax monoidal product. A triple graft product (F*G)*H(F \ast G) \ast H cab be decomposed as a coproduct of parts, the first being

F*(G*H)[S]= subintervalsT 1T 2F[S/T 2]G[T 2/T 1]H[T 1]F \ast (G \ast H)[S] = \sum_{subintervals T_1 \subseteq T_2} F[S/T_2] \otimes G[T_2/T_1] \otimes H[T_1]

and the second being

subint.T 1,T 2:T 1T 2=F[S/(T 1,T 2)]G[T 1]H[T 2]\sum_{subint. T_1, T_2: T_1 \cap T_2 = \emptyset} F[S/(T_1, T_2)] \otimes G[T_1] \otimes H[T_2]

where S/(T 1,T 2)S/(T_1, T_2) denotes the pushout of the inclusion T 1+T 2ST_1 + T_2 \hookrightarrow S along the quotient !+!:T 1+T 21+1! + !: T_1 + T_2 \to 1 + 1. By interchanging the roles of T 1T_1 and T 2T_2, we get a nontrivial involution on the second part.

To be continued…

Last revised on March 17, 2015 at 22:05:36. See the history of this page for a list of all contributions to it.