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 $m$ operations simultaneously into a single $m$-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 $F$ equipped with an associative product $F \circ F \to F$, 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.
Let $FB$ 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 $F, G: FB \to V$ be (Joyal) species valued in a bicomplete symmetric monoidal closed category $V$. The graft product $F \ast G$ is defined by the formula
where $S/T$ denotes the pushout of $T \to 1$ along the inclusion $T \subseteq S$ in the category of finite sets. This includes the degenerate case where $T = 0$; in this case $S/0$ is the result of freely adjoining a point to $S$. In the theory of species, there is a fundamental differentiation functor $F \mapsto F'$ defined by the formula $F'[S] = F[S/0]$.
The set $S/T$ has $T/T$ as distinguished basepoint, and if $U$ is complementary to $T$, we have natural bijections $S/T \cong U + \{T/T\} \cong U/0$. It follows that
where $\otimes$ refers to the usual Day convolution product on species, induced from the tensor product on $FB$. Since differentiation of species,
is cocontinuous, and since Day convolution is cocontinuous in each of its arguments, we see that
is separately cocontinuous in each of its arguments $F$, $G$. (Contrast with the plethystic or substitution product on species $F \circ G$, which is cocontinuous in the first argument $F$ but not in the second $G$.)
For $V = Set$, a structure of species $F \ast G$ is given by three data:
A tree obtained by grafting the root of a sprout (aka corolla) with leaf set $\tau$ to a leaf of another sprout with leaf set $\sigma$;
An element of $F[\sigma]$;
An element of $G[\tau]$.
The graft poduct is not associative up to isomorphism, but there is a lax associativity. Specifically, we calculate (using a categorified Leibniz rule)
so that there is a noninvertible associativity (an inclusion)
which is natural in each of its arguments $F$, $G$, and $H$, 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 $X$ for which $X[S]$ is the monoidal unit of $V$ if $card(S) = 1$, else $X[S]$ is initial, for which we have evident natural maps
The first map $\lambda_F$ is invertible, but the second is not: its component at a finite set $S$ is the codiagonal
However, the standard unit coherence conditions for monoidal categories hold. We may call such a structure, relaxing the condition of invertibility of $\alpha$ and $\rho$ but retaining the usual naturality and coherence conditions, a lax monoidal category.
In the sequel, $F^{\ast n}$ will denote the iterated graft product defined recursively by
so that all parentheses in $F^{\ast n}$ are to the left. We state without proof the following partial coherence theorem:
Any two maps
definable in the language of lax monoidal categories are equal. We denote this map by $\alpha_{m n}$.
A (permutative) non-unital operad in a cosmos $V$ is a species $F: FB^{op} \to V$ equipped with a multiplication $m: F \ast F \to F$, satisfying the following two associativity axioms:
commutes;
The two composites $F'' \otimes F^{\otimes 2} \to F$ named in
commute. Here $i$ denotes the inclusion complementary to $\alpha: F \ast (F \ast F) \to (F \ast F) \ast F$, and $\sigma$ is the involution
where $\sigma: F'' \to F''$ is the natural involution on the second derivative
induced by the nonidentity involution $2 \to 2$, and $\sigma_2$ is the symmetry isomorphism $F^{\otimes 2} \to F^{\otimes 2}$.
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.
A non-permutative species in a monoidally cocomplete category $V$ is simply an $\mathbb{N}$-graded object in $V$, which we may think of as equivalent to a functor $\mathbb{N} \to V$ from the monoidal groupoid of finite linear orders, with the monoidal product given by restriction of the ordinal sum to the core groupoid. If $S$ is a finite linear order and $T = [a, b] \subseteq S$ is a subinterval? of $S$, then we may again define $S/T$ via a pushout construction in the category of finite linear orders, and define a corresponding graft product by the formula
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 $(F \ast G) \ast H$ cab be decomposed as a coproduct of parts, the first being
and the second being
where $S/(T_1, T_2)$ denotes the pushout of the inclusion $T_1 + T_2 \hookrightarrow S$ along the quotient $! + !: T_1 + T_2 \to 1 + 1$. By interchanging the roles of $T_1$ and $T_2$, 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.