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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIn the paper "Ezra Getzler, M. M. Kapranov, Cyclic operads and cyclic homology" (linked at [[cyclic operad]]), I find the following definition of a monad $\mathbb{T}$ on [[symmetric sequences]] whose algebras are supposed to be operads: $$ \mathbb{T}(V) = \bigoplus_T V(T) $$ where $V(T) = \bigotimes_{v\in v(T)} V(in(v))$, with $v(T)$ the vertices of the tree $T$ and $in(v)$ the set of incoming edges at $v$. I am confused about the $\bigoplus$ in the definition of $\mathbb{T}(V)$. In their equation (1.9) it is labeled as over "rooted $n$-trees $T$" but in the sentence preceeding it is said to be over "isomorphism classes of rooted trees". But I can't make sense of either one. If it is over all trees, then there is a lot of duplication; but if it is over isomorphism classes of trees, then what does it mean to talk about "the set of vertices" of an isomorphism class of trees?

   In the paper “Ezra Getzler, M. M. Kapranov, Cyclic operads and cyclic homology” (linked at cyclic operad), I find the following definition of a monad $\mathbb{T}$ on symmetric sequences whose algebras are supposed to be operads:

   $$\mathbb{T}(V) = \bigoplus_T V(T)$$

   where $V(T) = \bigotimes_{v\in v(T)} V(in(v))$, with $v(T)$ the vertices of the tree $T$ and $in(v)$ the set of incoming edges at $v$.

   I am confused about the $\bigoplus$ in the definition of $\mathbb{T}(V)$. In their equation (1.9) it is labeled as over “rooted $n$-trees $T$” but in the sentence preceeding it is said to be over “isomorphism classes of rooted trees”. But I can’t make sense of either one. If it is over all trees, then there is a lot of duplication; but if it is over isomorphism classes of trees, then what does it mean to talk about “the set of vertices” of an isomorphism class of trees?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexHonestly, I've found that article pretty confusing in the past.

   Honestly, I’ve found that article pretty confusing in the past.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNow I am looking at our page [[free operad]] which has a similar formula, summing over isomorphism classes of trees: $$ \coprod_{[T] \in \pi_0(Trees)} \bar K(T) \otimes_{Aut(T)} \bar \lambda(T) $$ As written, this has the same problem that things like $\bar K(T)$ and $Aut(T)$ are not defined if we are given only an isomorphism class of trees $[T]$. Perhaps what is meant is to choose a representative of each isomorphism class? But even in this case, the two formulas don't agree: the second one is "tensored up" to get a $\Sigma_n$-action. Choosing representatives also makes me uneasy because we have to define the multiplication of the monad by "grafting trees", and that generally won't produce the chosen representative; so it seems we would have to choose an isomorphism from the result to our chosen representative of its equivalence class, and this won't be coherent. But perhaps the earlier definition as a coend $\int^{T \in Trees} \bar K(T) \otimes \bar \lambda(T)$ avoids this problem, and the point is that the version involving isomorphism classes is only related to the "real" definition by a noncanonical isomorphism so we can't transfer the monad structure to it?

   Now I am looking at our page free operad which has a similar formula, summing over isomorphism classes of trees:

   $$\coprod_{[T] \in \pi_0(Trees)} \bar K(T) \otimes_{Aut(T)} \bar \lambda(T)$$

   As written, this has the same problem that things like $\bar K(T)$ and $Aut(T)$ are not defined if we are given only an isomorphism class of trees $[T]$. Perhaps what is meant is to choose a representative of each isomorphism class? But even in this case, the two formulas don’t agree: the second one is “tensored up” to get a $\Sigma_n$-action.

   Choosing representatives also makes me uneasy because we have to define the multiplication of the monad by “grafting trees”, and that generally won’t produce the chosen representative; so it seems we would have to choose an isomorphism from the result to our chosen representative of its equivalence class, and this won’t be coherent. But perhaps the earlier definition as a coend $\int^{T \in Trees} \bar K(T) \otimes \bar \lambda(T)$ avoids this problem, and the point is that the version involving isomorphism classes is only related to the “real” definition by a noncanonical isomorphism so we can’t transfer the monad structure to it?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI have several half-written articles on my public nLab web and my private nLab web which goes into these matters. Somewhat in the manner of Richard Williamson's series of posts which walk through a development, I'll try the same thing here, drawing on one of the articles in my private web. We let $FB$ denote the symmetric monoidal groupoid of finite sets and bijections, where the monoidal product is obtained by restricting the coproduct on the category of finite sets. This is equivalent as a symmetric monoidal category to the permutation category $\mathbf{P}$, the free symmetric monoidal category on one generator. Let $F, G: FB \to V$ be two species valued in a complete cocomplete symmetric monoidal closed category $V$. The **graft product** $F \star G$ is defined by the formula $$(F \star G)[S] = \sum_{T \subseteq S} F[S/T] \otimes G[T]$$ where $S/T$ denotes the pushout of $T \to 1$ along the inclusion $T \subseteq S$. This includes the degenerate case where $T = 0$; in this case $S/0$ is the result of freely adjoining a point to $S$. The set $S/T$ has $T/T$ as distinguished basepoint, and if $U$ is complementary to $T$, we have a natural bijection $S/T \cong U + \{T/T\}$. It follows that $$\array{ (F \star G)[S] & \cong & \sum_{U + T = S} F[U + *] \otimes G[T] \\ & \cong & \sum_{U + T = S} F'[U] \otimes G[T] \\ & \cong & (F' \otimes G)[S] }$$ where $\otimes$ refers to the usual Day convolution product, induced from the tensor product on $FB$, and $F'$ is the derivative of the species $F$, defined by the formula $F'[S] = F[S + \ast]$. Since differentiation of species $$V^{(-)+\ast} \colon V^{FB} \to V^{FB}$$ is cocontinuous, and since Day convolution is separately cocontinuous (i.e., cocontinuous in each of its arguments), we see that $$F \star G \cong F' \otimes G$$ is also separately cocontinuous. For $V = Set$, a structure of species $F \star G$ 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 $G[\tau]$.

   I have several half-written articles on my public nLab web and my private nLab web which goes into these matters. Somewhat in the manner of Richard Williamson’s series of posts which walk through a development, I’ll try the same thing here, drawing on one of the articles in my private web.

   We let $FB$ denote the symmetric monoidal groupoid of finite sets and bijections, where the monoidal product is obtained by restricting the coproduct on the category of finite sets. This is equivalent as a symmetric monoidal category to the permutation category $\mathbf{P}$, the free symmetric monoidal category on one generator.

   Let $F, G: FB \to V$ be two species valued in a complete cocomplete symmetric monoidal closed category $V$. The graft product $F \star G$ is defined by the formula

   $$(F \star G)[S] = \sum_{T \subseteq S} F[S/T] \otimes G[T]$$

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

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

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

   where $\otimes$ refers to the usual Day convolution product, induced from the tensor product on $FB$, and $F'$ is the derivative of the species $F$, defined by the formula $F'[S] = F[S + \ast]$. Since differentiation of species

   $$V^{(-)+\ast} \colon V^{FB} \to V^{FB}$$

   is cocontinuous, and since Day convolution is separately cocontinuous (i.e., cocontinuous in each of its arguments), we see that

   $$F \star G \cong F' \otimes G$$

   is also separately cocontinuous.

   For $V = Set$, a structure of species $F \star G$ 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 $G[\tau]$.
5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexIn this post, I'll make some remarks on the "lax monoidal" structure of the graft product. The graft product is not associative up to isomorphism, but we do have a "lax" associativity. Specifically, we calculate $$\array{ (F \star G) \star H & \cong & (F' \otimes G)' \otimes H \\ & \cong & F'' \otimes G \otimes H + F' \otimes G' \otimes H \\ & \cong & F'' \otimes G \otimes H + F \star (G \star H) }$$ so that there is a noninvertible associativity $$\alpha_{F G H}: F \star (G \star H) \to (F \star G) \star H$$ given by inclusion, natural in each of its arguments $F$, $G$, and $H$, and satisfying an evident pentagon coherence condition. Additionally, there is a lax unit, defined as the species $X$ for which $X[S]$ is the monoidal unit of $V$ if $S$ is a 1-element set, else $X[S]$ is initial. Here we have evident natural maps $$\lambda_F: X \star F \to F, \qquad \rho_F: F \star X \to F$$ The first map $\lambda_F$ is invertible, but the second is not: its component at a finite set $S$ is the codiagonal $$\nabla: S \cdot F[S] \to F[S]$$ However, all coherence conditions for monoidal categories hold (I won't prove this here). We may call such a structure, relaxing the condition of invertibility of $\alpha$ and $\rho$ but retaining the naturality and coherence conditions, a **lax monoidal category**. In the sequel, $F^{\star n}$ will denote the iterated graft product defined recursively by $$F^{\star 0} = X, \qquad F^{\star (n+1)} = F^{\star n} \star F$$ so that all parentheses in $F^{\star n}$ are to the left. We have a partial coherence theorem: **Proposition:** Any two maps $$F^{\star m} \star F^{\star n} \to F^{\star (m+n)}$$ definable in the language of lax monoidal categories are equal. We denote this map by $\alpha_{m n}$. For $F$ a $Set$-valued species, an element of $F^{\star n}$ is described by data as follows: 1. A structure of tree constructed from $n$ sprouts _in a prescribed plugging order_; 1. An element of $F[\sigma]$ for each sprout $\sigma$ used in the construction of the tree in 1. Thus we expect some connection between the "geometric series" $\sum_n F^{\star n}$ and the free operad on $F$. The only difference between the two is that the standard construction of free operads via trees is in no way dependent on plugging order. Thus, to obtain the free operad, we have to mod out by isomorphisms on $F^{\star n}$ induced by "inessential" differences in plugging order. This will be described in detail in the section on trees and hierarchies.

   In this post, I’ll make some remarks on the “lax monoidal” structure of the graft product.

   The graft product is not associative up to isomorphism, but we do have a “lax” associativity. Specifically, we calculate

   $$\array{ (F \star G) \star H & \cong & (F' \otimes G)' \otimes H \\ & \cong & F'' \otimes G \otimes H + F' \otimes G' \otimes H \\ & \cong & F'' \otimes G \otimes H + F \star (G \star H) }$$

   so that there is a noninvertible associativity

   $$\alpha_{F G H}: F \star (G \star H) \to (F \star G) \star H$$

   given by inclusion, natural in each of its arguments $F$, $G$, and $H$, and satisfying an evident pentagon coherence condition. Additionally, there is a lax unit, defined as the species $X$ for which $X[S]$ is the monoidal unit of $V$ if $S$ is a 1-element set, else $X[S]$ is initial. Here we have evident natural maps

   $$\lambda_F: X \star F \to F, \qquad \rho_F: F \star X \to F$$

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

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

   However, all coherence conditions for monoidal categories hold (I won’t prove this here). We may call such a structure, relaxing the condition of invertibility of $\alpha$ and $\rho$ but retaining the naturality and coherence conditions, a lax monoidal category.

   In the sequel, $F^{\star n}$ will denote the iterated graft product defined recursively by

   $$F^{\star 0} = X, \qquad F^{\star (n+1)} = F^{\star n} \star F$$

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

   Proposition: Any two maps

   $$F^{\star m} \star F^{\star n} \to F^{\star (m+n)}$$

   definable in the language of lax monoidal categories are equal. We denote this map by $\alpha_{m n}$.

   For $F$ a $Set$-valued species, an element of $F^{\star n}$ is described by data as follows:

   1. A structure of tree constructed from $n$ sprouts in a prescribed plugging order;

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

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

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn this comment will be a brief interlude on presenting the notion of operad. There are various ways to describe operads and free operads. For example, * An *operad* $C$ is a monoid in the category of species with respect to a monoidal product known as the substitution product (denoted $\circ$). When unpacked, this means we have a collection of maps $$C[k] \otimes C[n_1] \otimes \ldots \otimes C[n_k] \to C[n_1 + \ldots + n_k]$$ and an identity element $I \to C[1]$, satisfying appropriate associativity and unit conditions. * Alternatively, an *operad* consists of objects $C(k)$ and a collection of morphisms $$plug_i \colon C[j] \otimes C[k] \to C[j+k-1]$$ satisfying suitable properties. In terms of sets, where elements $p \in C(j)$ are interpreted as $j$-ary operations $X^j \to X$, the meaning of $plug_i(p, q)$ is that outputs of the operation $q$ are plugged in as inputs for the $i^{th}$ argument of $p$, resulting in an operation $p(1, \ldots, q, \ldots, 1)$. Each description has its advantages. The first is concise and fits into clean categorical algebra. The second is easier to use for describing things like bar or cobar resolutions. The description in terms of graft products, given below, is essentially a way to repackage the second description; the plugging operations $$plug_1, \ldots, \plug_j \colon C[j] \otimes C[k] \to C[j+k-1]$$ are rolled together into a single operation $$plug \colon C'[j-1] \times C[k] \to C[j+k-1],$$ one for each pair $(j, k)$, and then the $plug$ operations are then rolled together into a single species map $C \star C \to C$. This map is to satisfy various associativity and unit conditions. With some qualification (related to mere lax associativity of the graft product) A key point for us is the following analogy. In one of the standard bar resolutions for groups or monoids $G$, one starts with the augmentation ideal $I G$ defined by an exact sequence $$0 \to I G \to \mathbb{Z} G \stackrel{\varepsilon}{\to} \mathbb{Z} \to 0$$ and then one defines a chain complex with components $\mathbb{Z}G \otimes (I G)^{\otimes n}$, or in other words a differential on $\mathbb{Z}G \otimes T(I G)$ where the tensor algebra $$T(I G) = \sum_{n \geq 0} I G^{\otimes n}$$ is seen as a graded algebra. Part of what makes this mechanism succeed, in particular the description of the tensor algebra, is the fact that the tensor product $\otimes$ is separately continuous in its arguments. One could try to develop a bar resolution for operads by analogy, and in fact that is what one does, but not in a simple-minded way suggested by the first description of operads. That is, one _does_ work with a resolution based on a free operad (free operads being analogous to tensor algebras) generated by a species analogue of an augmentation ideal, but the graded components are not iterated substitution products $M^{\circ n}$, and indeed, the free operad $\mathcal{O}(F)$ on a species $F$ is _not_ formed in a simple-minded way as $$\mathcal{O}(F) = \sum_{n \geq 0} F^{\circ n} \qquad (no)$$ because that is false (it would be true if the monoidal product $\circ$ were separately cocontinuous, or even if it preserves countable coproducts in its separate arguments). Instead, in the literature, the bar resolution for operads is developed more in accordance with the second description of operads. It is not _exactly_ based on a geometric series $$\sum_{n \geq 0} F^{\star n}$$ (because that is still not quite right for the free operad), but it's based on something close to that, as will be explained below. In any event, it is relevant for this development that $\star$ is separately cocontinuous in each of its arguments. Note that in this way of developing operad theory, the notion of _derivative_ of a species plays a more visible role. This is perhaps fortunate not just because we understand the calculus of species as a kind of categorification of the differential calculus, but also because it brings to light phenomena that usually go unremarked. Some examples: * The derivative $C'$ of an operad $C$ is naturally an algebra, in such a way that the operad structure according to the second description, $$C' \otimes C = C \star C \to C,$$ endows $C$ with a natural structure of $C'$-module. * The derivative of a cyclic operad is an ordinary operad.

   In this comment will be a brief interlude on presenting the notion of operad.

   There are various ways to describe operads and free operads. For example,

   • An operad $C$ is a monoid in the category of species with respect to a monoidal product known as the substitution product (denoted $\circ$). When unpacked, this means we have a collection of maps

     $$C[k] \otimes C[n_1] \otimes \ldots \otimes C[n_k] \to C[n_1 + \ldots + n_k]$$

     and an identity element $I \to C[1]$, satisfying appropriate associativity and unit conditions.

   • Alternatively, an operad consists of objects $C(k)$ and a collection of morphisms

     $$plug_i \colon C[j] \otimes C[k] \to C[j+k-1]$$

     satisfying suitable properties. In terms of sets, where elements $p \in C(j)$ are interpreted as $j$-ary operations $X^j \to X$, the meaning of $plug_i(p, q)$ is that outputs of the operation $q$ are plugged in as inputs for the $i^{th}$ argument of $p$, resulting in an operation $p(1, \ldots, q, \ldots, 1)$.

   Each description has its advantages. The first is concise and fits into clean categorical algebra. The second is easier to use for describing things like bar or cobar resolutions.

   The description in terms of graft products, given below, is essentially a way to repackage the second description; the plugging operations

   $$plug_1, \ldots, \plug_j \colon C[j] \otimes C[k] \to C[j+k-1]$$

   are rolled together into a single operation

   $$plug \colon C'[j-1] \times C[k] \to C[j+k-1],$$

   one for each pair $(j, k)$, and then the $plug$ operations are then rolled together into a single species map $C \star C \to C$. This map is to satisfy various associativity and unit conditions. With some qualification (related to mere lax associativity of the graft product)

   A key point for us is the following analogy. In one of the standard bar resolutions for groups or monoids $G$, one starts with the augmentation ideal $I G$ defined by an exact sequence

   $$0 \to I G \to \mathbb{Z} G \stackrel{\varepsilon}{\to} \mathbb{Z} \to 0$$

   and then one defines a chain complex with components $\mathbb{Z}G \otimes (I G)^{\otimes n}$, or in other words a differential on $\mathbb{Z}G \otimes T(I G)$ where the tensor algebra

   $$T(I G) = \sum_{n \geq 0} I G^{\otimes n}$$

   is seen as a graded algebra. Part of what makes this mechanism succeed, in particular the description of the tensor algebra, is the fact that the tensor product $\otimes$ is separately continuous in its arguments.

   One could try to develop a bar resolution for operads by analogy, and in fact that is what one does, but not in a simple-minded way suggested by the first description of operads. That is, one does work with a resolution based on a free operad (free operads being analogous to tensor algebras) generated by a species analogue of an augmentation ideal, but the graded components are not iterated substitution products $M^{\circ n}$, and indeed, the free operad $\mathcal{O}(F)$ on a species $F$ is not formed in a simple-minded way as

   $$\mathcal{O}(F) = \sum_{n \geq 0} F^{\circ n} \qquad (no)$$

   because that is false (it would be true if the monoidal product $\circ$ were separately cocontinuous, or even if it preserves countable coproducts in its separate arguments).

   Instead, in the literature, the bar resolution for operads is developed more in accordance with the second description of operads. It is not exactly based on a geometric series

   $$\sum_{n \geq 0} F^{\star n}$$

   (because that is still not quite right for the free operad), but it’s based on something close to that, as will be explained below. In any event, it is relevant for this development that $\star$ is separately cocontinuous in each of its arguments.

   Note that in this way of developing operad theory, the notion of derivative of a species plays a more visible role. This is perhaps fortunate not just because we understand the calculus of species as a kind of categorification of the differential calculus, but also because it brings to light phenomena that usually go unremarked. Some examples:

   • The derivative $C'$ of an operad $C$ is naturally an algebra, in such a way that the operad structure according to the second description,

     $$C' \otimes C = C \star C \to C,$$

     endows $C$ with a natural structure of $C'$-module.

   • The derivative of a cyclic operad is an ordinary operad.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNow we'll go a little deeper into the structure of free operads, by analyzing further the structure of $F^{\star n}$ in terms of (for us, useful) notions of *hierarchies* and *trees*. It is interesting to resolve $F^{\star n}$ into a sum of monomials, each of which is a tensor product of derivatives of $F$. For example, we have $$\array{ F \star F & \cong & F' \otimes F & & \\ (F \star F) \star F & \cong & (F \star F)' \otimes F & \cong & F'' \otimes F^{\otimes 2} + F' \otimes F' \otimes F \\ F^{\star 4} & \cong & (F^{\star 3})' \otimes F & \cong & F''' \otimes F^{\otimes 3} + 2 F'' \otimes F' \otimes F + F'' \otimes F \otimes F' \otimes F + F' \otimes F'' \otimes F^{\otimes 2} + (F')^{\otimes 3} \otimes F }$$ There are in fact $(n-1)!$ monomial terms in $F^{\star n}$, each a tensor product of $n$ higher derivative terms, and each corresponding to a function $f: [2, n] \to [1, n]$ (here $[k, n] \coloneqq \{k, k+1, \ldots, n\}$) such that $f(k) \lt k$ for all $k \in [2, n]$. Think of such a function as specifying a "plugging order", where each tensor factor $F^{(k)}$ occurring in a monomial is plugged into a vertex marked by a derivative symbol $^\prime$ which occurs prior in the monomial term. **Definition:** Let $[1, n]$ be the set of integers between $1$ and $n$. A **hierarchy** (of order $n$) is a function $f \colon [2, n] \to [1, n]$ such that $f(k) \lt k$ for all $k \in [2, n]$. If $(S, r)$ is a pointed set, then a **rooted tree** on $S$ is a morphism of pointed sets $f \colon S \to S$ such that every $s$ is taken to the basepoint $r$ by some iterate of $f$: for all $s \in S$ there exists $k$ such that $f^k(s) = r$. Note there are $(n-1)!$ possible hierarchies on the $n$-element set $[1, n]$. We denote this set by $Hier(n)$, or rather ${|Hier(n)|}$. There is an evident notion of morphism and isomorphism of rooted trees. Evidently every hierarchy $f \colon [2, n] \to [1, n]$ gives a rooted tree structure on the set $[1, n]$ with $1$ as basepoint. There is thus a groupoid $Hier(n)$ consisting of hierarchies on $[1, n]$ and isomorphisms between them. The disjoint union of groupoids $Hier = \sum_{n \geq 0} Hier(n)$ forms a full subgroupoid of the groupoid $Tree$ of finite rooted trees. In fact, the inclusion $$Hier \to Tree$$ is essentially surjective, hence an equivalence. For a set $S$, define the **$S^{th}$ derivative** $F^{(S)}$ of a species $F$ by the formula $F^{(S)}[T] = F[S + T]$. For each hierarchy $f$ of order $n$, define a species $F^{(f)}$ by $$F^{(f)} = \bigotimes_{k = 1}^n F^{(f^{-1}(k))}$$ **Proposition:** If $F$ is a species, then $$F^{\star n} = \sum_{f \in {|Hier(n)|}} F^{(f)}$$ for every $n \geq 1$. **Proof:** We proceed by induction. The case $n = 0$ is clear. We have $$F^{\star (n+1)} = (F^{\star n})' \otimes F = \sum_{f \in {|Hier(n)|}} \sum_{j=1}^n F^{(f^{-1}(1))} \otimes \ldots \otimes (F^{(f^{-1}(j))})' \otimes \ldots \otimes F^{(f^{-1}(n))} \otimes F$$ by the product rule for differentiation. Then, to each summand indexed by $(f, j)$, we associate a new hierarchy $g \in {|Hier(n+1)|}$ defined by $g(k) = f(k)$ for $k \leq n$, and $g(n+1) = j$. This means we can rewrite the last sum as $$F^{\star (n+1)} = \sum_{g \in {|Hier(n+1)|}} F^{(g)}$$ and we are done. $\Box$ Next, let $S_n$ be the set of permutations on $[1, n]$. If $\phi \in S_n$ defines an isomorphism $f \to g$ between hierarchies of order $n$, then there is an isomorphism (natural in species $F$) $$F^{(f)} \to F^{(g)}$$ and accordingly each species $F$ gives rise to a representation $F^{(-)}$ of the groupoid $Hier$. **Example:** Consider the hierarchy $f \colon [2, 3] \to [1, 3]$ that takes both $2$ and $3$ to $1$. There is an automorphism $\phi \in S_3$ from $f$ to itself, given by the simple transposition interchanging $2$ and $3$. The corresponding automorphism on $F^{(f)} = F'' \otimes F \otimes F$, evaluated at an object $W \in {|FB|}$, is a coproduct of evident isomorphisms \[F[T \sqcup \{2, 3\}] \otimes F[U] \otimes F[V] \to F[T \sqcup \{3, 2\}] \otimes F[V] \otimes F[U]\] ranging over the possible decompositions $W = T + U + V$. Thus the induced automorphism involves transport of $F$-structures along bijections of sets (as in the map $$F[T \sqcup \phi] \colon F[T \sqcup \{2, 3\}] \to F[T \sqcup \{3, 2\}]$$ where $\phi$ swaps $2$ and $3$), in addition to corresponding permutations of tensor factors, where we permute the factors $$F \otimes F = F^{(f^{-1}(2))} \otimes F^{(f^{-1}(3))} \to F^{(f^{-1}(3))} \otimes F^{(f^{-1}(2))} = F \otimes F.$$

   Now we’ll go a little deeper into the structure of free operads, by analyzing further the structure of $F^{\star n}$ in terms of (for us, useful) notions of hierarchies and trees.

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

   $$\array{ F \star F & \cong & F' \otimes F & & \\ (F \star F) \star F & \cong & (F \star F)' \otimes F & \cong & F'' \otimes F^{\otimes 2} + F' \otimes F' \otimes F \\ F^{\star 4} & \cong & (F^{\star 3})' \otimes F & \cong & F''' \otimes F^{\otimes 3} + 2 F'' \otimes F' \otimes F + F'' \otimes F \otimes F' \otimes F + F' \otimes F'' \otimes F^{\otimes 2} + (F')^{\otimes 3} \otimes F }$$

   There are in fact $(n-1)!$ monomial terms in $F^{\star n}$, each a tensor product of $n$ higher derivative terms, and each corresponding to a function $f: [2, n] \to [1, n]$ (here $[k, n] \coloneqq \{k, k+1, \ldots, n\}$) such that $f(k) \lt k$ for all $k \in [2, n]$. Think of such a function as specifying a “plugging order”, where each tensor factor $F^{(k)}$ occurring in a monomial is plugged into a vertex marked by a derivative symbol $^\prime$ which occurs prior in the monomial term.

   Definition: Let $[1, n]$ be the set of integers between $1$ and $n$. A hierarchy (of order $n$) is a function $f \colon [2, n] \to [1, n]$ such that $f(k) \lt k$ for all $k \in [2, n]$. If $(S, r)$ is a pointed set, then a rooted tree on $S$ is a morphism of pointed sets $f \colon S \to S$ such that every $s$ is taken to the basepoint $r$ by some iterate of $f$: for all $s \in S$ there exists $k$ such that $f^k(s) = r$.

   Note there are $(n-1)!$ possible hierarchies on the $n$-element set $[1, n]$. We denote this set by $Hier(n)$, or rather ${|Hier(n)|}$.

   There is an evident notion of morphism and isomorphism of rooted trees. Evidently every hierarchy $f \colon [2, n] \to [1, n]$ gives a rooted tree structure on the set $[1, n]$ with $1$ as basepoint. There is thus a groupoid $Hier(n)$ consisting of hierarchies on $[1, n]$ and isomorphisms between them. The disjoint union of groupoids $Hier = \sum_{n \geq 0} Hier(n)$ forms a full subgroupoid of the groupoid $Tree$ of finite rooted trees. In fact, the inclusion

   $$Hier \to Tree$$

   is essentially surjective, hence an equivalence.

   For a set $S$, define the $S^{th}$ derivative $F^{(S)}$ of a species $F$ by the formula $F^{(S)}[T] = F[S + T]$. For each hierarchy $f$ of order $n$, define a species $F^{(f)}$ by

   $$F^{(f)} = \bigotimes_{k = 1}^n F^{(f^{-1}(k))}$$

   Proposition: If $F$ is a species, then

   $$F^{\star n} = \sum_{f \in {|Hier(n)|}} F^{(f)}$$

   for every $n \geq 1$.

   Proof: We proceed by induction. The case $n = 0$ is clear. We have

   $$F^{\star (n+1)} = (F^{\star n})' \otimes F = \sum_{f \in {|Hier(n)|}} \sum_{j=1}^n F^{(f^{-1}(1))} \otimes \ldots \otimes (F^{(f^{-1}(j))})' \otimes \ldots \otimes F^{(f^{-1}(n))} \otimes F$$

   by the product rule for differentiation. Then, to each summand indexed by $(f, j)$, we associate a new hierarchy $g \in {|Hier(n+1)|}$ defined by $g(k) = f(k)$ for $k \leq n$, and $g(n+1) = j$. This means we can rewrite the last sum as

   $$F^{\star (n+1)} = \sum_{g \in {|Hier(n+1)|}} F^{(g)}$$

   and we are done. $\Box$

   Next, let $S_n$ be the set of permutations on $[1, n]$. If $\phi \in S_n$ defines an isomorphism $f \to g$ between hierarchies of order $n$, then there is an isomorphism (natural in species $F$)

   $$F^{(f)} \to F^{(g)}$$

   and accordingly each species $F$ gives rise to a representation $F^{(-)}$ of the groupoid $Hier$.

   Example: Consider the hierarchy $f \colon [2, 3] \to [1, 3]$ that takes both $2$ and $3$ to $1$. There is an automorphism $\phi \in S_3$ from $f$ to itself, given by the simple transposition interchanging $2$ and $3$. The corresponding automorphism on $F^{(f)} = F'' \otimes F \otimes F$, evaluated at an object $W \in {|FB|}$, is a coproduct of evident isomorphisms

   $$F[T \sqcup \{2, 3\}] \otimes F[U] \otimes F[V] \to F[T \sqcup \{3, 2\}] \otimes F[V] \otimes F[U]$$

   ranging over the possible decompositions $W = T + U + V$. Thus the induced automorphism involves transport of $F$-structures along bijections of sets (as in the map

   $$F[T \sqcup \phi] \colon F[T \sqcup \{2, 3\}] \to F[T \sqcup \{3, 2\}]$$

   where $\phi$ swaps $2$ and $3$), in addition to corresponding permutations of tensor factors, where we permute the factors

   $$F \otimes F = F^{(f^{-1}(2))} \otimes F^{(f^{-1}(3))} \to F^{(f^{-1}(3))} \otimes F^{(f^{-1}(2))} = F \otimes F.$$
8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNow for a final comment before taking a breather. Here we describe the free operad. Write $Der$ for the functor that takes a species $F$ to the representation $F^{(-)}$ of the groupoid $Hier$. Thus $Der$ defines a functor $$Der \colon V^\mathbf{P} \to V^{\mathbf{P} \times Hier}$$ An alternative view is that there is a fibering $$\pi_F: \sum_{n \geq 0} F^{\star n} \to {|Hier|} \cdot \mathbf{1}$$ which maps each monomial term in $F^{\star n}$ down to a copy of the terminal species $\mathbf{1}$, indexed by the appropriate hierarchy in ${|Hier(n)|}$. This fibering may be regarded as a span $\sum F^{\star n}$ from $\mathbf{1}$ to ${|Hier|} \cdot \mathbf{1}$ in $V^\mathbf{P}$. The groupoid $Hier$ may be regarded as a monad on ${|Hier|}$ in $Span(V^{\mathbf{P})$ that acts on $\sum F^{\star n}$ via the representation $F^{(-)}$. Eventually we will prove the following theorem. **Theorem:** Let $colim_{Hier}$ denote the colimit functor $$V^{\mathbf{P} \times Hier} \to V^\mathbf{P}.$$ Then $colim_{Hier} \circ Der \colon V^\mathbf{P} \to V^\mathbf{P}$ sends a species $F$ to the free operad $\mathcal{O}(F)$ on $F$. In alternative notation, \[\array{ \mathcal{O}(F) & \cong & \int^{f \in Hier} F^{(f)} \\ & \cong & (\sum_n F^{\star n}) \otimes_{Hier} 1 }\] This will be explored further in comments to come, perhaps.

   Now for a final comment before taking a breather. Here we describe the free operad.

   Write $Der$ for the functor that takes a species $F$ to the representation $F^{(-)}$ of the groupoid $Hier$. Thus $Der$ defines a functor

   $$Der \colon V^\mathbf{P} \to V^{\mathbf{P} \times Hier}$$

   An alternative view is that there is a fibering

   $$\pi_F: \sum_{n \geq 0} F^{\star n} \to {|Hier|} \cdot \mathbf{1}$$

   which maps each monomial term in $F^{\star n}$ down to a copy of the terminal species $\mathbf{1}$, indexed by the appropriate hierarchy in ${|Hier(n)|}$. This fibering may be regarded as a span $\sum F^{\star n}$ from $\mathbf{1}$ to ${|Hier|} \cdot \mathbf{1}$ in $V^\mathbf{P}$. The groupoid $Hier$ may be regarded as a monad on ${|Hier|}$ in that acts on $\sum F^{\star n}$ via the representation $F^{(-)}$.

   Eventually we will prove the following theorem.

   Theorem: Let $colim_{Hier}$ denote the colimit functor

   $$V^{\mathbf{P} \times Hier} \to V^\mathbf{P}.$$

   Then $colim_{Hier} \circ Der \colon V^\mathbf{P} \to V^\mathbf{P}$ sends a species $F$ to the free operad $\mathcal{O}(F)$ on $F$. In alternative notation,

   $$\array{ \mathcal{O}(F) & \cong & \int^{f \in Hier} F^{(f)} \\ & \cong & (\sum_n F^{\star n}) \otimes_{Hier} 1 }$$

   This will be explored further in comments to come, perhaps.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMar 13th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWow! What an impressive picture. I have one silly question: way back up in #4, isn't the Day convolution a coend rather than a coproduct? Is that coend getting wrapped up somewhere else in this presentation? The reason I'm asking about this is that I wanted to understand the analogous construction for cyclic operads, and then generalize it to cyclic multicategories. You mentioned cyclic operads at one point, so I suppose you've thought about how it works for them too; I'd be interested to hear whatever you have to say about them. And have you thought about how it works for multicategories (colored operads)? Presumably $\mathbf{P}$ gets replaced by the free symmetric strict monoidal category on the set of objects (colors)?

   Wow! What an impressive picture.

   I have one silly question: way back up in #4, isn’t the Day convolution a coend rather than a coproduct? Is that coend getting wrapped up somewhere else in this presentation?

   The reason I’m asking about this is that I wanted to understand the analogous construction for cyclic operads, and then generalize it to cyclic multicategories. You mentioned cyclic operads at one point, so I suppose you’ve thought about how it works for them too; I’d be interested to hear whatever you have to say about them.

   And have you thought about how it works for multicategories (colored operads)? Presumably $\mathbf{P}$ gets replaced by the free symmetric strict monoidal category on the set of objects (colors)?

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 14th 2016
    • (edited Mar 14th 2016)
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    Author: Todd_Trimble
    Format: MarkdownItexHi Mike; thanks for your feedback! That the tensor product $F \otimes G$, which abstractly is indeed a Day convolution coend induced from the monoidal structure of $\mathbf{P}$, can be reformulated as a sum $$(F \otimes G)[S] = \sum_{S = T + U} F[T] \otimes G[U]$$ is something I first learned from the Joyal theory of species papers. The idea is that from $(F \otimes G)[S] = \int^{V, W \in FB} F[V] \otimes G[W] \otimes \hom_{FB}(S, V \oplus W)$, each bijection $\phi: S \to V \oplus W$ induces a decomposition of $S$ into two components $T = \phi^{-1}(V)$, $U = \phi^{-1}(W)$, so that the coend may be rewritten $$\sum_{S = T + U} \int^{V, W} F[V] \otimes G[W] \otimes \hom_{FB}(T, V) \otimes \hom_{FB}(U, W) \cong \sum_{S = T + U} F[T] \otimes G[U]$$ where the last isomorphism comes about by the Yoneda (or coYoneda) lemma. (If you haven't gone through those species papers of Joyal and his coworkers (like Labelle), they are highly recommended! Really beautiful stuff.) I'll lead up to some ruminations on cyclic operads in the next couple or three comments to come (Williamson-like), but to be honest I haven't pursued those thoughts as hard as I ought to have. Mostly these are things which came out of considering the Lie operad, which is truly a key example. (No, I haven't thought much about the colored case.)

    Hi Mike; thanks for your feedback!

    That the tensor product $F \otimes G$, which abstractly is indeed a Day convolution coend induced from the monoidal structure of $\mathbf{P}$, can be reformulated as a sum

    $$(F \otimes G)[S] = \sum_{S = T + U} F[T] \otimes G[U]$$

    is something I first learned from the Joyal theory of species papers. The idea is that from $(F \otimes G)[S] = \int^{V, W \in FB} F[V] \otimes G[W] \otimes \hom_{FB}(S, V \oplus W)$, each bijection $\phi: S \to V \oplus W$ induces a decomposition of $S$ into two components $T = \phi^{-1}(V)$, $U = \phi^{-1}(W)$, so that the coend may be rewritten

    $$\sum_{S = T + U} \int^{V, W} F[V] \otimes G[W] \otimes \hom_{FB}(T, V) \otimes \hom_{FB}(U, W) \cong \sum_{S = T + U} F[T] \otimes G[U]$$

    where the last isomorphism comes about by the Yoneda (or coYoneda) lemma.

    (If you haven’t gone through those species papers of Joyal and his coworkers (like Labelle), they are highly recommended! Really beautiful stuff.)

    I’ll lead up to some ruminations on cyclic operads in the next couple or three comments to come (Williamson-like), but to be honest I haven’t pursued those thoughts as hard as I ought to have. Mostly these are things which came out of considering the Lie operad, which is truly a key example.

    (No, I haven’t thought much about the colored case.)

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexThanks! Now that you describe it, it kind of rings a bell, but it still seems weird to me, like something that shouldn't work. (-: Does that work for Day convolution over any, um, cocartesian monoidal extensive category?

    Thanks! Now that you describe it, it kind of rings a bell, but it still seems weird to me, like something that shouldn’t work. (-: Does that work for Day convolution over any, um, cocartesian monoidal extensive category?

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 14th 2016
    • (edited Mar 14th 2016)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexWithout thinking about it very hard, it might work for the groupoidal core of a cocartesian monoidal extensive category. <b>Edit:</b> And maybe for a cocartesian monoidal extensive category as well, although I've had a few drinks. If true, that would seem potentially interesting for a more Lawvere theory context, as opposed to an operad context.

    Without thinking about it very hard, it might work for the groupoidal core of a cocartesian monoidal extensive category.

    Edit: And maybe for a cocartesian monoidal extensive category as well, although I’ve had a few drinks. If true, that would seem potentially interesting for a more Lawvere theory context, as opposed to an operad context.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexProbably what I meant was the groupoidal core.

    Probably what I meant was the groupoidal core.

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 14th 2016
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    Author: Todd_Trimble
    Format: MarkdownItexLet me pick up again from the end of #8. I now want to reformulate the notion of permutative $V$-operad in terms of the graft product $\star$. Recall that for a species $F$, we have the decomposition $$\array{ F^{\star 3} & = & (F \star F) \star F \\ & = & F' \otimes F' \otimes F + F'' \otimes F \otimes F \\ & = & F^{(f)} + F^{(g)} }$$ where $f$ is the hierarchy $[2, 3] \to [1, 3]$ sending $j$ to $j-1$, and $g$ is the hierarchy considered at the end of #7, sending both $2$ and $3$ to $1$. The inclusion of $F^{(f)}$ in $F^{\star 3}$ is identified with the lax associativity $\alpha \colon F \star (F \star F) \to (F \star F) \star F$. **Definition:** An **$V$-valued operad** is a species $F \colon FB \to V$ equipped with maps $m \colon F \star F \to F$, $u: X \to F$ (recall $X$ from #5; it is Joyal's notation) such that the following conditions hold. * The restriction of $m \circ (m \star 1_F) \colon (F \star F) \star F \to F$ to $F^{(f)}$ is given by $$F \star (F \star F) \stackrel{1_F \star m}{\to} F \star F \stackrel{m}{\to} F.$$ In other words, the diagram $$\array{ F \star (F \star F) & \stackrel{1 \star m}{\to} & F \star F & \\ ^\mathllap{\alpha} \downarrow & & & \searrow^\mathrlap{m} \\ (F \star F) \star F & \underset{m \star 1}{\to} F \star F & \underset{m}{\to} & F }$$ commutes; * The restriction of $m \circ (m \star 1_F)$ to $F^{(g)}$ coequalizes the pair of automorphisms $F^{(g)} \to F^{(g)}$ induced by the two hierarchy automorphisms on $g$, namely the identity automorphism and the automorphism on $F^{(g)}$ described in the Example at the end of #7. * The composite $$X \star F \stackrel{u \star 1_F}{\to} F \star F \stackrel{m}{\to} F$$ is $\lambda_F$, and the composite $$F \star X \stackrel{1_F \star u}{\to} F \star F \stackrel{m}{\to} F$$ is $\rho_F$. (See #5 for the descriptions of the left and right unit constraints $\lambda_F$ and $\rho_F$.) The first two conditions correspond to the associativity condition in the usual definition of operad (monoid with respect to plehystic monoidal product on species -- what Joyal called the substitution product and denotes by $\circ$), and the last condition to the unit conditions. $\Box$ This reformulation is much closer to the idea of operad that is implicit in Stasheff's Homotopy of H-Spaces papers, where for example the associahedra $K_j$ carry operations $sub_i: K_j \times K_k \to K_{j+k-1}$ for $i = 1, \ldots, j$, which we called $plug_i$ in #6 above and which we have rolled into a single species map $K \star K \to K$. Also this reformulation is the right one to use if we want to consider *non-unital* operads as Stasheff did (he considered associahedra $K_2, K_3, \ldots$ but no $K_1$); if we define a non-unital operad instead as a species $F$ equipped with an associative multiplication $F \circ F \to F$, we get something much too weak or inexpressive for this purpose. Before we prove that these conditions are necessary and sufficient, I'd like to observe that if $F$ is an operad (*à la* the definition above), then its derivative $F'$ naturally carries a structure of algebra (i.e., monoid with respect to the convolution product $\otimes$), something I alluded to at the end of #6. For if we take the derivative of the species map $m: F' \otimes F \to F$ with the help of the product rule, we get a map $$F' \otimes F' + F'' \otimes F \to F'$$ which in turn given by a pair of maps $F' \otimes F' \to F'$, $F'' \otimes F \to F'$. By taking the derivative on the first associativity condition, one sees that the first of these maps $F' \otimes F' \to F'$ is an associative operation. The derivative of the operad unit $u: X \to F$ gives a unit $u': I \to F'$ where $I$ is the monoidal unit for $\otimes$, and the third condition implies that $u'$ satisfies the unit axioms for an algebra. Finally, note that the map $m: F' \otimes F \to F$ endows $F$ with a structure of module over this algebra $F'$; again the requisite axioms for a module follow from the first and third conditions of the definition. I'll give some examples of this later, perhaps. An important example is the Lie algebra operad as a $Vect$-enriched operad, whose derivative is the tensor algebra species $\sum_{n \geq 0} X^{\otimes n}$ whose algebra structure as free noncommutative algebra coincides with the derivative structure described above.

    Let me pick up again from the end of #8. I now want to reformulate the notion of permutative $V$-operad in terms of the graft product $\star$.

    Recall that for a species $F$, we have the decomposition

    $$\array{ F^{\star 3} & = & (F \star F) \star F \\ & = & F' \otimes F' \otimes F + F'' \otimes F \otimes F \\ & = & F^{(f)} + F^{(g)} }$$

    where $f$ is the hierarchy $[2, 3] \to [1, 3]$ sending $j$ to $j-1$, and $g$ is the hierarchy considered at the end of #7, sending both $2$ and $3$ to $1$. The inclusion of $F^{(f)}$ in $F^{\star 3}$ is identified with the lax associativity $\alpha \colon F \star (F \star F) \to (F \star F) \star F$.

    Definition: An $V$-valued operad is a species $F \colon FB \to V$ equipped with maps $m \colon F \star F \to F$, $u: X \to F$ (recall $X$ from #5; it is Joyal’s notation) such that the following conditions hold.

    • The restriction of $m \circ (m \star 1_F) \colon (F \star F) \star F \to F$ to $F^{(f)}$ is given by

      $$F \star (F \star F) \stackrel{1_F \star m}{\to} F \star F \stackrel{m}{\to} F.$$

      In other words, the diagram

      $$\array{ F \star (F \star F) & \stackrel{1 \star m}{\to} & F \star F & \\ ^\mathllap{\alpha} \downarrow & & & \searrow^\mathrlap{m} \\ (F \star F) \star F & \underset{m \star 1}{\to} F \star F & \underset{m}{\to} & F }$$

      commutes;

    • The restriction of $m \circ (m \star 1_F)$ to $F^{(g)}$ coequalizes the pair of automorphisms $F^{(g)} \to F^{(g)}$ induced by the two hierarchy automorphisms on $g$, namely the identity automorphism and the automorphism on $F^{(g)}$ described in the Example at the end of #7.

    • The composite

      $$X \star F \stackrel{u \star 1_F}{\to} F \star F \stackrel{m}{\to} F$$

      is $\lambda_F$, and the composite

      $$F \star X \stackrel{1_F \star u}{\to} F \star F \stackrel{m}{\to} F$$

      is $\rho_F$. (See #5 for the descriptions of the left and right unit constraints $\lambda_F$ and $\rho_F$.)

    The first two conditions correspond to the associativity condition in the usual definition of operad (monoid with respect to plehystic monoidal product on species – what Joyal called the substitution product and denotes by $\circ$), and the last condition to the unit conditions. $\Box$

    This reformulation is much closer to the idea of operad that is implicit in Stasheff’s Homotopy of H-Spaces papers, where for example the associahedra $K_j$ carry operations $sub_i: K_j \times K_k \to K_{j+k-1}$ for $i = 1, \ldots, j$, which we called $plug_i$ in #6 above and which we have rolled into a single species map $K \star K \to K$. Also this reformulation is the right one to use if we want to consider non-unital operads as Stasheff did (he considered associahedra $K_2, K_3, \ldots$ but no $K_1$); if we define a non-unital operad instead as a species $F$ equipped with an associative multiplication $F \circ F \to F$, we get something much too weak or inexpressive for this purpose.

    Before we prove that these conditions are necessary and sufficient, I’d like to observe that if $F$ is an operad (à la the definition above), then its derivative $F'$ naturally carries a structure of algebra (i.e., monoid with respect to the convolution product $\otimes$), something I alluded to at the end of #6. For if we take the derivative of the species map $m: F' \otimes F \to F$ with the help of the product rule, we get a map

    $$F' \otimes F' + F'' \otimes F \to F'$$

    which in turn given by a pair of maps $F' \otimes F' \to F'$, $F'' \otimes F \to F'$. By taking the derivative on the first associativity condition, one sees that the first of these maps $F' \otimes F' \to F'$ is an associative operation. The derivative of the operad unit $u: X \to F$ gives a unit $u': I \to F'$ where $I$ is the monoidal unit for $\otimes$, and the third condition implies that $u'$ satisfies the unit axioms for an algebra. Finally, note that the map $m: F' \otimes F \to F$ endows $F$ with a structure of module over this algebra $F'$; again the requisite axioms for a module follow from the first and third conditions of the definition.

    I’ll give some examples of this later, perhaps. An important example is the Lie algebra operad as a $Vect$-enriched operad, whose derivative is the tensor algebra species $\sum_{n \geq 0} X^{\otimes n}$ whose algebra structure as free noncommutative algebra coincides with the derivative structure described above.