Operads: first contact

An operad is a monoid for the substitution product discussed in lecture 3. As such, “the theory of operads” is the study of the category $\mathbf{Opd}={\bf Mon}({\bf Spc},\circ)$; but things are never as simple as they seem…

First, let’s flesh out what the definition prescribes. An operad consists of a functor $T : {\bf B} \to \bf Set$, equipped with

• multiplication, a natural transformation of type $\mu : T \circ T\Rightarrow T$, subject to the associativity axiom (the two compositions are mediated by the associator):

  

• unit, a natural transformation of type $\eta : J \Rightarrow T$ which satisfies the left and right unit axioms (mediated by the unitors):

  

It is worth to note explicitly the shape of $\mu,\eta$; those who already know it will recognize the form of an operad as defined in, say, algebraic topology textbooks.

• $\eta$, as a natural transformation of type $y1 \Rightarrow T$, corresponds by Yoneda to an element $u\in T1$.

• $\mu$ is a natural transformation in the set

  $$ \begin{aligned} {\bf Spc}\Big(\int^m Tm\times T^{*m}(-) , T\Big) &\cong \int_m {\bf Spc}\big(Tm\times T^{*m}(-) , T\big)\\ &\cong \int_{mr} {\bf Set}\big(Tm\times T^{*m}r , Tr\big)\\ &\cong \int_{mr, \vec n} {\bf Set}\Big(Tm\times Tn_1\times\dots\times Tn_m \times {\bf B}(\textstyle\sum n_i,r) , Tr\Big)\\ &\cong \int_{mr, \vec n} {\bf Set}\Big({\bf B}(\textstyle\sum n_i,r), {\bf Set}(Tm\times Tn_1\times\dots\times Tn_m , Tr)\Big)\\ &\cong {\bf Set}(Tm\times Tn_1\times\dots\times Tn_m , T(\textstyle\sum n_i)) \end{aligned} $$

  where the last equality is Yoneda, and thus $\mu$ has components

  

Remark. Naturality for $\eta$ is straightforward; note that naturality for $\mu$ amounts to the commutativity of the diagram

given permutations $\alpha, \beta_1,\dots, \beta_m$, where $\bar\beta=\beta_1 \mathbin{ш} \beta_2 \mathbin{ш} \dots \mathbin{ш} \beta_m$ is the shuffle (шафл) of the permutations $\beta_1,\dots, \beta_m$.

Associativity of $\mu$ is better appreciated through a picture: given an an $n$-ary term $f\in Tn$, terms $g_1\in Tp_1,\dots,g_n\in Tp_n$, and moreover terms $h_{1,1},\dots, h_{1,q_{p_1}},\dots, h_{n,1},\dots, h_{n,p_n}$, the two graftings depicted below are equal.

Some of the structure of $\mathbf{Opd}$ can be deduced from the fact that we are just looking at a particular instance of the category of monoids in a monoidal category:

Remark. There is an adjunction

whence we can derive (among other things) that

• the category of operads is complete, and limits are created by the forgetful functor $U$; in particular, the Hadamard product of two species who happen to be operads inherits an operad structure (it is a straightforward consequence of a general fact, but it’s cool to see the derivation): let $T_* : {\cal J} \to {\bf Opd}$ be a diagram of operads; then the limit $\text{lim}_{\cal J}\, T_J$ computed in species inherits a structure of operad as follows:

  $$ \begin{aligned} (\text{lim}_{\cal J}\, T_J)\circ (\text{lim}_{\cal J}\, T_J) & = \int^m (\text{lim}_{\cal J}\, T_J)m \times (\text{lim}_{\cal J}\, T_J)^{*m} \\ & = \int^m \text{lim}_{\cal J}\, (T_Jm) \times (\text{lim}_{\cal J}\, T_J)^{*m} \\ & \cong \int^m \text{lim}_{\cal J}\, (T_Jm) \times \int^{J_1,\dots, J_m} \text{lim}_{J_1}\dots \text{lim}_{J_m} T_{J_1}n_1 \times\dots \times T_{J_m}n_m\times {\bf B}(\sum n_i,-)\\ & \to \text{lim}_{\cal J}\,\text{lim}_{J_1}\dots \text{lim}_{J_m} \int^m (T_Jm) \times \int^{n_1,\dots, n_m} T_{J_1}n_1 \times\dots \times T_{J_m}n_m\times {\bf B}(\sum n_i,-)\\ (\dag)& \to \text{lim}_{\cal J} \int^m (T_Jm) \times \int^{n_1,\dots, n_m} T_J n_1 \times\dots \times T_J n_m\times {\bf B}(\sum n_i,-)\\ & = \text{lim}_{\cal J} \int^m (T_Jm) \times T_J^{*m}\\ (\ddag)& \to \text{lim}_{\cal J} T_J \end{aligned} $$

  where $(\dag)$ precomposes with the diagonal functor $\Delta : {\cal J} \to {\cal J}^m$, and $(\ddag)$ uses the multiplication of each $T_J$; the unit is way easier to obtain as $\eta_J : y1 \Rightarrow T_J$ is a cone, inducing a unique $\bar\eta : y1 \Rightarrow \text{lim}_{\cal J} T_J$ into the limit.

• colimits are complicated, but still they are reducible to certain reflexive coequalizers of free algebras, via the “Linton trick”.

One can define the free operad on a combinatorial species, i.e. a left adjoint to the forgetful functor $U$ above; the situation, however, is in stark contrast with the case of Cauchy monoids, because

• The Day convolution product is separately cocontinuous in both variables (compare this with the fact that the substitution product commutes with colimits in its left, but not in its right, component: $-\circ G$ is cocontinuous (and thus a left adjoint, given that $\bf Spc$ is a locally finitely presentable category), $F\circ-$ isn’t -and thus a right adjoint to it can’t exist).
• In the case of Day convolution, the free-forgetful adjunction is easy to characterize conceptually: the free Day-monoid on a species is $\sum_{n\ge 0} F^{*n}$, which reduces to the species $L\circ F$; two corollaries of this fact are important: first, the action of the forgetful functor $U : {\bf Mon}({\bf Spc},\otimes) \to {\bf Spc}$ coincides with the Day-hom ${}$

A not-so-helpful observation is that the free operad on $F$ has to be obtained as the initial algebra for the functor ${\bf Spc} \to {\bf Spc} : P\mapsto y1 + F\circ P$. Adamek’s theorem then ensures that such an initial algebra is the colimit of the chain

$$ y1 \to y1 + F \circ y1 = y1 + F \to y1 + F \circ (y1 + F) \to \dots $$

The problem is, we cannot develop such an expression simplifying, since the substitution product is only left cocontinuous.

Proposition. Every operad $T : {\bf B} \to {\bf Set}$ defines a monad $\flat T : {\bf Set} \to {\bf Set}$ as follows:

• the underlying endofunctor $\flat T$ is

  $$ (\flat T)X := \int^n X^n \times Tn $$

  evidently, $\flat T$ is the analytic functor associated to $T$, so the construction of this proposition can be summarized as: the analytic functor associated to an operad (qua species) is a monad, and thus an analytic monad.

• the unit (component) $\eta : X \to (\flat T)X$ is defined through the unit of $T$,

  

• the multiplication (component) $\mu : (\flat T)((\flat T)X) \to (\flat T)X$ is defined from the composition

  $$ \begin{aligned} \int^{m} (M_T(X))^m \times Tm &\cong \int^{m} \Bigl[ \int^{n} X^n \times Tn \Bigr]^m \times Tm \\ &\cong \int^{m} \left(\int^{\vec n} X^{n_1+\cdots+n_m} \times T{n_1} \times \cdots \times T{n_m} \right) \times Tm \\ &\cong \int^{m,\vec n} X^{n_1+\cdots+n_m} \times T{n_1} \times \cdots \times T{n_m} \times Tm \\ &\to \int^k X^k\times Tk \end{aligned} $$

  where the last arrow is obtained, in turn, composing

  

Examples of operads

As it is obvious, on the same set there can be non-isomorphic monoid structures; similarly, even if usually there is just one important implicitly understood operad structure on a species, there can be many. One example is the species of linear orders $L$: it is the associative operad $As$ operad with respect to a natural composition operation, but it is also an operad $Zinb$, the Zinbiel operad, with respect to a different structure.

Example. Every species defines an operad, the endomorphism operad.

Example. The operad $Com$ with carrier $E$

Example. The operad $As$ with carrier $L$

Example. A different operad structure (the operad $Zinb$) over $L_+$

Example. The operad $Perm$ with carrier $El$

Example. The $Lie$ operad.

We summarize the situation in a table:

Algebra type	Description
$Com, Com_+$	$E, E_+$
$As, As_+$	$L, L_+$
$Zinb$	$L_+$
$Perm$	species of elements
$Lie$

Algebras for an operad