*Guest post by Simon Cho*

We continue the Kan Extension Seminar II with Max Kelly’s On the operads of J. P. May. As we will see, the main message of the paper is that (symmetric) operads enriched in a suitably nice category $\mathcal{V}$ arise naturally as monoids for a “substitution product” in the monoidal category $[\mathbf{P}, \mathcal{V}]$ (where $\mathbf{P}$ is a category that keeps track of the symmetry). Before we begin, I want to thank the organizers and participants of the Kan Extension Seminar (II) for the opportunity to read and discuss these nice papers with them.

Some time ago, in her excellent post about Hyland and Power’s paper, Evangelia described what Lawvere theories are about. We might think of Lawvere theories as a way to frame algebraic structure by stratifying the different components of an algebraic structure into roughly three ascending levels of specificity: the product structure, the specific algebraic operations (meaning, other than projections, etc.), and the models of that algebraic structure. These structures are manifested categorically through (respectively) the category $\aleph_0^{\text{op}}$ of finite sets and (the duals of) maps between them, a category $\mathcal{L}$ with finite products that has the same objects as $\aleph_0$, and some other category $\mathcal{C}$ with finite products. Then a Lawvere theory is just a strict product preserving functor $I: \aleph_0^{\text{op}} \rightarrow \mathcal{L}$, and a model or interpretation of a Lawvere theory is a (non-strict) product preserving functor $M: \mathcal{L} \rightarrow \mathcal{C}$.

Thus $\aleph_0^{\text{op}}$ specifies the bare product structure (with the attendant projections, etc.) which gives us a notion of what it means to be “$n$-ary” for some given $n$; $I$ then transfers this notion of arity to the category $\mathcal{L}$, whose shape describes the specific algebraic structure in question (think of the diagrams one uses to categorically define the group axioms, for example); $M$ then gives a particular manifestation of the algebraic structure $\mathcal{L}$ on an object $M \circ I (1) \in \mathcal{C}$.

The reason I bring this up is that I like to think of operads as what results when we make the following change of perspective on Lawvere theories: whereas models of Lawvere theories are essentially given by specifying a “ground set of elements” $A \in \mathcal{C}$ and taking as the $n$-ary operations morphisms $A^n \rightarrow A$, we now consider a hypothetical category whose ($n$-indexed) objects themselves are the homsets $\mathcal{C}(A^n, A)$, along with some machinery that keeps track of what happens when we permute the argument slots.

#### Cosmos structure on $[\mathbf{P}, \mathcal{V}]$

More precisely, consider the category $\mathbf{P}$ with objects the natural numbers, and morphisms $\mathbf{P}(m,n)$ given by $\mathbf{P}(n,n) = \Sigma_n$ (the symmetric group on $n$ letters) and $\mathbf{P}(m,n) = \emptyset$ for $m \neq n$.

Let $\mathcal{V}$ be a cosmos, that is, a complete and cocomplete symmetric monoidal closed category with identity $I$ and internal hom $[-,-]$.

Fix $A \in \mathcal{V}$. The assignment $n \mapsto [A^{\otimes n}, A]$ defines a functor $\mathbf{P} \rightarrow \mathcal{V}$ (where functoriality in $\mathbf{P}$ comes from the symmetry of the tensor product in $\mathcal{V}$). This turns out to be a typical example of a $\mathcal{V}$-operad, which we call the “endomorphism operad” on $A$. In order to actually define what an operad is, we need to lay some groundwork.

(A point of notation: we will henceforth denote $A^{\otimes n}$ by $A^n$.)

We’ll need the fact that the functor $\mathcal{V}(I, -): \mathcal{V} \rightarrow \textbf{Sets}$ has a left adjoint $F$ given by $FX = \coprod_X I$. $F$ takes the product to the tensor product (since it’s a left adjoint and tensor products in $\mathcal{V}$ distributes over coproducts), and in fact we can assume that it does so strictly. Henceforth for $X \in \textbf{Sets}$ and $A \in \mathcal{V}$ we write $X \otimes A$ to actually mean $FX \otimes A$.

We then get a cosmos structure on $\mathcal{F}$ which is given by Day convolution: for $T,S \in \mathcal{F}$ we have
$T \otimes S = \int^{m,n} \mathbf{P}(m+n, - ) \otimes Tm \otimes Sn$
Since we are thinking of a given $T \in \mathcal{F}$ as a collection of operations (indexed by arity) on which we can act by permuting the argument slots, we can think of $(T \otimes S) k$ as a collection of the $k$-ary operations that we obtain by freely permuting $m$ argument slots of type $T$ and $n$ argument slots of type $S$ (where $m,n$ range over all pairs such that $m+n = k$), modulo respecting the previously given actions of $\Sigma_m$ (resp. $\Sigma_n$) on $Tm$ (resp. $Sn$).

The identity is then given by $\mathbf{P}(0,-) \otimes I$.

**Associativity and symmetry of the cosmos structure.** Now let $T,S, R \in \mathcal{F}$. If we unpack the definition, draw out some diagrams, and apply some abstract nonsense, we find that
$T \otimes (S \otimes R) \simeq (T \otimes S) \otimes R \simeq \int^{m+n+k} \mathbf{P}(m+n+k, - ) \otimes Tm \otimes Sn \otimes Rk$
which we can again assume are actually equalities.

Before we address the symmetry of this monoidal structure, we make a technical point. $\mathbf{P}$ itself has a symmetric monoidal structure, given by addition. Thus for $n_1, \dots, n_m \in \mathbf{P}$ we have $n_1 + \cdots + n_m \in \mathbf{P}$. There is evidently an action of $\Sigma_m$ on this term, which we require to be in the “wrong” direction, so that $\xi \in \Sigma_m$ induces $\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m$ rather than the other way around.

(However, for the symmetry of the monoidal structure on $\mathcal{V}$, given a product $A_1 \otimes \cdots \otimes A_m$ we require that the action of $\Sigma_m$ on this term is in the “correct” direction, i.e. $\xi \in \Sigma_m$ induces $\langle \xi \rangle: A_1 \otimes \cdots \otimes A_m \rightarrow A_{\xi 1} \otimes \cdots \otimes A_{\xi m}$.)

We thus have:

$\begin{matrix}
T_1 \otimes \cdots \otimes T_m &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_1 + \cdots n_m, - ) \otimes T_1 n_1 \otimes \cdots \otimes T_{m} n_m\\
&&\\
{\langle \xi \rangle} \Big \downarrow && \Big \downarrow {\mathbf{P}(\langle \xi \rangle, -) \otimes \langle \xi \rangle}\\
&&\\
T_{\xi 1} \otimes \cdots \otimes T_{\xi m} &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_{\xi 1} + \cdots n_{\xi m}, - ) \otimes T_{\xi 1} n_{\xi 1} \otimes \cdots \otimes T_{\xi m} n_{\xi m}\\
\end{matrix}$

Now $\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m$ extends to an action $\langle \xi \rangle: T_1 \otimes \cdots \otimes T_m \rightarrow T_{\xi 1} \otimes \cdots \otimes T_{\xi m}$ as we saw previously. Therefore we now have a functor $\mathbf{P}^{\text{op}} \times \mathcal{F} \rightarrow \mathcal{F}$ given by $(m, T) \mapsto T^m$, a fact which we will later use.

**$\mathcal{F}$ as a $\mathcal{V}$-category.** There is a way in which we can regard $\mathcal{V}$ as a full coreflective subcategory of $\mathcal{F}$: consider the functor $\phi: \mathcal{F} \rightarrow \mathcal{V}$ given by $\phi T = T0$. This has a right adjoint $\psi: \mathcal{V} \rightarrow \mathcal{F}$ given by $\psi A = \mathbf{P}(0, -) \otimes A$.

The inclusion $\psi$ preserves all of the relevant monoidal structure, so we are justified in considering $A \in \mathcal{V}$ as either an object of $\mathcal{V}$ or of $\mathcal{F}$ (via the inclusion $\psi$). With this notation we can write, for $A \in \mathcal{V}$ and $T,S \in \mathcal{F}$:
$\mathcal{F}(A \otimes T, S) \simeq \mathcal{V}(A, [T,S])$
If $T, S \in \mathcal{F}$ then their $\mathcal{F}$-valued hom is given by $[[T,S]]$, where for $k \in \mathbf{P}$ we have
$[[T,S]]k = \int_n [Tn, S(n+k)]$
and their $\mathcal{V}$-valued hom, which makes $\mathcal{F}$ into a $\mathcal{V}$-category, is given by
$[T,S] = \phi [[T,S]] = \int_n [Tn, Sn]$

#### The substitution product

Let us return to our motivating example of the endomorphism operad (which we denote by $\{A,A\}$) on $A$, for a fixed $A \in \mathcal{V}$. For now it’s just an object $\{A, A\} \in \mathcal{F}$; but it contains more structure than we’re currently using. Namely, for each $m, n_1, \dots, n_m \in \mathbf{P}$ we can give a morphism
$[A^m, A] \otimes \left ( [A^{n_1}, A] \otimes \cdots \otimes [A^{n_m}, A] \right ) \rightarrow [A^{n_1 + \cdots + n_m}, A]$
coming from evaluation (see the section below about the little $n$-disks operad for details). We would like a general framework for expressing such a notion of composing operations.

**Definition of an operad.** Recall from the previous section that, for given $T \in \mathcal{F}$, we can consider $n \mapsto T^n$ as a functor $\mathbf{P}^{\text{op}} \rightarrow \mathcal{F}$. We can thus define a (non-symmetric!) product $T \circ S = \int^n Tn \otimes S^n$. It is easy to check that if $S \in \mathcal{V}$ then in fact $T \circ S \in \mathcal{V}$, so that $\circ$ can be considered as a functor either of type $\mathcal{F} \times \mathcal{F} \rightarrow \mathcal{F}$ or of type $\mathcal{F} \times \mathcal{V} \rightarrow \mathcal{V}$.

The clarity with which Kelly’s paper demonstrates the various important properties of this substitution product would be difficult for me to improve upon, so I simply list here the punchlines, and refer the reader to the original paper for their proofs:

For $T,S \in \mathcal{F}$ and $n \in \mathbf{P}$, we have $(T \circ S)^n \simeq T^n \circ S$ which is natural in $T, S, n$. Using this and a Fubini style argument we get associativity of $\circ$.

$J = \mathbf{P}(1, - )\otimes I$ is the identity for $\circ$.

For $S \in \mathcal{F}$, $- \circ S: \mathcal{F} \rightarrow \mathcal{F}$ has the right adjoint $\{S, -\}$ given by $\{S, R\}m = [S^m, R]$. Moreover if $A \in \mathcal{V}$ then we in fact have $\mathcal{V}(T \circ A, B) \simeq \mathcal{F} (T, \{A, B\})$.

We can now define an *operad* as a monoid for $\circ$, i.e. some $T \in \mathcal{F}$ equipped with $\mu: T \circ T \rightarrow T$ and $\eta: J \rightarrow T$ satisfying the monoid axioms. Operad morphisms are morphisms $T \rightarrow T^\prime$ that respect $\mu$ and $\eta$.

**$\{A, A\}$ as an operad.** Once again we turn back to the example of $\{A, A\} \in \mathcal{F}$. Note that our choice to denote the endomorphism operad $(n \mapsto [A^n, A])$ by $\{A, A\}$ agrees with the construction of $\{A, -\}$ as the right adjoint to $- \circ A$.

There is an evident evaluation map $\{A, A\} \circ A \xrightarrow{e} A$, so that we have the composition
$\{A, A\} \circ \{A, A\} \circ A \xrightarrow{1 \circ e} \{A,A\} \circ A \xrightarrow{e} A$
which by adjunction gives us $\mu:\{A,A\} \circ \{A,A\} \rightarrow \{A,A\}$ which we take as our monoid multiplication. Similarly $J \circ A \simeq A$ corresponds by adjunction to $\eta: J \rightarrow \{A, A\}$. We thus have that $\{A,A\}$ is an operad. In fact it is the “universal” operad, in the following sense:

Every operad $T \in \mathcal{F}$ gives a monad $T \circ -$ on $\mathcal{F}$, or on $\mathcal{V}$ via restriction. Given $A \in \mathcal{F}$, algebra structures $h^{\prime}: T \circ A \rightarrow A$ for the monad $T \circ -$ on $A$ correspond precisely to operad morphisms $h: T \rightarrow \{A,A\}$. In this case we say that $h$ gives an algebra structure on $A$ for the operad $T$.

#### The little $n$-disks operad

There are some other aspects of operads that the paper looks at, but for this post I will abuse artistic license to talk about something else that isn’t exactly in the paper (although it is indirectly referenced): May’s little $n$-disks operad. For a great introduction to the following material I recommend Emily Riehl’s notes on Kathryn Hess’s two-part (I,II) talk on operads in algebraic topology.

Let $\mathcal{V} = (\mathbf{Top}_{\text{nice}}, \times, \{*\})$ where $\mathbf{Top}_{\text{nice}}$ is one’s favorite cartesian closed category of topological spaces, with $\times$ the appropriate product in this category.

Fix some $n \in \mathbb{N}$. For $k \in \mathbf{P}$, we let $d_n(k) = \text{sEmb}(\coprod_{k} D^n, D^n)$, the space of standard embeddings of $k$ copies of the closed unit $n$-disk in $\mathbb{R}^n$ into the closed unit $n$-disk in $\mathbb{R}^n$. By the space of standard embeddings we mean the subspace of the mapping space consisting of the maps which restrict on each summand to affine maps $x \mapsto \lambda x + c$ with $0 \leq \lambda \leq 1$.

Given $\xi \in \mathbf{P}(k, k)$ we have the evident action $\langle \xi \rangle: \text{sEmb}(\coprod_{k} D^n, D^n) \rightarrow \text{sEmb}(\coprod_{\xi k} D^n, D^n)$, which gives us a functor $d_n: \mathbf{P} \rightarrow \mathbf{Top}_{\text{nice}}$, so $d_n \in \mathcal{F}$.

Fix some $k,l \in \mathbf{P}$; then $d_n^k(l) = \int^{m_1, \dots, m_k} \mathbf{P}(m_1 + \cdots + m_k, l) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k)$, which we can roughly think of as all the different ways we can partition a total of $l$ disks into $k$ blocks, with the $i^{\text{th}}$ block having $m_i$ disks, and then map each block of $m_i$ disks into a single disk, all the while being able to permute the $l$ disks amongst themselves (without necessarily having to respect the partitions).

We then get $\mu: d_n \circ d_n \rightarrow d_n$ by composing the disk embeddings. More precisely, for each $l$ we get a morphism $\mu_l: (d_n(k) \otimes d_n^k)l \simeq d_n(k) \otimes (d_n^k(l)) \rightarrow d_n(l)$ from the following considerations:

First we note that
$\begin{aligned}
d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &= \text{sEmb}(\coprod_k D^n, D^n) \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\
&\simeq \text{sEmb}(D^n, D^n)^k \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\
&\simeq \prod_{1 \leq i \leq k} (\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n)).
\end{aligned}$
Now for each $i$ there is a map $\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n) \rightarrow \text{sEmb}(\coprod_{m_i}D^n, D^n)$ induced from iterated evaluation by adjunction. Then by the above, this gives a morphism
$\begin{aligned}
d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &\rightarrow \prod_{1 \leq i \leq k} \text{sEmb} (\coprod_{m_i} D^n, D^n)\\
&\simeq \text{sEmb}(\coprod_{m_1 + \cdots + m_k} D^n, D^n)\\
&= d_n(m_1 + \cdots + m_k).
\end{aligned}$

A big reason that the little $n$-disks operad is relevant to algebraic topology is that there is a big theorem stating that a space is weakly equivalent to an $n$-fold loop space if and only if it’s an algebra for $d_n$.

One direction is straightforward: consider a space $A$ and its $n$-fold loop space $\Omega^n A$. Given an element of $d_n (k)$ and $k$ choices of “little maps” $(D^n, \partial D^n) \rightarrow (A, \ast)$, we can stitch together these little maps into one large map $(D^n, \partial D^n) \rightarrow (A,\ast)$ according to the instructions specified by the chosen element of $d_n(k)$ (where we map everything in the complement of the $k$ little disks to the basepoint in $A$). Doing this for each $k$, we get an operad morphism $d_n \rightarrow \{\Omega^n A, \Omega^n A\}$.

The other direction is much harder, and Maru gave an absolutely fantastic sketch of the basic story in our group discussions, which I hope she will post in the comments; I refrain from including it in the body of this post, partially for reasons of length and partially because I would just end up repeating verbatim what she said in the discussion.