The n-Category Café

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.