Published .  algtop  higher-cat  math  operads

The purpose of this post is to record the definition of ∞-operads and explain why it works like that. For this I’m using Lurie’s definition of ∞-operads; there is also a definition by Cisinski–Moerdijk–Weiss using dendroidal sets, about which I might talk later.

Indeed, the definition on an ∞-operad is a bit mysterious taken “as-is” – see [HA, §2.1.1.10]. My goal is to explain how to reach this definition, mostly for my own sake. Most of what follows is taken either from the book Higher Algebra, the $n$Lab, or the semester-long workshop about hgiher category theory in Lille in 2015.

## Cartesian morphisms#

### Grothendieck fibrations#

The first thing to explain would be Cartesian (and coCartesian) morphisms. They are generalizations of Grothendieck fibrations from ordinary category theory. The basic idea of a Grothendieck fibration $p : E \to B$ is that the fibers $E_b = p^{-1}(b)$ depend contravariantly on $b$, i.e. given a morphism $f : b \to b'$, there exists a lift (a functor) $\bar{f} : E_{b'} \to E_b$. The definition of a Grothendieck fibration is exactly what’s needed for all this to work correctly.

Following the $n$Lab, a morphism $\phi : e' \to e$ is said to be **$p$-Cartesian lift $\phi : e' \to e$. Then $\hat{p}(f)(b) := e' \in E_{b'}$. The unique factorizations in the definition of a fibration makes this into a functor $\hat{p}(f)$, and this becomes a (pseudo)functor $B \to \mathsf{Cat}^{\rm op}$.

### Quasi-categories#

The study of ∞-operads obviously involves higher category theory, and in Lurie’s setting higher categories are quasi-categories (I think some people call them quategories). There are plenty of explanations of quasi-categories online, so I will just record the bare definitions here.

A quasi-category is a simplicial set $C_\bullet \Delta^{\rm op} \to \mathsf{Set}$ such that all inner horns can be filled, i.e. every morphism $\Lambda_k^n \to C$ can be extended to $\Delta^n$ for $1 < k < n$ (compare with Kan complexes, where outer horns can be filled too).

The set $C_0$ is to be understood as the set of objects of the quasi-categories; the set $C_1$ is to be understood as the set of morphisms. Faces $d_0, d_1 : C_1 \to C_0$ give the target and the source of a morphism, and the degeneracy $s_0 : C_0 \to C_1$ gives the identity of an object. The set $C_2$ gives information about composition: a composable pair of arrows is the same thing as a morphism $\sigma : \Lambda_1^2 \to C$, which extends to $\tilde\sigma \in C_2$; then $d_1\tilde\sigma$ is “a composite” of the pair of composable morphisms. Higher $C_n$ give coherence conditions for composition, and everything works out (the space of possible compositions for a pair of composable morphisms is contractible, $- \circ \operatorname{id}_x$ is homotopic to the identity and so on).

The nerve of a plain category is an example of a quasi-category.

### (co)Cartesian morphisms#

The definition of a Cartesian morphism essentially mimics the definition of a Grothendieck fibration, except that the categories are replaced with quasi-categories. I won’t dwell too much on this (see the $n$Lab article), since as far as I can tell there’s no hidden surprise here, the most difficult thing being to determine how to translate the axioms in terms of quasi-categories. Let’s just note that a morphism $p : E \to B$ is said to be a coCartesian morphism (AKA opfibration) if $p^{\rm op} : E^{\rm op} \to B^{\rm op}$ is Cartesian. This means that the fibers $E_b$ depends covariantly instead of contravariantly on $b$. I am not 100% sure why a different name is used for plain categories and for quasi-categories; maybe it’s just a historical artefact?

## Symmetric monoidal ∞-categories#

Rather than plain operads, ∞-operads are a generalization of colored operads, AKA multicategories. As such, they have multiple objects, and behave more like a ∞-category where morphisms can have multiple inputs instead of just one. Let’s first define symmetric monoidal ∞-categories, where a morphism $c_1 \otimes \dots \otimes c_n \to d$ can be thought of as a morphism with multiple inputs. (This section is taken straight from the introduction of Chapter 2 of Higher Algebra and adapted to my notations).

Given a symmetric monoidal category $\mathsf{C}$, one can construct its category of operators $\mathsf{C}^\otimes$. This is a category over $\Gamma$, the category of finite pointed sets with objects $n_+ = \{ *, 1, \dots, n \}$.

The objects of $\mathsf{C}^\otimes$ are finite sequences $[c_1, \dots, c_n]$ of objects of $\mathsf{C}$. The morphisms $[c_1, \dots, c_n] \to [d_1, \dots, d_m]$ consist of:

• a morphism $\alpha : n_+ \to m_+$ in $\Gamma$;
• morphisms $\bigotimes_{\alpha(i) = j} c_i \to d_j$ for $1 \le j \le m$.

Then $p[c_1, \dots, c_n]$ is $n_+$, and the image of a morphism by $p$ is the ”$\alpha$” part. Consistenly with the previous notation for fibers, let $\mathsf{C}^\otimes_n := p^{-1}(n_+)$. Note that $\Gamma$ is the category of operators of the terminal category (equipped with its unique symmetric monoidal structure).

Then $\mathsf{C}^\otimes \xrightarrow{p} \Gamma$ satisfies the two fundamental properties:

• (M1) $p$ is a coCartesian morphism (opfibration).

If we unroll the definition, this means that for every $[c_1, \dots, c_n] \in \mathsf{C}^\otimes$ and for every $f : n_+ \to m_+$, there is some morphism $\bar{f} : [c_1, \dots, c_n] \to [d_1, \dots, d_m]$ that covers $f$ and such that for all morphism $\bar{g} : [c_1, \dots, c_n] \to [d_1, \dots, d_k]$ and every factorization of $f$ through $g = p(\bar{g})$, there is a factorization of $\bar{f}$ through $\bar{g}$ inducing it. Indeed, we can choose $d_j$ to be $\bigotimes_{f(i) = j} c_i$ and see that everything works. Recall that this means the fibers $\mathsf{C}^\otimes_n$ depend covariantly on $n$.

• (M2) The fiber $\mathsf{C}^\otimes_n$ is isomorphic to $\mathsf{C}^{\times n}$ via the product of the functors induced by $\rho^i : n_+ \to 1_+$ given by $\rho^i(i) = 1$ and $\rho^i(j \neq i) = *$.

Indeed, a morphism $[c_1, \dots, c_n] \to [d_1, \dots, d_n]$ in $\mathsf{C}^\otimes$ which covers the identity of $n_+$ is uniquely determined by morphism $c_i \to d_i$; one just has to see that this morphism is given by $\hat{p}(\rho^i) = \rho^i_*$.

Conversely, if $\mathsf{D} \xrightarrow{p} \Gamma$ is a functor that satisfies (M1) and (M2), let $\mathsf{C} = \mathsf{D}_1 = p^{-1}(1_+)$. Then $\mathsf{C}$ becomes a symmetric monoidal category.

• The fold map $\alpha : 2_+ \to 1_+$ ($\alpha(1) = \alpha(2) = 1$) induces, by (M1) and (M2), a functor (well defined up to equivalence) $\otimes : \mathsf{C}^{\times 2} \xleftarrow{\sim} \mathsf{D}_2 \xrightarrow{\alpha_*} \mathsf{C}$.
• By (M2), $\mathsf{D}_0 = *$ is the terminal category, and the unique morphism $0_+ \to 1_+$ induces $* \to \mathsf{C}$, which gives the unit object of $\mathsf{C}$.
• Since $\alpha$ is “symmetric” (meaning $\alpha(1) = \alpha(2)$), “unital” (meaning $\rho^1 \alpha = \rho^2 \alpha = \operatorname{id}_{1_+}$) and “associative”, then so are the corresponding functors, always up to isomorphism.

So axioms (M1) and (M2) capture exactly what it means to be a symmetric monoidal category. There are many possible equivalent definitions of monoidal categories, but Lurie’s insight was to find one that could be adapted to quasi-categories. Indeed, he defines:

Definition. [HA, 2.0.0.7] A symmetric monoidal ∞-category is a coCartesian fibration $p : \mathsf{C}^\otimes \to N\Gamma$ such that the maps $\rho^i : n_+ \to 1_+$ induce an equivalence $\mathsf{C}^\otimes_n \to (\mathsf{C}^\otimes_1)^{\times n}$.

In this definition, $\mathsf{C}^\otimes$ should be thought of as the ∞-category of operators, and the underlying ∞-category is really $\mathsf{C}^\otimes_1 = p^{-1}(1_+)$. The monoidal product and the unit are induced as above.

Following the same pattern, Lurie defines the category of operators of a colored operad. This satisfies a bunch of axioms, which allow one to recover the colored operad from the category of operators. The axioms are also laid out in such a way that generalizing them to ∞-categories is possible, thus giving the definition of ∞-operads.

Let $\mathtt{P}$ be a colored operad. Its category of operators $\mathsf{C}_\mathtt{P}$ is given by:

• objects are sequences of colors $[c_1, \dots, c_n]$;
• morphisms $[c_1, \dots, c_n] \to [d_1, \dots, d_m]$ are given by a morphism $f : n_+ \to m_+$, and for all $1 \le j \le m$, an element in $\mathtt{P}((c_i)_{f(i) = j}; d_j)$.

Like before, there’s a functor $p : \mathsf{C}_\mathtt{P} \to \Gamma$ given by $p[c_1, \dots, c_n] = n_+$. This is a Grothendieck opfibration, which allows us the recover $\mathtt{P}$ from $\mathsf{C}_\mathtt{P}$.

Let $\mathtt{P}_n := p^{-1}(n_+)$. In particular, $\mathtt{P}_1$ is the category of unary operations in $\mathtt{P}$ (also called “underlying category” of $\mathtt{P}$). The colors of $\mathtt{P}$ are given by the objects of $\mathtt{P}_1$. As before, $\prod_i \rho^i_* : \mathtt{P}_n \to (\mathtt{P}_1)^{\times n}$ is an isomorphism. The operations of type $\mathtt{P}(c_1, \dots, c_n; d)$ are recovered as the morphisms $\mathsf{C}_\mathtt{P}([c_1, \dots, c_n], d)$ which cover $n_+ \to 1_i$, $i \mapsto 1$.

Two types of morphisms appear in the previous discussion, which lead to the following definition: a morphism $f : n_+ \to m_+$ in $\Gamma$ is

• inert if the preimage of every $j \in \underline{m} := m_+ \setminus \{*\}$ has exactly one element. Such a morphism induces an injection $\underline{m} \hookrightarrow \underline{n}$. Basically, it’s a morphism that “forgets” a bunch of points; think ”$\rho^i$“.
• active if $f^{-1}(*) = \{ * \}$. For any $n$, there is a unique active morphism $n_+ \to 1_+$; this is the morphism that allows us to recover the $n$-ary operations of $\mathtt{P}$.

Definition. [HA, 2.1.1.10] An ∞-operad (really, the category of operators of an ∞-operad) is a functor of quasi-categories $p : \mathsf{C}^\otimes \to N\Gamma$ such that:

1. For every inert morphism $f : m_+ \to n_+$ and every $C \in \mathsf{C}^\otimes_m$, there is a $p$-coCartesian morphism $\bar{f} : C \to C'$ lifting $f$, which induces a functor $f_! : \mathsf{C}^\otimes_m \to \mathsf{C}^\otimes_n$. Recall that the inert morphisms are those who “forget” points. The functor $f_!$ is the functor which, given an $m$-uple, forgets some of the factors. The object $C$ is a sequence of colors, and $C'$ is the same sequence with some colors forgotten.
2. For $f : n_+ \to m_+$, let $\mathsf{C}^\otimes_f(-,-) \subset \mathsf{C}^\otimes(-,-)$ be the connected components lying over $f$. Then
$\mathsf{C}^\otimes_f(C, C') \to \prod_{1 \le k \le m} \mathsf{C}^\otimes_{\rho^i \circ f}(C, C'_i)$

is a homotopy equivalence. This means that an “operation” $[c_1, \dots, c_n] \to [d_1, \dots, d_n]$ is “the same” as a collection of operations $\mathtt{P}((c_i)_{f(i) = j}, d_j)$. 3. For every collection of objects $c_1, \dots, c_n \in \mathsf{C}^\otimes_1$, there exists an object $C \in \mathsf{C}^\otimes_n$ and $p$-Cartesian morphisms $C \to c_i$ covering $\rho^i$. This means that $\prod \rho^i_! : \mathsf{C}^\otimes_n \to (\mathsf{C}^\otimes_1)^{\times n}$ is an equivalence.

From this data, the quategory of unary operations of the ∞-operad is given by $\mathsf{C}^\otimes_1$. The colors of the ∞-operad are the object of this quategory. Finally, the operations of type $(c_1, \dots, c_n; d)$ are the morphisms $c_1 \oplus \dots c_n \to d$ lying over the unique active morphism $n_+ \to 1_+$, where $c_1 \oplus \dots \oplus c_n$ is “the” object of $\mathsf{C}^\otimes_n$ corresponding to $(c_1, \dots, c_n) \in (\mathsf{C}^\otimes_1)^{\times n}$ under the equivalence of (3).

And voilà! An ∞-operad. I’m not as scared of the definition as I was when I first saw it, and I hope you aren’t anymore either.