Infinity-Operads Demystified

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.

∞-operads

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.

References

• [HA] Lurie, Jacob. Higher Algebra. Version of March 2016.
• [nLab] $n$Lab, $(\infty,1)$-operad.