# Infinity-Operads Demystified

#math #algtop #operads #higher-cat

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

Indeed, the definition on an
\(\infty\)-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 \(\infty\)-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 \(\infty\)-categories

Rather than plain operads,
\(\infty\)-operads are a generalization of *colored* operads, AKA multicategories. As such, they have multiple objects, and behave more like a
\(\infty\)-category where morphisms can have multiple inputs instead of just one. Let’s first define symmetric monoidal
\(\infty\)-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
\(\infty\)-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 \(\infty\)-category of operators, and the underlying \(\infty\)-category is really \(\mathsf{C}^\otimes_1 = p^{-1}(1_+)\). The monoidal product and the unit are induced as above.

## \(\infty\)-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 \(\infty\)-categories is possible, thus giving the definition of \(\infty\)-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
\(\infty\)-operad (really, the category of operators of an
\(\infty\)-operad) is a functor of quasi-categories
\(p : \mathsf{C}^\otimes \to N\Gamma\) such that:

- 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. - 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 \(\mathtt{P}((c_i)_{f(i) = j}, d_j)\).
- 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 \(\infty\)-operad is given by \(\mathsf{C}^\otimes_1\). The colors of the \(\infty\)-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 \(\infty\)-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](https://ncatlab.org/nlab/show/(infinity,1)-operad).