# 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 $p:E→B$ is that the fibers $E_{b}=p_{−1}(b)$ depend contravariantly on $b$, i.e. given a morphism $f:b→b_{′}$, there exists a lift (a functor) $fˉ :E_{b_{′}}→E_{b}$. The definition of a Grothendieck fibration is exactly what’s needed for all this to work correctly.

Following the $n$Lab, a morphism $ϕ:e_{′}→e$ is said to be **$p$-Cartesian lift $ϕ:e_{′}→e$. Then $p^ (f)(b):=e_{′}∈E_{b_{′}}$. The unique factorizations in the definition of a fibration makes this into a functor $p^ (f)$, and this becomes a (pseudo)functor $B→Cat_{op}$.

### 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 $C_{∙}Δ_{op}→Set$ such that all inner horns can be filled, i.e. every morphism $Λ_{k}→C$ can be extended to $Δ_{n}$ 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 $d_{0},d_{1}:C_{1}→C_{0}$ give the target and the source of a morphism, and the degeneracy $s_{0}:C_{0}→C_{1}$ 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 $σ:Λ_{1}→C$, which extends to $σ~∈C_{2}$; then $d_{1}σ~$ 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, $−∘id_{x}$ 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 $p:E→B$ is said to be a **coCartesian morphism** (AKA opfibration) if $p_{op}:E_{op}→B_{op}$ 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 $c_{1}⊗⋯⊗c_{n}→d$ 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 $C$, one can construct its **category of operators** $C_{⊗}$. This is a category over $Γ$, the category of finite pointed sets with objects $n_{+}={∗,1,…,n}$.

The objects of $C_{⊗}$ are finite sequences $[c_{1},…,c_{n}]$ of objects of $C$. The morphisms $[c_{1},…,c_{n}]→[d_{1},…,d_{m}]$ consist of:

- a morphism $α:n_{+}→m_{+}$ in $Γ$;
- morphisms $⨂_{α(i)=j}c_{i}→d_{j}$ for $1≤j≤m$.

Then $p[c_{1},…,c_{n}]$ is $n_{+}$, and the image of a morphism by $p$ is the ”$α$” part. Consistenly with the previous notation for fibers, let $C_{n}:=p_{−1}(n_{+})$. Note that $Γ$ is the category of operators of the terminal category (equipped with its unique symmetric monoidal structure).

Then $C_{⊗}p Γ$ satisfies the two fundamental properties:

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

If we unroll the definition, this means that for every $[c_{1},…,c_{n}]∈C_{⊗}$ and for every $f:n_{+}→m_{+}$, there is some morphism $fˉ :[c_{1},…,c_{n}]→[d_{1},…,d_{m}]$ that covers $f$ and such that for all morphism $gˉ :[c_{1},…,c_{n}]→[d_{1},…,d_{k}]$ and every factorization of $f$ through $g=p(gˉ )$, there is a factorization of $fˉ $ through $gˉ $ inducing it. Indeed, we can choose $d_{j}$ to be $⨂_{f(i)=j}c_{i}$ and see that everything works. Recall that this means the fibers $C_{n}$ depend covariantly on $n$.

**(M2)**The fiber $C_{n}$ is isomorphic to $C_{×n}$ via the product of the functors induced by $ρ_{i}:n_{+}→1_{+}$ given by $ρ_{i}(i)=1$ and $ρ_{i}(j=i)=∗$.

Indeed, a morphism $[c_{1},…,c_{n}]→[d_{1},…,d_{n}]$ in $C_{⊗}$ which covers the identity of $n_{+}$ is uniquely determined by morphism $c_{i}→d_{i}$; one just has to see that this morphism is given by $p^ (ρ_{i})=ρ_{∗}$.

Conversely, if $Dp Γ$ is a functor that satisfies (M1) and (M2), let $C=D_{1}=p_{−1}(1_{+})$. Then $C$ becomes a symmetric monoidal category.

- The fold map $α:2_{+}→1_{+}$ ($α(1)=α(2)=1$) induces, by (M1) and (M2), a functor (well defined up to equivalence) $⊗:C_{×2}∼ D_{2}α_{∗} C$.
- By (M2), $D_{0}=∗$ is the terminal category, and the unique morphism $0_{+}→1_{+}$ induces $∗→C$, which gives the unit object of $C$.
- Since $α$ is “symmetric” (meaning $α(1)=α(2)$), “unital” (meaning $ρ_{1}α=ρ_{2}α=id_{1_{+}}$) 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 $p:C_{⊗}→NΓ$ such that the maps $ρ_{i}:n_{+}→1_{+}$ induce an equivalence $C_{n}→(C_{1})_{×n}$.

In this definition, $C_{⊗}$ should be thought of as the ∞-category of operators, and the underlying ∞-category is really $C_{1}=p_{−1}(1_{+})$. 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 $P$ be a colored operad. Its category of operators $C_{P}$ is given by:

- objects are sequences of colors $[c_{1},…,c_{n}]$;
- morphisms $[c_{1},…,c_{n}]→[d_{1},…,d_{m}]$ are given by a morphism $f:n_{+}→m_{+}$, and for all $1≤j≤m$, an element in $P((c_{i})_{f(i)=j};d_{j})$.

Like before, there’s a functor $p:C_{P}→Γ$ given by $p[c_{1},…,c_{n}]=n_{+}$. This is a Grothendieck opfibration, which allows us the recover $P$ from $C_{P}$.

Let $P_{n}:=p_{−1}(n_{+})$. In particular, $P_{1}$ is the category of unary operations in $P$ (also called “underlying category” of $P$). The colors of $P$ are given by the objects of $P_{1}$. As before, $∏_{i}ρ_{∗}:P_{n}→(P_{1})_{×n}$ is an isomorphism. The operations of type $P(c_{1},…,c_{n};d)$ are recovered as the morphisms $C_{P}([c_{1},…,c_{n}],d)$ which cover $n_{+}→1_{i}$, $i↦1$.

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

**inert**if the preimage of every $j∈m :=m_{+}∖{∗}$ has exactly one element. Such a morphism induces an injection $m ↪n $. Basically, it’s a morphism that “forgets” a bunch of points; think ”$ρ_{i}$“.**active**if $f_{−1}(∗)={∗}$. For any $n$, there is a unique active morphism $n_{+}→1_{+}$; this is the morphism that allows us to recover the $n$-ary operations of $P$.

**Definition. [HA, 2.1.1.10]** An ∞-operad (really, the category of operators of an ∞-operad) is a functor of quasi-categories $p:C_{⊗}→NΓ$ such that:

- For every inert morphism $f:m_{+}→n_{+}$ and every $C∈C_{m}$, there is a $p$-coCartesian morphism $fˉ :C→C_{′}$ lifting $f$, which induces a functor $f_{!}:C_{m}→C_{n}$. 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 $f:n_{+}→m_{+}$, let $C_{f}(−,−)⊂C_{⊗}(−,−)$ be the connected components lying over $f$. Then

is a homotopy equivalence. This means that an “operation” $[c_{1},…,c_{n}]→[d_{1},…,d_{n}]$ is “the same” as a collection of operations $P((c_{i})_{f(i)=j},d_{j})$. 3. For every collection of objects $c_{1},…,c_{n}∈C_{1}$, there exists an object $C∈C_{n}$ and $p$-Cartesian morphisms $C→c_{i}$ covering $ρ_{i}$. This means that $∏ρ_{!}:C_{n}→(C_{1})_{×n}$ is an equivalence.

From this data, the quategory of unary operations of the ∞-operad is given by $C_{1}$. The colors of the ∞-operad are the object of this quategory. Finally, the operations of type $(c_{1},…,c_{n};d)$ are the morphisms $c_{1}⊕…c_{n}→d$ lying over the unique active morphism $n_{+}→1_{+}$, where $c_{1}⊕⋯⊕c_{n}$ is “the” object of $C_{n}$ corresponding to $(c_{1},…,c_{n})∈(C_{1})_{×n}$ 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, [$(∞,1)$-operad](https://ncatlab.org/nlab/show/(infinity,1\)-operad).