The goal is to develop of theory of symmetric monoidal categories.
Symmetric monoidal categories#
Recall the definition of a symmetric monoidal category in $1$-category theory is a category $\mathcal{C}$ together with
- a functor $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$,
- a collection of isomorphisms $$ \alpha_{X,Y,Z} \colon X \otimes (Y \otimes Z) \cong (X \otimes Y) Z $$ that are functorial,
- $1 \in \mathcal{C}$ an object,
- an isomorphism $\nu \colon 1 \otimes 1 \cong 1$,
- an isomorphism $$ \beta_{X,Y} \colon X \otimes Y \cong Y \otimes X $$ that are functorial,
satisfying a bunch of conditions like
- the functors $\mathcal{C} \to \mathcal{C}$ given by $C \mapsto 1 \otimes C$ and $C \mapsto C \otimes 1$ are fully faithful,
- some associativity axiom,
- some unital axiom,
- some commutativity axiom.
This is all very annoying to keep track of. So we want another way of thinking about all these data with these axioms.
Let $(\mathcal{C}, \otimes)$ be a symmetric monoidal category. Then we can define a new category $\mathcal{C}^\otimes$ where objects are given by tuples $$ [C_1, \dotsc, C_n] $$ for $n \ge 0$ and $C_i \in \mathcal{C}$. A morphism from $[C_1, \dotsc, C_m]$ to $[C_1^\prime, \dotsc, C_m^\prime]$ is given by
- $S \subseteq \lbrace 1, \dotsc, n \rbrace$ a subset,
- $\alpha \colon S \to \lbrace 1, \dotsc, m \rbrace$ a map, and
- morphisms $$ f_j \colon \bigotimes_ {i \in \alpha^{-1}(j)} C_i \to C_j. $$
There is also a natural way of composing morphisms. The point is that this is fibered over the following category, and this fibration can actually recover $(\mathcal{C}, \otimes)$.
Definition 1. The category $\mathsf{Fin}_\ast$ has object pointed finite sets $$ \langle n \rangle = \lbrace 1, \dotsc, n \rbrace \amalg \lbrace \ast \rbrace $$ with morphisms being maps that send $\ast$ to $\ast$.
There these morphisms $$ \rho^i \colon \langle n \rangle \to \langle 1 \rangle $$ sending $i \mapsto 1$ and all else to $\ast$. We also write $$ \langle n \rangle^\circ = \lbrace 1, \dotsc, n \rbrace = \langle n \rangle \setminus \lbrace \ast \rbrace. $$
Proposition 2. There is a functor $\mathcal{C}^\otimes \to \mathsf{Fin}_\ast$ defined by $$ \lbrack C_1, \dotsc, C_n \rbrack \mapsto \langle n \rangle $$ and this functor contains all the data to recover $(\mathcal{C}, \otimes)$.
For example, we can recover the underlying category $\mathcal{C}$ by looking at the fiber over $\langle 1 \rangle$. At the end, we will be able to make the following definition.
Definition 3. A symmetric monoidal ∞-category is a coCartesian fibration $p \colon \mathcal{C}^\otimes \to N(\mathsf{Fin}_ \ast)$ satisfying the property that if $\rho^i$ induces functors $$ \rho_ !^i \colon \mathcal{C}_ {\langle n \rangle}^\otimes \to \mathcal{C}_ {\langle 1 \rangle}^\otimes $$ then the functor $$ \prod_ {1 \le i \le n}^{} \rho_!^i \colon \mathcal{C}_ {\langle n \rangle}^\otimes \to (\mathcal{C}_ {\langle 1 \rangle}^\otimes)^n $$ is an equivalence.
Colored operads#
Let’s try to motivate this definition.
Definition 4. A colored operad is the data of
- a collection of objects $\lbrace X, Y, \dotsc \rbrace$,
- for each finite set $I$ and a collection $X_i$ for $i \in I$ and $Y$, a set $\operatorname{Mult}( \lbrace X_i \rbrace_ {i \in I}, Y)$
- an identity map $\id_Y \in \operatorname{Mult}( \lbrace Y \rbrace, Y)$
- for each $I \to J$ a map of finite sets, a composition map $$ \prod_{j \in J}^{} \operatorname{Mult}( \lbrace X_i \rbrace_{i \in I_j}, Y_j) \times \operatorname{Mult}( \lbrace Y_j \rbrace_ {j \in J}, Z) \to \operatorname{Mult}( \lbrace X_i \rbrace_{i \in I}, Z) $$
satisfying some associativity law.
This should be thought of as multi-version of a category. From any colored operad $\mathcal{O}$, we can construct a $1$-category where
- objects are just objects of $\mathcal{O}$,
- $\Hom(X, Y)$ is just $\operatorname{Mult}_ \mathcal{O}(\lbrace X \rbrace, Y)$.
Example 5. Given any $1$-category $\mathcal{C}$, we can define an operad $\mathcal{O}$ where objects are the objects of $\mathcal{C}$ and $$ \operatorname{Mult}(\lbrace X_i \rbrace, Y) = \begin{cases} \emptyset & \lvert I \rvert \neq 1, \br \Hom(X_i, Y) & I = \lbrace i \rbrace. \end{cases} $$
Example 6. Given a symmetric monoidal $1$-category $\mathcal{C}$, we can define an operad where objects are objects of $\mathcal{C}$ and $$ \operatorname{Mult}(\lbrace X_i \rbrace, Y) = \Hom(\bigotimes_{i \in I} X_i, Y). $$
Given an operad $\mathcal{O}$, we can define a category $\mathcal{O}^\otimes$ where objects are $\lbrace X_i \rbrace_{i \in I}$ for $I$ a finite set, and morphisms are given by a map $I \to J$ together with an element of $$ \prod_{j \in J}^{} \operatorname{Mult}( \lbrace X_i \rbrace_{i \in I_j}, Y_j). $$ Similarly, composition can be defined naturally.
Again, the point is that there is a natural functor $$ \mathcal{O}^\otimes \to \mathsf{Fin}_ \ast, $$ and $\mathcal{O}$ can be completely recovered from it. We will have $\mathcal{O} = \mathcal{O}_ {\langle 1 \rangle}$, and then multiplication can be recovered.
∞-operads#
Definition 7. A map $f \colon \langle m \rangle \to \langle n \rangle$ in $\mathsf{Fin}_\ast$ is inert if every $i \in \langle n \rangle^\circ$ has exactly one preimage.
Definition 8. An ∞-operad is a functor $p \colon \mathcal{O}^\otimes \to N(\mathsf{Fin}_\ast)$ of ∞-categories satisfying the following conditions.
- Write $\mathcal{O}_ {\langle n \rangle}^\otimes$ for the fiber over $\langle m \rangle$. We require that for every inert $f \colon \langle m \rangle \to \langle n \rangle$ and every object $C \in \mathcal{O}_{\langle m \rangle}^\otimes$, there exists a $p$-coCartesian morphism $\bar{f}$ in $\mathcal{O}^\otimes$ lifting $f$,
- For $C \in \mathcal{O}_ {\langle n \rangle}^\otimes$, $C^\prime \in \mathcal{O}_ {\langle m \rangle}^\otimes$, and $f \colon \langle m \rangle \to \langle n \rangle$, denote by $\Hom^f(C, C^\prime)$ the space of maps lying over $f$. Then we require $$ \Hom^f(C, C^\prime) \simeq \prod_ {1 \le i \le n}^{} \Hom^{\rho_i \circ f}(C, C_i^\prime) $$ to be a homotopy equivalence.
- For every $n \ge 0$, the functors $\rho_ !^i \colon \mathcal{O}_ {\langle n \rangle}^\otimes \to \mathcal{O}_ {\langle 1 \rangle}^\otimes$ induce an equivalence of ∞-categories $$ \mathcal{O}_ {\langle n \rangle}^\otimes \simeq (\mathcal{O}_ {\langle 1 \rangle}^\otimes)^n. $$
We write $\mathcal{O} = \mathcal{O}_ {\langle 1 \rangle}^\otimes$ for the underlying ∞-category of $\mathcal{O}^\otimes$.
Here a map $X \to Y$ being $p$-coCartesian means that $$ \begin{CD} \Hom(Y, Z) @>>> \Hom(X, Z) \br @VVV @VVV \br \Hom(\bar{Y}, \bar{Z}) @>>> \Hom(\bar{X}, \bar{Z}) \end{CD} $$ is a Cartesian diagram for every $Z \in \mathcal{O}^\otimes$.
Remark 9. If $\mathcal{O}$ is a colored operad as before, then $N(\mathcal{O}^\otimes) \to N(\mathsf{Fin}_\ast)$ is an ∞-operad.
Example 10. There is the trivial operad $\mathrm{Triv}$ where the objects are $\langle n \rangle$, and morphisms are inert maps in $\mathsf{Fin}_\ast$.
Example 11. The operad $\mathbb{E}_0$ has objects $\langle n \rangle$ and morphisms $f \colon \langle m \rangle \to \langle n \rangle$ such that $f^{-1}(i)$ all have at most one element.
Example 12. The commutative operad $\mathbb{E}_ \infty = \mathrm{Comm}$ is just the identity functor $N(\mathsf{Fin}_ \ast) \to N(\mathsf{Fin}_ \ast)$.
Definition 13. A morphism between two operads $\mathcal{O}^\otimes$ and $\mathcal{O}^{\otimes\prime}$ is a diagram $$ \begin{CD} \mathcal{O}^\otimes @>>> \mathcal{O}^{\otimes\prime} \br @VVV @VVV \br N(\mathsf{Fin}_ \ast) @= N(\mathsf{Fin}_ \ast) \end{CD} $$ satisfying some extra conditions.