ScalingStacks

Example 1.2. Let us illustrate just a hint of the bookkeeping power which results from this reduction of category level. First of all, the datum of a monoidal category $(\mathcal{C},\otimes)$ can be encoded as a certain functor

namely its bar construction (as a monoid object in the symmetric monoidal category $({\mathcal{C}\textup{at}},\times)$): this is given on objects by $\textup{Bar}(\mathcal{C})_{n}=\mathcal{C}^{\times n}$, while its structure maps encode the monoidal structure on $\mathcal{C}$ and the unit map $\textup{pt}_{\mathcal{C}\textup{at}}\simeq\{\textbf{1}_{\mathcal{C}}\}\hookrightarrow\mathcal{C}$. Similarly, we can encode the datum of a symmetric monoidal category $(\mathcal{C},\otimes)$ as a functor

$$\mathcal{F}\textup{in}_{*}\rightarrow{\mathcal{C}\textup{at}}$$

from the category of finite pointed sets: this takes an object $T_{+}=T\sqcup\{*\}$ to the category $\mathcal{C}^{\times T}$, and it takes a morphism $T_{+}\xrightarrow{\alpha}U_{+}$ to the functor $\mathcal{C}^{\times T}\rightarrow\mathcal{C}^{\times U}$ described by the formula

$$(c_{t})_{t\in T}\mapsto\left(\bigotimes_{t\in\alpha^{-1}(u)}c_{t}\right)_{u\in U}$$

where, by convention, a monoidal product indexed over an empty set is defined to be the unit object $\textbf{1}_{\mathcal{C}}\in\mathcal{C}$. (We note in passing that if $\alpha(t)=*\in U_{+}$ for some $t\in T$, then $c_{t}$ does not appear in any of the resulting monoidal products indexed by the elements $u\in U$: it is simply “thrown away”.) It follows that we can equivalently consider a symmetric monoidal category $(\mathcal{C},\otimes)$ as a cocartesian fibration

over the category of finite pointed sets. For instance, writing ${\langle{n}\rangle}=\{1,\ldots,n\}_{+}$, the unique map ${\langle{2}\rangle}\xrightarrow{\mu}{\langle{1}\rangle}$ in $\mathcal{F}\textup{in}_{*}$ satisfying $\mu^{-1}(*)=\{*\}$ has cocartesian lifts of the form

$$(c_{1},c_{2})\rightarrow(c_{1}\otimes c_{2}),$$

which morphisms therefore encode the monoidal product $\mathcal{C}\times\mathcal{C}\xrightarrow{-\otimes-}\mathcal{C}$.

Now, in this language, a symmetric monoidal functor

$$(\mathcal{C},\otimes)\xrightarrow{F}(\mathcal{D},\boxtimes)$$

is equivalent data to that of a morphism of cocartesian fibrations

over $\mathcal{F}\textup{in}_{*}$: over an object $T_{+}\in\mathcal{F}\textup{in}_{*}$ the induced map on fibers is given by $\mathcal{C}^{\times n}\xrightarrow{F^{\times n}}\mathcal{D}^{\times n}$, and the fact that the functor respects the monoidal products is encoded by the fact that it preserves cocartesian morphisms. For instance, the $p$-cocartesian morphism

$$(c_{1},c_{2})\rightarrow(c_{1}\otimes c_{2})$$

of $\mathcal{C}^{\otimes}$ lying over ${\langle{2}\rangle}\xrightarrow{\mu}{\langle{1}\rangle}$ is taken to a morphism

$$(F(c_{1}),F(c_{2}))\rightarrow F(c_{1}\otimes c_{2})$$

of $\mathcal{D}^{\boxtimes}$ also lying over ${\langle{2}\rangle}\xrightarrow{\mu}{\langle{1}\rangle}$, and the assertion that this is $q$-cocartesian guarantees a unique isomorphism

$$F(c_{1})\boxtimes F(c_{2})\cong F(c_{1}\otimes c_{2})$$

in $\mathcal{D}\cong\mathcal{D}^{\times 1}$.

However, more is true: even if $F$ is now only a lax symmetric monoidal functor, it still defines a morphism among cocartesian fibrations (in constrast to “a morphism of cocartesian fibrations”), but in general it will not preserve the cocartesian morphisms. For instance, the $p$-cocartesian morphism $(c_{1},c_{2})\rightarrow(c_{1}\otimes c_{2})$ in $\mathcal{C}^{\otimes}$ will be sent to an arbitrary morphism $(F(c_{1}),F(c_{2}))\rightarrow F(c_{1}\otimes c_{2})$, which then admits a unique cocartesian/fiber factorization

in $\mathcal{D}^{\boxtimes}$, in which the fiber morphism is the “structure map” witnessing the laxness of $F$ (at the pair of objects $c_{1},c_{2}\in\mathcal{C}$).

As a special case, note that the identity map of $\mathcal{F}\textup{in}_{*}$ is a cocartesian fibration, which corresponds to the canonical (and unique) symmetric monoidal structure on the terminal category $\textup{pt}_{\mathcal{C}\textup{at}}\in{\mathcal{C}\textup{at}}$. Then, a commutative algebra object in the symmetric monoidal category $(\mathcal{C},\otimes)$ is nothing but a lax symmetric monoidal functor

$$\textup{pt}_{\mathcal{C}\textup{at}}\xrightarrow{A}(\mathcal{C},\otimes).$$

Moreover, the functoriality of commutative algebra objects for a lax symmetric monoidal functor

$$(\mathcal{C},\otimes)\xrightarrow{F}(\mathcal{D},\boxtimes)$$

is encoded simply by the composition

of morphisms among cocartesian fibrations. (Of course, we can equivalently write the map $A$ as a section of the cocartesian fibration $\mathcal{C}^{\otimes}\rightarrow\mathcal{F}\textup{in}_{*}$; then, the functoriality of commutative algebras for lax symmetric monoidal functors is encoded by the functoriality of sections.)