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

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

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

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

Now, in this language, a symmetric monoidal functor

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

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

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

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

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

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.)

Example 1.5. Consider the category ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}$ of vector bundles: its objects are the pairs $(M,V)$ of a manifold $M$ and a vector bundle $V\downarrow M$, and its morphisms are commutative squares

(of a morphism of manifolds and a compatible morphism of (total spaces of) vector bundles). Then, the forgetful functor

to the category of manifolds is a cartesian fibration. Given a morphism $M\xrightarrow{\varphi}N$ in ${\mathcal{M}\textup{fld}}$ and a vector bundle $W\downarrow N$, a cartesian lift is provided by the pullback vector bundle, which is defined by a pullback square

of underlying topological spaces. The fiber/cartesian factorization system in this cartesian fibration translates into the assertion that for any vector bundle $V\downarrow M$, the induced diagram

is a pullback square in ${\mathcal{S}\textup{et}}$, where ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}(M)$ denotes the fiber $\textup{U}_{\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}^{-1}(M)$ of the forgetful functor over the object $M\in{\mathcal{M}\textup{fld}}$ (or equivalently, the value at $M$ of the functor

classifying our cartesian fibration).

There is a canonical section

of the forgetful functor, called the tangent bundle: this takes a manifold $M\in{\mathcal{M}\textup{fld}}$ to its tangent bundle $TM\downarrow M$. Note that this is not a cartesian section: it does not take morphisms in ${\mathcal{M}\textup{fld}}$ to cartesian morphisms of the cartesian fibration. (Correspondingly, this does not arise from a natural transformation

between ${\mathcal{C}\textup{at}}$-valued functors on ${\mathcal{M}\textup{fld}}^{op}$.) In other words, given an arbitrary morphism $M\xrightarrow{\varphi}N$ of manifolds, we obtain a canonical map $TM\rightarrow\varphi^{*}(TN)$ in ${\mathcal{V}\textup{ect}\mathcal{B}\textup{dl}}(M)$, but this map is not generally an isomorphism. However, it is an isomorphism (and the morphism $TM\rightarrow\varphi^{*}(TN)$ is a cartesian lift of $\varphi$) whenever the morphism $\varphi$ is an open embedding.

Example 1.6. The forgetful functor

is a cartesian fibration. Given a morphism $U\rightarrow Y$ in ${\mathcal{S}\textup{et}}$ and a topological space $\mathcal{Y}\in{\mathcal{T}\textup{op}}$ equipped with an isomorphism $\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{Y})\cong Y$ in ${\mathcal{S}\textup{et}}$, a $\textup{U}_{\mathcal{T}\textup{op}}$-cartesian lift is provided by endowing the set $U\in{\mathcal{S}\textup{et}}$ with the induced topology: this yields a topological space $\mathcal{U}\in{\mathcal{T}\textup{op}}$ equipped with a map $\mathcal{U}\rightarrow\mathcal{Y}$ in ${\mathcal{T}\textup{op}}$ and an isomorphism $\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{U})\cong U$ in ${\mathcal{S}\textup{et}}$, which has the universal property that for any $\mathcal{Z}\in{\mathcal{T}\textup{op}}$ with underlying set $Z=\textup{U}_{\mathcal{T}\textup{op}}(\mathcal{Z})\in{\mathcal{S}\textup{et}}$, the resulting diagram

is a pullback square in ${\mathcal{S}\textup{et}}$. (If the morphism $U\rightarrow Y$ is actually the inclusion of a subset, this specializes to define the subspace topology on the set $U$.)