ScalingStacks

Remark 1.3. In Example 1.2, we were careful not to say that a lax symmetric monoidal functor (or, in particular, an algebra object) was exactly characterized as a morphism

among cocartesian fibrations. This was only because we had not yet introduced a certain bit of terminology. First of all, let us say that a morphism $T_{+}\xrightarrow{\alpha}U_{+}$ in $\mathcal{F}\textup{in}_{*}$ is inert if for all $u\in U$, the preimage $\alpha^{-1}(u)$ has exactly one element. Inasmuch as the basepoints of objects in $\mathcal{F}\textup{in}_{*}$ might be thought of as “trash receptacles”, such an inert morphism should therefore be thought of as parametrizing the operation of “throwing away” some specified subset of the $T$-indexed list of objects. For instance, the unique map ${\langle{2}\rangle}\xrightarrow{\rho}{\langle{1}\rangle}$ in $\mathcal{F}\textup{in}_{*}$ satisfying $\rho^{-1}(1)=\{2\}$ has $p$-cocartesian lifts of the form

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

Then, a morphism in $\mathcal{C}^{\otimes}$ is called $p$-inert (or simply inert) if it is $p$-cocartesian and lies over an inert morphism in $\mathcal{F}\textup{in}_{*}$. Now, we can define a lax symmetric monoidal functor

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

to be a commutative triangle as above which preserves inert morphisms (i.e.​ which takes $p$-inert morphisms to $q$-inert morphisms).