ScalingStacks

Proof.

The tensor unit of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is the category \(\mathrm{Set}^{\mathrm{fin}}\) of finite sets, and since \(\mathrm{CProj}_k \in \mathrm{CAlg}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}})\), the unit induces a finite coproduct preserving functor \(\mathrm{Set}^{\mathrm{fin}} \rightarrow\mathrm{CProj}_k\) which sends a finite set \(X\) to the coproduct \(\sqcup_X k\) and is therefore dominant by lemma 3.5.7.([005W]). Hence, since the tensor product in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) of dominant functors is again dominant (since its (dominant, fully faithful)-factorization system is compatible with its monoidal structure, as follows from the proof of proposition 6.2.2), it follows that for any \(\mathcal A\in \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), the unit \(\mathcal A\simeq \mathrm{Set}^{\mathrm{fin}} \otimes \mathcal A\rightarrow\mathrm{CProj}_k \otimes \mathcal A\) of the adjunction between \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{add}_k\) is dominant. In particular, it immediately follows that for any \(\mathcal B\in \mathrm{add}_k\), a functor \(\mathcal A\rightarrow\mathcal B\) in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is dominant if and only if its adjunct (drawn horizontally) is:

Hence, a functor \(F\) as in the statement of the lemma is dominant, if and only if the functor \((\mathcal C\times \mathbb{Z})^{\sqcup, \mathrm{idem}} \rightarrow\mathcal D\) is, equivalently if every object of \(\mathcal D\) is a retract of a finite coproduct of objects in the image of \(\mathcal C\times \mathbb{Z}\), i.e. of objects which are \(\mathbb{Z}\)-shifts of objects in the image of \(\mathcal C\). ◻