ScalingStacks

Further, assume there is a functor \(\iota \colon \mathcal B\rightarrow\mathcal C\) in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty,2)})\) which is surjective on objects and such that for every two objects \(b,b' \in \mathcal B\), any object in \(\underline{\mathrm{Hom}}_{\mathcal C}(\iota b, \iota b') \in \mathrm{add}_{k}^{B\mathbb{Z}}\) is a retract of a finite coproduct of \(\mathbb{Z}\)-shifts of objects in the image of \(\underline{\mathrm{Hom}}_{\mathcal B}(b, b') \in \mathrm{Cat}_{(\infty,1)}\). Then, the pre-composition map \[\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\] is an equivalence of spaces.