ScalingStacks

Let \(F \colon \mathcal P\rightarrow\mathcal Q\) be an operad map such that for all \(n\geq 0\) and colors \(X_1,\ldots, X_n, Y\) in \(\mathcal O\), the map of spaces \[\mathrm{Mul}_{\mathcal P}(b_{X_1}, \ldots, b_{X_{n}}; c_{Y}) \xrightarrow{F(-)} \mathrm{Mul}_{\mathcal Q}(Fb_{X_1}, \ldots, Fb_{X_{n}}; Fc_{Y})\] is an equivalence. Then, for any \(m \geq 0\), the map of multi-hom spaces \[\mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{b,\ldots, b}_{m}; c) \xrightarrow{F(-)} \mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal Q)}(\underbrace{Fb,\ldots, Fb}_{m}; Fc)\] is an equivalence.