ScalingStacks

Proof. Set $d=d_{1}+d_{2}+2$. By 3.2.6, it is enough to show that for every $\left(d+1\right)$-topos $\mathcal{C}$ with the Cartesian symmetric monoidal structure, the map

$$\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{Q}\otimes\mathcal{R},\mathcal{C}\right)\to\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{P}\otimes\mathcal{R},\mathcal{C}\right),$$

induced by pre-composition with $f$, is a homotopy equivalence. Using the tensor-hom adjunction, it is the same as showing that the map

$$\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{Q},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)\to\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{P},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)$$

is an equivalence. The underlying category functor gives a commutative diagram:

$$\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.12514pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\halign{\entry@#!@&&\entry@@#!@\cr&\cr&\crcr}}}\ignorespaces{\hbox{\kern-50.12514pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{Q},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 74.12514pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-24.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 74.12514pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{P},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 123.76416pt\raise-24.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-44.92546pt\raise-32.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Cat}_{\infty}}(\underline{\mathcal{Q}},\underline{\operatorname{Alg}}_{\mathcal{R}}\left(\mathcal{C}\right))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 77.44981pt\raise-32.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 77.44981pt\raise-32.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Cat}_{\infty}}(\underline{\mathcal{P}},\underline{\operatorname{Alg}}_{\mathcal{R}}\left(\mathcal{C}\right)).}$}}}}}}}\ignorespaces}}}}\ignorespaces\ \left(*\right)$$

As $\underline{P}\to\underline{\mathcal{Q}}$ is an equivalence of $\infty$-categories (both are equivalent to $\Delta^{0}$), the bottom map is a homotopy equivalence. Hence, it suffices to show that the induced map on the homotopy fibers is a homotopy equivalence for each choice of a base point. A point in the space $\operatorname{Map}\left(\Delta^{0},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)$ is just an $\mathcal{R}$-algebra $X$ in $\mathcal{C}$. We denote by $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)_{X}$ the $\infty$-operad $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$ pointed by $X$ viewed as an object of $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$. With this notation, we see that the homotopy fiber of the right vertical map is equivalent to

$$\operatorname{Map}_{\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}}\left(\mathcal{P},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)_{X}\right).$$

By 2.2.5, the $\infty$-operad $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$ is unital. Therefore, by the adjunction

$$\iota\colon\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\leftrightarrows\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\colon\left(-\right)^{\operatorname{\scriptsize{red}}},$$

the above mapping space is also equivalent to

$$\operatorname{Map}_{\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}}\left(\mathcal{P},\operatorname{End}_{\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)}^{\operatorname{\scriptsize{red}}}\left(X\right)\right)\simeq\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{P},\operatorname{End}_{\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)}^{\operatorname{\scriptsize{red}}}\left(X\right)\right),$$

since $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\subseteq\mathbf{Op}_{\infty}$ is a full subcategory. The induced map on the fibers of the vertical maps in $\left(*\right)$ over $X$, is therefore equivalent to

$$\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{Q},\operatorname{End}_{\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)}^{\operatorname{\scriptsize{red}}}\left(X\right)\right)\to\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{P},\operatorname{End}_{\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)}^{\operatorname{\scriptsize{red}}}\left(X\right)\right).$$

Finally, since $\mathcal{\mathcal{C}}$ is a $\left(d+1\right)$-topos and $\mathcal{R}$ is $d_{2}$-connected, 5.1.4 implies that the $\infty$-operad $\operatorname{End}_{\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)}^{\operatorname{\scriptsize{red}}}\left(X\right)$ is an essentially $\left(d-d_{2}-1=d_{1}+1\right)$-operad. Since $\mathcal{P}\to\mathcal{Q}$ is a $d_{1}$-equivalence, by 3.1.8 the above map is a homotopy equivalence and this completes the proof. ∎