ScalingStacks

Proof. For $d=-1,0$, both assertions are trivial to check and so we assume that $d\geq 1$. The argument that $h_{d}\left(p\right)$ is an inner fibration is similar to the argument that $h_{d}\left(f\right)$ is coCartesian and so we shall prove them together. Using T.2.4.1.4, we need to consider the lifting problem

for some $m\geq 2$ and either

1. (1)

   $0<i<m$ or

2. (2)

   $i=0$ and $\Delta^{\left\{0,1\right\}}\subseteq\Lambda_{0}^{m}$ is mapped in $h_{d}\mathcal{C}$ to $h_{d}\left(f\right)$.

For $m\geq d+3$, we have $\mbox{sk}^{j}\Lambda_{i}^{m}=\mbox{sk}^{j}\Delta^{m}$ for all $j\leq d+1$, and so the map

is a bijection and there is nothing to prove. For $m\leq d+2$, we have $\Lambda_{i}^{m}=\mbox{sk}^{d+1}\Lambda_{i}^{m}$, and so the map

is surjective, hence the map $\Lambda_{i}^{m}\to h_{d}\mathcal{C}$ factors through $\Lambda_{i}^{m}\to\mathcal{C}$. Now, the functor $\mathcal{C}\to h_{d}\mathcal{C}$ identifies only homotopic morphisms (for $d\geq 1$); hence in (2) the image of $\Delta^{\left\{0,1\right\}}$ in $\mathcal{C}$ is coCartesian. Thus, in both cases we can solve the corresponding lifting problem in $\mathcal{C}$, which induces a lift in the original square. ∎

Proof. For $d=-1$, there is nothing to prove in (1)–(3) and so we assume that $d\geq 0$.

(1) For $d=0$, it is clear that $\left(h_{0}\mathcal{\mathcal{O}}\right)^{\otimes}$ is a skeletal $1$-category, with $p_{0}$ fully faithful; and for $d\geq 1$, it is clear that $\left(h_{d}\mathcal{\mathcal{O}}\right)^{\otimes}$ is a $d$-category. Hence, we only need to show that $\left(h_{d}\mathcal{\mathcal{O}}\right)^{\otimes}$ is an $\infty$-operad. For this we need to check the three conditions of Definition A.2.1.1.10.

• •

  Since $p\colon\mathcal{O}^{\otimes}\to\mathbf{Fin}_{*}$ is an $\infty$-operad, for every inert morphism $f\colon\left\langle m\right\rangle\to\left\langle n\right\rangle$ and an object $\overline{X}\in h_{d}\mathcal{O}_{\left\langle m\right\rangle}^{\otimes}$, we can lift $\overline{X}$ to $X\in\mathcal{O}_{\left\langle m\right\rangle}^{\otimes}$ and find a coCartesian lift $g\colon X\to Y$ of $f$ in $\mathcal{O}^{\otimes}$. For $d\geq 1$, the image $\overline{g}$ of $g$ in $\left(h_{d}\mathcal{O}\right)^{\otimes}$ is a coCartesian lift of $f$ by 3.3. For $d=0$, we use the dual of T.2.4.4.3 to show that $\overline{g}$ is coCartesian. $\left(h_{0}\mathcal{O}\right)^{\otimes}\to\mathbf{Fin}_{*}$ is an inner fibration (as the nerve of a functor of ordinary categories) and for every $\overline{Z}\in\left(h_{0}\mathcal{O}\right)_{\left\langle m\right\rangle}^{\otimes}$, pre-composition with $\overline{g}$ induces a diagram

  $$\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 42.02347pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\halign{\entry@#!@&&\entry@@#!@\cr&\cr&\crcr}}}\ignorespaces{\hbox{\kern-36.83408pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\left(h_{0}\mathcal{O}\right)^{\otimes}}\left(\overline{Y},\overline{Z}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\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}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 69.82396pt\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}}{\hbox{\kern 69.82396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\left(h_{0}\mathcal{O}\right)^{\otimes}}\left(\overline{X},\overline{Z}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 106.65804pt\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-42.02347pt\raise-32.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Fin}_{*}}\left(\left\langle m\right\rangle,\left\langle k\right\rangle\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 66.02347pt\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 66.02347pt\raise-32.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise-2.5pt\hbox{$\textstyle{\operatorname{Map}_{\mathbf{Fin}_{*}}\left(\left\langle n\right\rangle,\left\langle k\right\rangle\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$$

  and it is easy to verify that it is a homotopy pullback.

• •

  Let $\overline{X}\in\left(h_{d}\mathcal{O}\right)_{\left\langle m\right\rangle}^{\otimes}$ and $\overline{Y}\in\left(h_{d}\mathcal{O}\right)_{\left\langle n\right\rangle}^{\otimes}$ and let $f\colon\left\langle m\right\rangle\to\left\langle n\right\rangle$ be a morphism in $\mathbf{Fin}_{*}$. We first observe that

  $$\operatorname{Map}_{\left(h_{d}\mathcal{O}\right)^{\otimes}}^{f}\left(X,Y\right)\simeq h_{d-1}\left(\operatorname{Map}_{\mathcal{O}^{\otimes}}^{f}\left(X,Y\right)\right).$$

  For $d\geq 1$ this follows from 2.13 and for $d=0$ it follows directly from the definition. Hence,

  	$\displaystyle\operatorname{Map}_{\left(h_{d}\mathcal{O}\right)^{\otimes}}^{f}\left(X,Y\right)$	$\displaystyle\simeq$	$\displaystyle h_{d-1}\left(\operatorname{Map}_{\mathcal{O}^{\otimes}}^{f}\left(X,Y\right)\right)\simeq h_{d-1}\left(\prod_{1\leq i\leq n}\operatorname{Map}_{\mathcal{O}^{\otimes}}^{\rho^{i}\circ f}\left(X,Y_{i}\right)\right)$
  		$\displaystyle\simeq$	$\displaystyle\prod_{1\leq i\leq n}h_{d-1}\left(\operatorname{Map}_{\mathcal{O}^{\otimes}}^{\rho^{i}\circ f}\left(X,Y_{i}\right)\right)\simeq\prod_{1\leq i\leq n}\operatorname{Map}_{\left(h_{d}\mathcal{O}\right)^{\otimes}}^{\rho^{i}\circ f}\left(X,Y_{i}\right).$

  Note that we use the fact that $h_{d}$ preserves finite products of spaces.

• •

  For every finite collection of objects $\overline{X}_{1},\dots,\overline{X}_{n}\in\left(h_{d}\mathcal{O}\right)_{\left\langle 1\right\rangle}^{\otimes}$ that are lifted to objects of $\mathcal{O}_{\left\langle 1\right\rangle}^{\otimes}$, there is an object $X\in\mathcal{O}_{\left\langle n\right\rangle}^{\otimes}$ and coCartesian morphisms $f_{i}\colon X\to X_{i}$ covering $\rho^{i}\colon\left\langle n\right\rangle\to\left\langle 1\right\rangle$. The images of those maps in $h_{d}\mathcal{O}^{\otimes}$ are coCartesian as well and satisfy the analogous property.

(2) From the proof of (1), $\theta_{d}$ maps inert morphisms in $\mathcal{O}^{\otimes}$ to inert morphisms in $h_{d}\mathcal{O}^{\otimes}$.

Proof. (2) Assume that $d\geq 1$. By the analogous fact for $\infty$-categories, the composition with $\theta_{d}$ induces an isomorphism

The simplicial set $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)$ is the full subcategory of $\operatorname{Fun}_{\mathbf{Fin}_{*}}\left(\mathcal{O}^{\otimes},\mathcal{U}^{\otimes}\right)$ spanned by maps of $\infty$-operads (and similarly for $h_{d}\mathcal{O}$ instead of $\mathcal{O}$). The claim now follows from the fact that the image of a coCartesian edge in $\mathcal{O}^{\otimes}$ is coCartesian in $\left(h_{d}\mathcal{O}\right)^{\otimes}$ and, conversely, every inert morphism in $\left(h_{d}\mathcal{O}\right)^{\otimes}$ is up to equivalence the image of an inert morphism in $\mathcal{O}^{\otimes}$ (lift the source to some object $X\in\mathcal{O}^{\otimes}$ and choose any inert map with domain $X$).

For $d=0$, essentially the same argument works, only now the inert maps of $\left(h_{0}\mathcal{O}\right)^{\otimes}$ are precisely those whose image in $\mathbf{Fin}_{*}$ is inert and therefore the inert maps of $\left(h_{0}\mathcal{O}\right)^{\otimes}$ are again precisely the images of inert maps in $\mathcal{O}^{\otimes}$. For $d=-1$, the claim is obvious.