Proof. For $d\geq 1$ this is the content of T.2.3.4.12. For $d=-1$ this is trivial. For $d=0$, (1) and (2) are obvious from the definition. For (3) observe that we have a factorization of the map in question:

where the second map is an isomorphism (from the claim for $d=1$). Therefore, we can assume that $\mathcal{C}$ is an ordinary category and $\mathcal{D}$ is a poset and hence both simplicial sets are discrete. The result now follows from the observation that every functor $\mathcal{C}\to\mathcal{D}$ factors uniquely through $h_{0}\mathcal{C}$. ∎

Proof. By T.2.3.4.18, every essentially $d$-category is equivalent to a $d$-category and for every $d$-category $\mathcal{D}$, the map

is an isomorphism by 2.11. Restricting to the maximal Kan sub-complexes, the map of simplicial sets

Proof. It is clear that $\theta_{d}$ is essentially surjective since it is surjective on objects. Let $X,Y\in\mathcal{C}$ be two objects. Since the map

is represented by the map

it will be enough to show that for every Kan complex $X$, the map $X\to h_{d-1}X$ is a $\left(d-1\right)$-truncation map. We prove this by induction. For $d\leq 0$ it is clear. For $d\geq 1$, recall that $\hom_{X}^{R}\left(p,q\right)$ has the homotopy type of the path space $P_{p,q}X$ between $p$ and $q$ in $X$ when viewed as a space. Thus, by induction, $\theta$ is a map of spaces that is surjective on $\pi_{0}$ and induces the $\left(d-2\right)$-truncation map on path spaces

It follows that $\theta$ is a $\left(d-1\right)$-truncation map. ∎

Proof. Consider the following diagram (with the obvious maps) and compute the colimit, starting once with the rows and once with the columns:

∎

Proof. We start with the equivalence $\left(1\right)\iff\left(2\right)$. The first part follows from the fact that $\Phi$ is a monomorphism and the second part follows from the fact that $\Phi$ is a homotopy equivalence of Kan complexes. In the equivalence $\left(3\right)\iff\left(4\right)$, the first part follows from the fact that $\Sigma A\to J\left(A\right)$ is an epimorphism and the second part can be seen as follows: the maps $\overline{f},\overline{g}\colon J\left(B\right)\to\mathcal{C}$ are homotopic rel $J\left(A\right)$ if and only if they are equivalent as elements of the $\infty$-category that is the fiber over $\overline{f}|_{J\left(A\right)}=\overline{g}|_{J\left(A\right)}$ (which is also a homotopy fiber) of the categorical fibration $\mathcal{C}^{J\left(B\right)}\to\mathcal{C}^{J\left(A\right)}$. Since we have functorial categorical equivalences $\Sigma\left(A\right)\overset{\sim}{\longrightarrow}J\left(A\right)$ and $\Sigma\left(B\right)\overset{\sim}{\longrightarrow}J\left(B\right)$, this is the same as showing that the corresponding maps $\overline{F},\overline{G}\colon\Sigma\left(B\right)\to\mathcal{C}$ are equivalent in the fiber of $\mathcal{C}^{\Sigma\left(B\right)}\to\mathcal{C}^{\Sigma\left(A\right)}$ (which is also the homotopy fiber). This in turn is the same as having $\overline{F},\overline{G}$ homotopic rel. $\Sigma A$. It is left to show the equivalence $\left(2\right)\iff\left(4\right)$. The first part is clear. The second part amounts to showing the equivalence of two extension problems. If $F|_{A}=G|_{A}$, we get a map $F\cup_{A}G$ from $B\cup_{A}B\simeq B\rtimes_{A}\partial\Delta^{1}$ to $\hom_{\mathcal{C}}^{M}\left(X,Y\right)$ and $F$ and $G$ are homotopic rel. $A$ if and only if $F\cup_{A}G$ extends to the relative cylinder $B\rtimes_{A}\Delta^{1}$. In terms of maps to $\mathcal{C}$, this is equivalent to the extension problem

On the other hand, from $\overline{F}|_{\Sigma A}=\overline{G}|_{\Sigma A}$ we get a map $\overline{F}\cup_{\Sigma A}\overline{G}$ from $\Sigma B\rtimes_{\Sigma A}\partial\Delta^{1}$ to $\mathcal{C}$ and $\overline{F}$ and $\overline{G}$ are homotopic rel. $\Sigma A$ if and only if it extends to the relative cylinder $\Sigma B\rtimes_{\Sigma A}\Delta^{1}$. By 2.16 for $D=\Delta^{1},\partial\Delta^{1}$, the two extension problems are isomorphic. ∎

Proof (of 2.13). For $d\leq 0$ this follows directly from the definitions, and so we assume that $d\geq 1$. Let $K$ be a simplicial set. On the one hand,

where subscript $\left(X,Y\right)$ indicates that we take only the subset of maps that restrict to $\left(X,Y\right)$ on $\partial\Delta^{1}\hookrightarrow J\left(K\right)$ (observe that this is independent of the representative as $\partial\Delta^{1}\subseteq J\left(K\right)^{d-1}$). On the other hand,

We will argue that this last set is in natural bijection with the set

First, by definition of the right mapping space we have a natural bijection between maps of the form $f\colon K^{d-1}\to\hom_{\mathcal{C}}^{R}\left(X,Y\right)$ and maps of the form $\overline{f}\colon J\left(K^{d-1}\right)\to\mathcal{C}$ restricting to $\left(X,Y\right)$ on $\partial\Delta^{1}\subseteq J\left(K^{d-1}\right)$. Second, $f$ extends to $K^{d}$ if and only if $\overline{f}$ extends to $J\left(K^{d}\right)$. Likewise, it is clear that two maps $f,g\colon K^{d-1}\to\hom_{\mathcal{C}}^{R}\left(X,Y\right)$ agree on $K^{d-2}$ if and only if the corresponding maps $\overline{f},\overline{g}\colon J\left(K^{d-1}\right)\to\mathcal{C}$ agree on $J\left(K^{d-2}\right)$. Hence, the only thing we need to show is that $f$ and $g$ are homotopic rel. $K^{d-2}$ if and only if $\overline{f}$ and $\overline{g}$ are homotopic rel. $J\left(K^{d-2}\right)$ and this follows from $\left(1\right)\iff\left(3\right)$ in 2.17. It remains to observe that for every simplicial set $K$ and every $d\geq 1$ we have a canonical isomorphism $J\left(K^{d-1}\right)\overset{\sim}{\longrightarrow}J\left(K\right)^{d}.$ Hence, we get a natural bijection

and therefore an isomorphism $\alpha\colon\hom_{h_{d}\mathcal{C}}^{R}\left(X,Y\right)\simeq h_{d-1}\hom_{\mathcal{C}}^{R}\left(X,Y\right)$.

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.