Notation 2.1 Let $Cat_{(\infty ,n-1)}$ denote the $\infty $ -category of $(\infty ,n-1)$ -categories in the sense of [Reference Barwick and Schommer-PriesBSP21, Definition 7.2].

Notation 2.8 For $m\geq 0$ , we also denote by $[m]$ the image of the object ${[m]\in \Delta }$ under the canonical inclusion $\Delta \hookrightarrow Fun(\Delta ^{\operatorname {\mathrm {op}}},\mathcal S)\hookrightarrow Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-1)})$ , where the first map is Yoneda. Heuristically, the object $[m]\in Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-1)})$ represents the $(\infty ,n)$ -category free on the diagram

$$\begin{align*}0\to 1\to \ldots \to m-1\to m. \end{align*}$$

Notation 2.9 Given a simplicial $(\infty ,n-1)$ -category J and an $(\infty ,n)$ -category C, we denote by $\Delta _{C}^{J}\colon C\to [J,C]$ the diagonal map in $DblCat_{(\infty ,n-1)}$ induced by the unique map $J\to [0]$ .

Notation 2.10 Given an $(\infty ,n)$ -diagram $K\colon J\to C$ , the double $(\infty ,n-1)$ -category of cones $C\mathrm {one}^{(\infty ,n)}_{J}K$ is the following pullback in the $\infty $ -category $DblCat_{(\infty ,n-1)}$ :

Notation 2.11 Given a double $(\infty ,n-1)$ -category D and an object d of D, the slice double $(\infty ,n-1)$ -category of D over d is the following pullback in the $\infty $ -category $DblCat_{(\infty ,n-1)}$ :

Construction 2.14 We construct an inclusion $\iota _{n-1}\colon Cat_{(\infty ,n-2)}\hookrightarrow Cat_{(\infty ,n-1)}$ by induction that

1. (1) commutes with the canonical inclusions from $\mathcal S$ ;

2. (2) is a right adjoint; in particular, it preserves limits;

3. (3) is fully faithful.

• • When $n=1$ , we set $\iota _0\colon {\mathcal {S}}\to Cat_{(\infty ,1)}$ to be the canonical inclusion of the $\infty $ -category of spaces into the $\infty $ -category of $(\infty ,1)$ -categories, which satisfies (1)–(3).

• • When $n>0$ , we define $\iota _{n-1}\colon Cat_{(\infty ,n-2)}\hookrightarrow Cat_{(\infty ,n-1)}$ to be the restriction to full reflective subcategories (using Remark 2.13) of the functor induced by postcomposing with $\iota _{n-2}$ :

  

  This can be seen to restrict appropriately using (1) and (2) for $\iota _{n-2}$ . Moreover, (2) and (3) then hold for $\iota _{n-1}$ as they hold for $\iota _{n-2}$ . The fact that (1) holds for $\iota _{n-1}$ follows from the fact that the following square of functors commutes and (2) for $\iota _{n-2}$ :

  

Proof We prove this by induction on $n\geq 1$ . When $n=1$ , this is clear.

So, let $n> 1$ . Let $CS(Cat_{(\infty ,n-2)})$ be the full reflective subcategory of $Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-2)})$ spanned by the complete Segal objects in $Cat_{(\infty ,n-2)}$ , and denote by $L_{CS}$ the left adjoint of the inclusion. As a first step, we show that if X is a simplicial object in $Cat_{(\infty ,n-2)}$ such that $X_0$ is a space, then $(L_{CS}X)_0$ is a space as well.

Define $P(Cat_{(\infty ,n-2)})$ to be the full reflective subcategory of $Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-2)})$ spanned by those $X\colon \Delta ^{\operatorname {\mathrm {op}}}\to Cat_{(\infty ,n-2)}$ such that $X_0$ is a space. We then have the following adjunctions:

The left adjoints of the inclusions are given by the corresponding localization functors, and the right adjoint $R\colon Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-2)})\to P(Cat_{(\infty ,n-2)})$ of the inclusion sends a simplicial object $X\colon \Delta ^{\operatorname {\mathrm {op}}}\to Cat_{(\infty ,n-2)}$ to the pullback

where $\mathrm {cosk}\colon Cat_{(\infty ,n-2)}\to Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-2)})$ is the right adjoint of the evaluation at $0$ functor ${(-)_0\colon Fun(\Delta ^{\operatorname {\mathrm {op}}},Cat_{(\infty ,n-2)})\to Cat_{(\infty ,n-2)}}$ and ${(-)^\simeq \colon Cat_{(\infty ,n-2)}\to {\mathcal {S}}}$ is the right adjoint of the canonical inclusion ${\mathcal {S}}\hookrightarrow Cat_{(\infty ,n-2)}$ . In particular, since it preserves complete Segal objects and $Cat_{(\infty ,n-1)}$ is the full subcategory of $P(Cat_{(\infty ,n-2)})$ spanned by the complete Segal objects, it restricts to a right adjoint $R'\colon CS(Cat_{(\infty ,n-2)})\to Cat_{(\infty ,n-1)}$ of the inclusion. Since R and $R'$ commute with the inclusions, the corresponding left adjoints commute. Hence, for $X\in P(Cat_{(\infty ,n-2)})$ , there is an equivalence $L_{CS}X\simeq L^{\prime }_{CS}X$ in $CS(Cat_{(\infty ,n-2)})$ , showing that $(L_{CS} X)_0$ is a space.

Now consider the following diagram of adjunctions:

By induction, the left adjoint

$$\begin{align*}(L_{n-2})_*\colon Fun(\Delta^{\operatorname{\mathrm{op}}},Cat_{(\infty,n-2)})\to Fun(\Delta^{\operatorname{\mathrm{op}}},Cat_{(\infty,n-3)}) \end{align*}$$

preserves products. By [Reference Bergner and RezkBR20, Proposition 5.9], the localization functor

$$\begin{align*}L_{CS}\colon Fun(\Delta^{\operatorname{\mathrm{op}}},Cat_{(\infty,n-2)})\to CS(Cat_{(\infty,n-2)}) \end{align*}$$

preserves products, and so the left adjoint

$$\begin{align*}L_{CS}(L_{n-2})_*\colon CS(Cat_{(\infty,n-2)})\to CS(Cat_{(\infty,n-3)}) \end{align*}$$

also preserves products. Finally, since $L_{CS}$ and $(L_{n-2})_*$ preserve the property that level $0$ is a space, the left adjoint $L_{n-1}\colon Cat_{(\infty ,n-1)}\to Cat_{(\infty ,n-2)}$ coincides with the restriction of $L_{CS}(L_{n-2})_*$ to full reflective subcategories, and so it preserves products.

Proof This follows directly from Construction 2.14 and Proposition 2.15.

Proof This follows directly from Proposition 2.16(4).

Proof By definition, we have a pullback in $ DblCat_{(\infty ,n-2)}$

Hence, by Proposition 2.16(3), we also have a pullback in $ DblCat_{(\infty ,n-1)}$

On the other hand, we have, by definition, a pullback in $DblCat_{(\infty ,n-1)}$

Using Proposition 2.17 and the fact that $I_n [1]\simeq [1]$ , there is a natural comparison map between the two cospans defining the above pullbacks in $DblCat_{(\infty ,n-1)}$ , which is pointwise an equivalence of double $(\infty ,n-1)$ -categories. Hence, this induces a unique map between the two double $(\infty ,n-1)$ -categories presenting the pullback

$$\begin{align*}C\mathrm{one}^{(\infty,n)}_{ I_n J}( I_nK)\longrightarrow I_n(C\mathrm{one}^{(\infty,n-1)}_{J}K) ,\end{align*}$$

and this map is an equivalence of double $(\infty ,n-1)$ -categories, as desired.

Proof The proof follows the same steps as the one of Lemma 2.18.

Proof We have that an object $\ell $ of C is an $(\infty ,n-1)$ -limit object of the ${(\infty ,n-1)}$ -diagram $K\colon J\to C$ if and only if, by definition, there is an object $(\ell ,\lambda )$ in $C\mathrm {one}^{(\infty ,n-1)}_JK$ such that the canonical projection is an equivalence of double ${(\infty ,n-2)}$ -categories

if and only if, by Proposition 2.16(2), its image under $I_n$ is an equivalence of double $(\infty ,n-1)$ -categories

if and only if, by Lemma 2.19, the canonical projection is an equivalence of double $(\infty ,n-1)$ -categories

if and only if, by Lemma 2.18, the canonical projection is an equivalence of double $(\infty ,n-1)$ -categories

if and only if, by definition, the object $\ell $ of C is an $(\infty ,n)$ -limit object of the induced $(\infty ,n)$ -diagram $ I_nK\colon I_n J\to I_n C$ , as desired.

Notation 3.5 We denote by $[DblCat]_\infty $ the underlying $\infty $ -category (in the sense of [Reference BergnerBer09, Theorem 6.2]) of the model structure on $DblCat$ from [Reference RasekhMSV23b, Theorem 3.26] (see Section 4.1 for further details).

Notation 3.6 Given a $2$ -diagram $K\colon {\mathcal {J}}\to {\mathcal {C}}$ , the double category ${\mathbb {C}}\mathrm {one}^h_{\widetilde {{\mathbb {H}}}{\mathcal {J}}}(\widetilde {{\mathbb {H}}} K)$ of cones over K is the following pullback in $[Dbl{\mathcal {C}}\!\textit {at}]_\infty $ :

Notation 3.7 Given a double category ${\mathbb {D}}$ in $[DblCat]_\infty $ and an object d of ${\mathbb {D}}$ , the slice double category of ${\mathbb {D}}$ over d is the following pullback in $[Dbl{\mathcal {C}}\!\textit {at}]_\infty $ :

Notation 3.9 We denote by

$$\begin{align*}{\mathbb{N}}\colon [Dbl{\mathcal{C}}\!\textit{at}]_\infty \to Dbl{\mathcal{C}}\!\textit{at}_{(\infty,1)}\end{align*}$$

the functor induced by the nerve functor constructed in [Reference MoserMos20, Section 5.1] (see Section 4.3 for further details).

Proof First, note that (1) is a direct consequence of Theorem 4.36(3), while (2) and (3) follow from the fact that ${\mathbb {N}}\colon [DblCat]_\infty \to DblCat_{(\infty ,1)}$ has a left adjoint and the counit of this adjunction is an equivalence in $[DblCat]_\infty $ by Theorem 4.36(1) and (2). Finally, (4) is proven in Corollary 4.44.

Proof This follows directly from Proposition 3.10(4).

Proof By definition, we have a pullback in $[Dbl{\mathcal {C}}\!\textit {at}]_{\infty }$

Hence, by Proposition 3.10(3), we also have a pullback in $DblCat_{(\infty ,1)}$

On the other hand, we have, by definition, a pullback in $ DblCat_{(\infty ,1)}$

Using Proposition 3.11 and the fact that ${\mathbb {N}}\widetilde {{\mathbb {H}}} [0]\cong [0]$ and ${\mathbb {N}}\widetilde {{\mathbb {H}}} [1]\cong [1]$ , there is a natural comparison map between the two cospans defining the above pullbacks in $ DblCat_{(\infty ,1)}$ , which is pointwise an equivalence of double $(\infty ,1)$ -categories. Hence, this induces a unique map between the two double $(\infty ,1)$ -categories presenting the pullback

$$\begin{align*}{\mathbb{N}}({\mathbb{C}}\mathrm{one}^{h}_{\widetilde{{\mathbb{H}}} {\mathcal{J}}}(\widetilde{{\mathbb{H}}} K))\longrightarrow C\mathrm{one}^{(\infty,2)}_{{\mathbb{N}}\widetilde{{\mathbb{H}}} {\mathcal{J}}}({\mathbb{N}}\widetilde{{\mathbb{H}}} K),\end{align*}$$

and this map is an equivalence of double $(\infty ,1)$ -categories, as desired.

Proof The proof follows the same steps as the one of Lemma 3.12.

Proof We have that an object $\ell $ of ${\mathcal {C}}$ is a homotopy $2$ -limit object of the $2$ -diagram $K\colon {\mathcal {J}}\to {\mathcal {C}}$ if and only if, by Theorem 4.19, there is an object $(\ell ,\lambda )$ in ${\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{\widetilde {{\mathbb {H}}} {\mathcal {J}}}(\widetilde {{\mathbb {H}}} K)$ such that the canonical projection is an equivalence in $[DblCat]_\infty $

if and only if, by Proposition 3.10(2), its image under ${\mathbb {N}}$ is an equivalence of double $(\infty ,1)$ -categories

if and only if, using $2$ -out-of- $3$ and Lemma 3.13, the canonical projection is an equivalence of double $(\infty ,1)$ -categories

if and only if, by Lemma 3.12, the canonical projection is an equivalence of double $(\infty ,1)$ -categories

if and only if, by definition, the object $\ell $ of ${\mathcal {C}}$ is an $(\infty ,2)$ -limit object of the induced $(\infty ,2)$ -diagram ${\mathbb {N}}\widetilde {{\mathbb {H}}} K\colon {\mathbb {N}}\widetilde {{\mathbb {H}}} {\mathcal {J}}\to {\mathbb {N}}\widetilde {{\mathbb {H}}} {\mathcal {C}}$ , as desired.

Construction 4.1 There is a canonical fully faithful functor

$$\begin{align*}{\mathbb{H}}\colon 2{\mathcal{C}}\!\textit{at}\to Dbl{\mathcal{C}}\!\textit{at,} \end{align*}$$

which assigns to a $2$ -category ${\mathcal {C}}$ the double category ${\mathbb {H}}{\mathcal {C}}$ whose objects are the objects of ${\mathcal {C}}$ , whose horizontal morphisms are the morphisms of ${\mathcal {C}}$ , whose vertical morphisms are all trivial, and whose squares are the $2$ -morphisms in ${\mathcal {C}}$ .

Construction 4.7 We denote by

$$\begin{align*}\widetilde{{\mathbb{H}}} \colon 2{\mathcal{C}}\!\textit{at}\to Dbl{\mathcal{C}}\!\textit{at}, \end{align*}$$

the functor from [Reference RasekhMSV23b, Definition 2.13]. The functor $\widetilde {{\mathbb {H}}}$ assigns to a $2$ -category ${\mathcal {C}}$ the double category $\widetilde {{\mathbb {H}}} {\mathcal {C}}$ whose objects are the objects of ${\mathcal {C}}$ , whose horizontal morphisms are the morphisms of ${\mathcal {C}}$ , whose vertical morphisms are the adjoint equivalence data in ${\mathcal {C}}$ , and whose squares are the $2$ -morphisms in ${\mathcal {C}}$ of the form

Proof By Theorem 4.11, the $\infty $ -category $[DblCat]_\infty $ has a closed monoidal structure induced by the pseudo Gray tensor product $\boxtimes ^{\mathrm {ps}}$ , and by Lemma 4.12, this monoidal structure is equivalent to the one induced by the Cartesian product.

Proof Let $F\colon {\mathcal {J}}'\to {\mathcal {J}}$ be a biequivalence of $2$ -categories. Then F admits a pseudo-inverse, that is, there is a pseudo-functor $G\colon {\mathcal {J}}\to {\mathcal {J}}'$ together with pseudo-natural equivalences $\eta \colon \operatorname {\mathrm {id}}_{\mathcal {J}}\simeq F\circ G$ and $\varepsilon \colon G\circ F\simeq \operatorname {\mathrm {id}}_{{\mathcal {J}}'}$ . In particular, this induces a double biequivalence $\widetilde {{\mathbb {H}}}F\colon \widetilde {{\mathbb {H}}}{\mathcal {J}}'\to \widetilde {{\mathbb {H}}}{\mathcal {J}}$ that admits a horizontal pseudo-inverse given by the data of the induced double pseudo-functor $\widetilde {{\mathbb {H}}}G\colon \widetilde {{\mathbb {H}}}{\mathcal {J}}\to \widetilde {{\mathbb {H}}}{\mathcal {J}}'$ and the induced horizontal pseudo-natural equivalences $\widetilde {{\mathbb {H}}}\eta \colon \operatorname {\mathrm {id}}_{\widetilde {{\mathbb {H}}}{\mathcal {J}}}\simeq \widetilde {{\mathbb {H}}}F\circ \widetilde {{\mathbb {H}}}G$ and $\widetilde {{\mathbb {H}}}\varepsilon \colon \widetilde {{\mathbb {H}}}G\circ \widetilde {{\mathbb {H}}}F\simeq \operatorname {\mathrm {id}}_{\widetilde {{\mathbb {H}}}{\mathcal {J}}'}$ . This in turn implies that, for every double category ${\mathbb {D}}$ , the induced double functor has a horizontal pseudo-inverse given by the data $((\widetilde {{\mathbb {H}}}G)^*,(\widetilde {{\mathbb {H}}}\eta )^*,(\widetilde {{\mathbb {H}}}\varepsilon )^*)$ . Hence, it is a horizontal biequivalence in the sense of [Reference RasekhMSV23b, Definition 8.8]. Then it is a double biequivalence by [Reference RasekhMSV23b, Proposition 8.11], and so a weak equivalence in the model structure $DblCat_{\mathrm {whi}}$ by [Reference RasekhMSV23b, Proposition 3.25].

Proof Given a horizontal pseudo-natural equivalence , we construct a vertical pseudo-natural transformation $\beta \colon F\Rightarrow G$ as follows:

• • for an object i in ${\mathbb {J}}$ , set to be the component of $\alpha $ at the object i, which is an equivalence and hence a vertical morphism in $\widetilde {{\mathbb {H}}} {\mathcal {C}}$ ;

• • for a horizontal morphism $f\colon i\to j$ in ${\mathbb {J}}$ , set to be the pseudo-naturality constraint of $\alpha $ ;

• • for a vertical morphism $u\colon i\to j$ in ${\mathbb {J}}$ , set to be the component on vertical morphism of $\alpha $ , which is an invertible $2$ -morphism by [Reference MoserMos20, Lemma A.2.3] as $\alpha _u$ is a weakly horizontally invertible square.

The horizontal pseudo-naturality of $\alpha $ then implies the vertical pseudo-naturality of $\beta $ . Moreover, it is not hard to see that $(\alpha ,\beta )$ forms a companion pair.

Proof Since F is surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares, one can construct a normal pseudo double functor $G\colon {\mathbb {J}}'\to {\mathbb {J}}$ such that $FG=\operatorname {\mathrm {id}}_{{\mathbb {J}}'}$ and a horizontal pseudo-natural equivalence $\eta \colon \operatorname {\mathrm {id}}_{\mathbb {J}}\simeq GF$ . Applying the functor , this induces data $(F^*,G^*,\eta ^*)$ with a double functor, a normal pseudo double functor such that , and a horizontal pseudo-natural transformation. This shows that $F^*$ is a horizontal biequivalence in the sense of [Reference RasekhMSV23b, Definition 8.8], and so a double biequivalence by [Reference RasekhMSV23b, Proposition 8.11].

Proof Recall from Proposition 4.9 that the inclusion $I_{\mathcal {J}}\colon {{\mathbb {H}}}{\mathcal {J}}\to \widetilde {{\mathbb {H}}} {\mathcal {J}}$ is a double biequivalence. We factor $I_{\mathcal {J}}$ as a trivial cofibration followed by a trivial fibration in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$

By applying the functor , we get a factorization of $I_{\mathcal {J}}^*$

where the left-hand double functor is a trivial fibration in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ by Theorem 4.11, and the right-hand one is a double biequivalence by Lemma 4.22. This shows that $I_{\mathcal {J}}^*$ is a weak equivalence in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ and so a double biequivalence as all objects involved are weakly horizontally invariant by Lemma 4.21.

Notation 4.24 Given a double functor $K\colon {\mathbb {J}}\to {\mathbb {D}}$ , let ${\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathbb {J}}}(K)$ denote the pullback in $Dbl{\mathcal {C}}\!\textit {at}$

Proof We have, by definition, a pullback in the model category $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$

On the other hand, using Lemma 4.21, we have, by definition, a pullback in the model category $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$

Using Lemma 4.23, there is a natural comparison map between the two cospans defining the above pullbacks in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ , which is levelwise a double biequivalence. Hence, it induces a unique map between the two weakly horizontally invariant double categories presenting the pullback

$$\begin{align*}{\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{\widetilde{{\mathbb{H}}} {\mathcal{J}}}(\widetilde{{\mathbb{H}}} K)\longrightarrow {\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathbb{H}} {\mathcal{J}}}(\widetilde{{\mathbb{H}}} K\circ I_{\mathcal{J}}) \end{align*}$$

and this double functor is a double biequivalence, as desired.

Notation 4.28 Given a $2$ -functor $K\colon {\mathcal {J}}\to {\mathcal {C}}$ , the $2$ -category ${\mathcal {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathcal {J}}}(K)$ of pseudo-cones over K is the following pullback in $2{\mathcal {C}}\!\textit {at}$ :

Proof By applying the limit-preserving functor ${\textbf {H}} $ to the pullback in $Dbl{\mathcal {C}}\!\textit {at}$ from Notation 4.24 applied to $K\colon {\mathbb {H}} {\mathcal {J}}\to {\mathbb {D}}$ , we get the following pullback diagram in $2{\mathcal {C}}\!\textit {at}$ :

We can then compute the top and bottom right-hand $2$ -categories as follows. By [Reference Moser, Sarazola and VerdugoMSV22, Lemma 2.14], there is a natural $2$ -isomorphism

Next, there are also natural $2$ -isomorphisms

Hence, the above pullback square in $2{\mathcal {C}}\!\textit {at}$ becomes

But, by Notation 4.28, this pullback is exactly ${\mathcal {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathcal {J}}}(K^\sharp )$ .

Proof By applying the limit-preserving functor to the pullback in $Dbl{\mathcal {C}}\!\textit {at}$ from Notation 4.24 applied to $K\colon {\mathbb {H}} {\mathcal {J}}\to {\mathbb {D}}$ , we get the following pullback diagram in $2{\mathcal {C}}\!\textit {at}$ :

where $e_K$ denotes the vertical identity at K. We can then compute the top and bottom right-hand $2$ -categories as follows. By [Reference Moser, Sarazola and VerdugoMSV22, Lemma 2.14], there is a natural $2$ -isomorphism

and we also have natural $2$ -isomorphisms

Hence, the above pullback square in $2{\mathcal {C}}\!\textit {at}$ becomes

But, by Notation 4.28, this pullback is exactly ${\mathcal {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathcal {J}}}({\textbf {H}} E_{\mathbb {D}}\circ K^\sharp )$ .

Proof of Proposition 4.27

We need to show that applying the functors ${\textbf {H}} $ and to the double functor

$$\begin{align*}{\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathbb{H}} {\mathcal{J}}}({{\mathbb{H}}} K)\to {\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{{{\mathbb{H}}}{\mathcal{J}}}(I_{\mathcal{C}}\circ {{\mathbb{H}}} K) \end{align*}$$

yield two biequivalences.

First, note that ${\textbf {H}} {\mathbb {H}} {\mathcal {C}}={\mathcal {C}}={\textbf {H}} \widetilde {{\mathbb {H}}} {\mathcal {C}}$ so that we have

$$\begin{align*}({\mathbb{H}} K)^\sharp=K=(I_{\mathcal{C}}\circ {\mathbb{H}} K)^\sharp.\end{align*}$$

Hence, by Lemma 4.29, we have $2$ -isomorphisms

$$\begin{align*}{\boldsymbol{\rm{H}}} {\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathbb{H}} {\mathcal{J}}}({{\mathbb{H}}} K)\cong {\mathcal{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathcal{J}}}(K)\cong {\boldsymbol{\rm{H}}} {\mathbb{C}}\mathrm{one}^{\mathrm{ps}}_{{{\mathbb{H}}}{\mathcal{J}}}(I_{\mathcal{C}}\circ {{\mathbb{H}}} K) \end{align*}$$

and so the $2$ -functor ${\textbf {H}} {\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathbb {H}} {\mathcal {J}}}({{\mathbb {H}}} K)\to {\textbf {H}} {\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{{{\mathbb {H}}}{\mathcal {J}}}(I_{\mathcal {C}}\circ {{\mathbb {H}}} K)$ is a $2$ -isomorphism and hence a biequivalence.

Now, by Lemma 4.30, we have that the $2$ -functor

is given by the $2$ -functor

$$\begin{align*}{\mathcal{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathcal{J}}}({\boldsymbol{\rm{H}}} E_{{\mathbb{H}} {\mathcal{C}}} \circ K)\to {\mathcal{C}}\mathrm{one}^{\mathrm{ps}}_{{\mathcal{J}}}({\boldsymbol{\rm{H}}} E_{\widetilde{{\mathbb{H}}} {\mathcal{C}}}\circ K). \end{align*}$$

Since ${\textbf {H}} E_{\widetilde {{\mathbb {H}}} {\mathcal {C}}}$ can be seen as the composite of $2$ -functors

the above $2$ -functor corresponds to the $2$ -functor

induced by . As $I_{\mathcal {C}}$ is a double biequivalence by Proposition 4.9, is a biequivalence. Hence,

is a biequivalence since the two spans defining these pullbacks are levelwise equivalent, and the pullbacks are in fact homotopy pullbacks as every $2$ -category is fibrant.

Proof This follows from Proposition 4.25.

Notation 4.33 Given a $2$ -category ${\mathcal {C}}$ and an object c of ${\mathcal {C}}$ , the pseudo-slice $2$ -category of ${\mathcal {C}}$ over c is the following pullback in $2{\mathcal {C}}\!\textit {at}$ :

Proof Note that, by considering an object d of a double category ${\mathbb {D}}$ as a diagram $d\colon {\mathbb {H}} [0]\to {\mathbb {D}}$ , we have an isomorphism in $Dbl{\mathcal {C}}\!\textit {at}$

and, by considering an object c of ${\mathcal {C}}$ as a $2$ -diagram $c\colon [0]\to {\mathcal {C}}$ , we have a $2$ -isomorphism

The result is then obtained by specializing Lemmas 4.29 and 4.30 to the case ${K=d\colon {\mathbb {H}} [0]\to {\mathbb {D}}}$ .

Proof of Proposition 4.32

We need to show that applying the functors ${\textbf {H}} $ and to the double functor

yields two biequivalences.

First, by Lemma 4.34, we have that the $2$ -functor

is given by the $2$ -functor

Since the $2$ -functor ${\textbf {H}} {\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{{\mathbb {H}} {\mathcal {J}}}({{\mathbb {H}}} K)\to {\textbf {H}} {\mathbb {C}}\mathrm {one}^{\mathrm {ps}}_{{{\mathbb {H}}}{\mathcal {J}}}(I_{\mathcal {C}}\circ {{\mathbb {H}}} K)$ is a biequivalence by Proposition 4.27, so is the above $2$ -functor between slices.

Now, by Lemma 4.34, we have that the $2$ -functor

is given by the $2$ -functor

Since the $2$ -functor is a biequivalence by Proposition 4.27, so is the above $2$ -functor between slices.

Proof of Theorem 4.19

Combining Propositions 4.31 and 4.32, there are zig-zags of double biequivalences fitting into a commutative diagram of the following form:

where $F=I_{\mathcal {C}}\circ {{\mathbb {H}}} K=\widetilde {{\mathbb {H}}} K\circ I_{\mathcal {J}}\colon {\mathbb {H}} {\mathcal {J}}\to \widetilde {{\mathbb {H}}} {\mathcal {C}}$ using Remark 4.26. Hence, by $2$ -out-of- $3$ , the double functor

is a double biequivalence if and only if the double functor

is a double biequivalence. Hence, the desired result follows from Theorem 4.20.

Notation 4.35 We denote by $\textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ the model structure on the category $\textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}$ for Segal objects in complete Segal spaces. It follows from [Reference Barwick and Schommer-PriesBSP21, Theorem 13.15] and the definition of $Dbl{\mathcal {C}}\!\textit {at}_{(\infty ,1)}$ as the $\infty $ -category of Segal objects in $Cat_{(\infty ,1)}$ that there is an equivalence between the underlying $\infty $ -category of $\textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ and $Dbl{\mathcal {C}}\!\textit {at}_{(\infty ,1)}$ .

Notation 4.37 Given $m,t\geq 0$ , we denote by $[m,t]$ the double category ${\mathbb {H}} [m]\times {\mathbb {V}}[t]$ and we denote by $F[m,t]$ the representable in $\textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}$ at the object $([m],[t])\in \Delta \times \Delta $ . We have an identification ${\mathbb {N}} [m,t]\cong F[m,t]$ .

Proof First, recall that $F[-,-]\colon \Delta ^{\times 2}\to \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ given at $m,t\geq 0$ by the representable $F[m,t]$ is projectively cofibrant. As the projection $\pi _{1,2}\colon \Delta ^{\times 4}\to \Delta ^{\times 2}$ onto the first two components is left adjoint to the inclusion $\Delta ^{\times 2}\to \Delta ^{\times 4}$ given at $m,t\geq 0$ by $([m],[t],[0],[0])$ , it induces by pre-composition a left Quillen functor between projective model structures

$$\begin{align*}\pi_{1,2}^*\colon (\textit{s}{\mathcal{S}}\!\textit{et}^{\Delta^{\operatorname{\mathrm{op}}}\times \Delta^{\operatorname{\mathrm{op}}}}_{\mathrm{dbl}(\infty,1)})^{\Delta^{\times 2}}\to (\textit{s}{\mathcal{S}}\!\textit{et}^{\Delta^{\operatorname{\mathrm{op}}}\times \Delta^{\operatorname{\mathrm{op}}}}_{\mathrm{dbl}(\infty,1)})^{\Delta^{\times 4}}. \end{align*}$$

In particular, this left Quillen functor preserves cofibrant objects and so the functor ${F[-,-]\colon \Delta ^{\times 4}\to \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}}$ given at $m,t,m',t'\geq 0$ by the representable $F[m,t]$ is also projectively cofibrant. Similarly, the functor $F[-,-]\colon \Delta ^{\times 4}\to \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ given at $m,t,m',t'\geq 0$ by the representable $F[m',t']$ is projectively cofibrant.

Next, since the product $-\times -\colon \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}\times \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}\to \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ is a left Quillen bifunctor, it is also a left Quillen functor as all objects are cofibrant in $\textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}$ . Therefore, it induces by postcomposition a left Quillen functor between projective model structures

In particular, this left Quillen functor preserves cofibrant objects and so the functor ${F[-,-]\times F[-,-]\colon \Delta ^{\times 4}\to \textit {s}{\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}_{\mathrm {dbl}(\infty ,1)}}$ given at $m,t,m',t'\geq 0$ by the product $F[m,t]\times F[m',t']$ is also projectively cofibrant.

Proof This follows directly from Lemma 4.38 and the fact that

$$\begin{align*}{\mathbb{C}}\colon \textit{s}{\mathcal{S}}\!\textit{et}^{\Delta^{\operatorname{\mathrm{op}}}\times \Delta^{\operatorname{\mathrm{op}}}}_{\mathrm{dbl}(\infty,1)}\to Dbl{\mathcal{C}}\!\textit{at}_{\mathrm{whi}} \end{align*}$$

is left Quillen.

Proof This natural map is obtained as a lift in the following diagram in the projective model structure on $(Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}})^{\Delta ^{\times 4}}$ :

using Corollary 4.39 and the fact that the right-hand natural transformation is levelwise a trivial fibration in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ by Lemma 4.12.

Proof We write

$$\begin{align*}X\cong \operatorname{\mathrm{colim}}_{F[m,t]\to X} F[m,t]\quad\text{and} \quad Y\cong \operatorname{\mathrm{colim}}_{F[m',t']\to Y} F[m',t']. \end{align*}$$

Then there are natural isomorphisms in $Dbl{\mathcal {C}}\!\textit {at}$

$$ \begin{align*} {\mathbb{C}}( X\times Y) &\cong {\mathbb{C}}((\operatorname{\mathrm{colim}}_{F[m,t]\to X} F[m,t])\times (\operatorname{\mathrm{colim}}_{F[m',t']\to Y} F[m',t'])) & \\ &\cong {\mathbb{C}}(\operatorname{\mathrm{colim}}_{F[m,t]\to X}\operatorname{\mathrm{colim}}_{F[m',t']\to Y} (F[m,t]\times F[m',t'])) & \\ & \cong\operatorname{\mathrm{colim}}_{F[m,t]\to X}\operatorname{\mathrm{colim}}_{F[m',t']\to Y} {\mathbb{C}}(F[m,t]\times F[m',t']) & \end{align*} $$

using that $\times $ and ${\mathbb {C}}$ preserve colimits. Moreover, there are natural isomorphisms in $Dbl{\mathcal {C}}\!\textit {at}$

$$ \begin{align*} {\mathbb{C}} X\boxtimes^{\mathrm{ps}} {\mathbb{C}} Y &\cong {\mathbb{C}}(\operatorname{\mathrm{colim}}_{F[m,t]\to X} F[m,t])\boxtimes^{\mathrm{ps}} {\mathbb{C}}(\operatorname{\mathrm{colim}}_{F[m',t']\to Y} F[m',t']) & \\ &\cong (\operatorname{\mathrm{colim}}_{F[m,t]\to X} {\mathbb{C}} F[m,t])\boxtimes^{\mathrm{ps}} (\operatorname{\mathrm{colim}}_{F[m',t']\to Y}{\mathbb{C}} F[m',t']) & \\ & \cong\operatorname{\mathrm{colim}}_{F[m,t]\to X}\operatorname{\mathrm{colim}}_{F[m',t']\to Y} ({\mathbb{C}} F[m,t]\boxtimes^{\mathrm{ps}} {\mathbb{C}} F[m',t']) & \end{align*} $$

using that ${\mathbb {C}}$ and $\boxtimes ^{\mathrm {ps}}$ preserve colimits. Then the double functor

$$\begin{align*}{\mathbb{C}}(F[m,t]\times F[m',t'])\to {\mathbb{C}} F[m,t]\boxtimes^{\mathrm{ps}} {\mathbb{C}} F[m',t'] \end{align*}$$

from Lemma 4.40, which is natural in $m,t,m',t'$ , induces a double functor between colimits

$$\begin{align*}{\mathbb{C}}( X\times Y)\to {\mathbb{C}} X\boxtimes^{\mathrm{ps}} {\mathbb{C}} Y. \end{align*}$$

Naturality in X and Y follows from the naturality in the indexing shapes of the colimits.

Proof First recall from Theorem 4.36(2) that the components of the counit

$$\begin{align*}\varepsilon_{[m,t]} \colon {\mathbb{C}} F[m,t]\cong {\mathbb{C}}{\mathbb{N}}[m,t]\stackrel\simeq\longrightarrow [m,t] \end{align*}$$

and

$$\begin{align*}\varepsilon_{[m,t]\times [m',t']}\colon {\mathbb{C}}(F[m,t]\times F[m',t'])\cong {\mathbb{C}}{\mathbb{N}}([m,t]\times [m',t'])\to [m,t]\times [m',t'] \end{align*}$$

are double biequivalences. Hence, by [Reference RasekhMSV23b, Remark 7.5], so is the product

$$\begin{align*}\varepsilon_{[m,t]}\times \varepsilon_{[m',t']}\colon {\mathbb{C}} F[m,t]\times {\mathbb{C}} F[m',t'] \stackrel\simeq\longrightarrow [m,t]\times [m',t']. \end{align*}$$

Moreover, recall from [Reference RasekhMSV23b, Lemma 7.3] that the natural projection double functor

$$\begin{align*}{\mathbb{C}} F[m,t]\boxtimes^{\mathrm{ps}} {\mathbb{C}} F[m',t']\stackrel\simeq\longrightarrow {\mathbb{C}} F[m,t]\times {\mathbb{C}} F[m',t'] \end{align*}$$

is also a double biequivalence.

Hence, we have the following commutative diagram in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ :

Note that the diagram indeed commutes by naturality of the counit and by construction of the double functor from Lemma 4.40. So the result follows by $2$ -out-of- $3$ .

Proof Given $m,t\geq 0$ , we first show that the double functor from Proposition 4.41 is a weak equivalence in the model structure $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$

(4.2) $$ \begin{align} {\mathbb{C}}(F[m,t]\times Y)\stackrel{\simeq}{\longrightarrow}{\mathbb{C}} F[m,t]\boxtimes^{\mathrm{ps}}{\mathbb{C}} Y, \end{align} $$

for all bisimplicial sets Y. By Theorems 4.11 and 4.36(1) and Remark 2.4, the functors

$$\begin{align*}{\mathbb{C}}(F[m,t]\times-),~{\mathbb{C}} F[m,t]\boxtimes^{\mathrm{ps}}{\mathbb{C}}(-)\colon DblCat_{(\infty,1)}\to Dbl{\mathcal{C}}\!\textit{at}_{\mathrm{whi}}\end{align*}$$

are left Quillen, so the set of $(\Delta \times \Delta )$ -sets Y for which the double functor (4.2) is a weak equivalence in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ is saturated by monomorphisms in the sense of [Reference CisinskiCis19, Definition 1.3.9]. Furthermore, by Lemma 4.42, this holds when $Y=F[m',t']$ is a representable in ${\mathcal {S}}\!\textit {et}^{\Delta ^{\operatorname {\mathrm {op}}}\times \Delta ^{\operatorname {\mathrm {op}}}}$ . Hence, by [Reference CisinskiCis19, Corollary 1.3.10], we obtain that the double functor (4.2) is a weak equivalence in $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$ for all bisimplicial sets Y.

Now, given a bisimplicial set Y, a similar argument using the above result shows that the double functor from Proposition 4.41 is a weak equivalence in the model structure $Dbl{\mathcal {C}}\!\textit {at}_{\mathrm {whi}}$

$$\begin{align*}{\mathbb{C}}(X\times Y)\stackrel{\simeq}{\longrightarrow}{\mathbb{C}} X\boxtimes^{\mathrm{ps}}{\mathbb{C}} Y, \end{align*}$$

for all bisimplicial sets X, as desired.

Proof By Theorem 4.43, we have that the left adjoint of ${\mathbb {N}}$ sends Cartesian products to pseudo Gray tensor products. Then, by Lemma 4.12, the monoidal structure induced by the pseudo Gray tensor product in $[DblCat]_\infty $ is equivalent to the one given by the Cartesian product. Hence, the left adjoint of ${\mathbb {N}}$ preserves products.