Notation 2.1 For any $k\geq 0$, we will write $\vec {n}_k=([n_1], \dots, [n_k])\in \mathbf {\Delta }^{\times k}$ for a generic (but fixed) $k$-tuple of objects in $\mathbf {\Delta }$. Furthermore, we will abbreviate $\vec {0}_k=([0], \dots, [0])$. For example, if $X\colon \mathbf {\Delta }^{\times d}\longrightarrow \mathscr {S}$ is a $d$-fold simplicial space, then

denotes the simplicial space obtained by fixing the first $k$ inputs to be $\vec {n}_k$ and the last $d-k-1$ inputs to be $0$, varying only the $(k+1)$th input. Finally, we will write $X(n)\colon \mathbf {\Delta }^{\times d-1}\longrightarrow \mathscr {S}$ for the $(d-1)$-fold simplicial space given by $X(n, -, \dots, -)$.

Warning 2.3 Because we impose completeness conditions in all directions, this notion of a $d$-fold category is not quite the $\infty$-categorical analogue of that of a $d$-uple category in the sense of Ehresmann [Reference EhresmannEhr63], i.e. a (strict) category object in $(d-1)$-uple categories. The latter type of objects (which are sometimes also called $d$-fold categories) corresponds instead to the more general notion of a $d$-uple Segal space [Reference HaugsengHau17].

For instance, for a double category in the sense of Definition 2.2, the completeness conditions in the horizontal and vertical direction imply that the horizontal equivalences and vertical equivalences coincide. Many examples of double (and, more generally, $d$-uple) Segal spaces do not satisfy this condition: as an example, one can take the double Segal space of associative algebras, with horizontal morphisms given by bimodules and vertical morphisms given by algebra homomorphisms [Reference HaugsengHau17].

Notation 2.4 (Opposites)

Let $\mathbb {C}$ be a $d$-fold category and $1\leq k\leq d$. We will write $\mathbb {C}^{k-\mathrm {op}}$ for the $d$-fold category given by taking opposites in the $k$th variable:

When $k=1$, we will simply write $\mathbb {C}^{\mathrm {op}}$. If $\mathbb {C}$ is a double category, we will furthermore write $\mathbb {C}^{\mathrm {rev}}$ for the double category with its coordinates reflected, i.e. $\mathbb {C}^\mathrm {rev}(m, n)=\mathbb {C}(n, m)$.

Proof. It suffices to verify that for each $d$-fold category $\mathbb {C}$ and $m_1, \dots, m_d\geq 0$, the $d$-fold simplicial space given by

\[ \mathbb{C}^{[m_1, \dots, m_d]}(n_1, \dots, n_d)=\mathrm{Map}\big([m_1, \dots, m_d]\times [n_1, \dots, n_d], \mathbb{C}\big) \]

remains a $d$-fold category. Using symmetry and the fact that $[m_1, \dots, m_d]=[m_1, 0, \dots, 0]\times \dots \times [0,\dots, 0, m_d]$, it suffices to verify that $\mathbb {C}^{[m, 0, \dots, 0]}$ is a $d$-fold category, i.e. satisfies the complete Segal conditions in all simplicial directions.

For the first direction, note that each $\mathbb {C}^{[m, 0, \dots, 0]}(-, \vec {n}_{d-1})$ agrees with $\mathbb {C}(-, \vec {n}_{d-1})^{[m]}$, which is a $1$-category by [Reference RezkRez01, Theorem 7.2]. For the remaining simplicial directions, note that the $(d-1)$-fold simplicial space

sends each $(n_2, \dots, n_d)$ to the space of maps between simplicial spaces

\begin{align*} \mathrm{Map}_{s\mathscr{S}}\Big([m]\times [n], \mathbb{C}(-, n_2, \dots, n_d)\Big) &\simeq \mathrm{Map}_{s\mathscr{S}}\Big(\mathop {{\rm colim}}\limits_{_{[k]\in \mathbf{\Delta}/[m]\times [n]}} \ [k], \ \mathbb{C}(-, n_2, \dots, n_d)\Big)\\ &\simeq \lim_{[k]\in (\mathbf{\Delta}/[m]\times [n])^{\mathrm{op}}} \mathbb{C}(k, n_2, \dots, n_d). \end{align*}

In other words, $\mathbb {C}^{[m, 0, \dots, 0]}(n, -)$ is naturally equivalent to the limit of a certain diagram consisting of the $(d-1)$-fold categories $\mathbb {C}(k, -)$. Since ${{\rm Cat}}^{\otimes d-1}\subseteq \mathrm {Fun}\big ((\mathbf {\Delta }^{\mathrm {op}})^{\times d-1}, \mathscr {S}\big )$ is closed under limits, each $\mathbb {C}^{[m, 0, \dots, 0]}(n, -)$ is a $(d-1)$-fold category, as desired.

Proof. To see condition (1) of Definition 2.9, consider the pullback square of categories

where $\Delta$ takes the constant diagram and $\iota$ is the inclusion of the space of objects. Since $(p_1, p_2)$ is a local orthofibration, the induced map $(p'_1, p'_2)\colon \mathrm {Fun}([n], \mathscr {X})\longrightarrow \mathrm {Fun}([n], \mathscr {C})\times \mathrm {Fun}([n], \mathscr {D})$ is a local orthofibration as well: a natural transformation in $\mathrm {Fun}([n], \mathscr {X})$ is $p'_1$-cocartesian if and only if it is given pointwise by $p_1$-cocartesian maps in $\mathscr {X}$, and similarly for $p'_2$-cartesian natural transformations (this follows, e.g., from Lemma 5.4(3)). In particular, the base change $q\colon \mathrm {Fun}'([n], \mathscr {X})\longrightarrow \mathscr {C}\times \mathrm {Map}_{{{\rm Cat}}}([n], \mathscr {D})$ is a cocartesian fibration.

Unraveling the definitions, one sees that the space of maps $[m]\longrightarrow \mathrm {Fun}'([n], \mathscr {X})$ is equivalent to the space of diagrams $\sigma \colon [m]\times [n]\longrightarrow \mathscr {X}$ of the form (2.13). Furthermore, $[m]\longrightarrow \mathrm {Fun}'([n], \mathscr {X})$ sends each arrow to a $q$-cocartesian arrow if and only if $\sigma$ sends each $[m]\times \{i\}$ to $p_1$-cocartesian arrows in $\mathscr {X}$. It follows that there is a factorization

where the first map is the inclusion of the wide subcategory spanned by the $q$-cocartesian arrows. In particular, this shows that each $\mathbb {X}(-, n)\longrightarrow \big (\mathscr {C}\boxtimes \mathscr {D}\big )(-, n)$ is a left fibration, as desired.

For condition (2) of Definition 2.9, it suffices to observe that $\mathbb {X}(0, -)\longrightarrow \big (\mathscr {C}\boxtimes \mathscr {D}\big )(0, -)$ coincides with top horizontal map in the following pullback diagram.

This map is a cartesian fibration by condition (ii) (cf. [Reference Haugseng, Hebestreit, Linskens and NuitenHHLN23b, Observation 2.3.2]).

Proof. Since $p$ and $q$ are left fibrations in the horizontal direction, the map $f$ is an equivalence if and only if the map $f_0\colon \mathbb {D}(0, -)\longrightarrow \mathbb {E}(0, -)$ is an equivalence of cartesian fibrations over $\mathbb {C}(0, -)$; in turn, the map of cartesian fibrations $f_0$ is an equivalence if and only if it induces an equivalence on fibers [Reference LurieLur09a, Corollary 2.4.4.4].

Variant 2.16 Taking opposites in the horizontal or vertical direction, and exchanging the vertical and horizontal direction, one obtains seven more types of fibrations. For example, a map $\mathbb {D}\longrightarrow \mathbb {C}$ is a (cocart, right)-fibration if $\mathbb {D}(-, 0)\longrightarrow \mathbb {C}(-, 0)$ is a cocartesian fibration and each $\mathbb {D}(m, -)\longrightarrow \mathbb {C}(m, -)$ is a right fibration. Strong maps between such fibrations are always required to preserve (co)cartesian arrows, either in the horizontal or vertical direction.

Proof. To see that $X$ is a category in the vertical direction, we use that $\mathrm {Cat}^{\otimes 2}$ is cartesian closed (Proposition 2.8): the simplicial space $X(m, -)$ can then be identified with the category $\big (\mathbb {D}^{\mathbb {A}\mathrm {r}[m]}\times _{\mathbb {C}^{\mathbb {A}\mathrm {r}[m]}} \mathbb {C}^{[m, 0]}\big )(0, -)$.

To see that each $X(-, n)$ is a category, it suffices to treat the case $n=0$: indeed, the general case then follows by considering the functor $\mathbb {D}^{[0, n]}\longrightarrow \mathbb {C}^{[0, n]}$, which is still a left fibration in the horizontal direction. Unraveling the definitions shows that the Segal condition for $X(-, 0)$ translates into the following unique lifting problem.

In other words, given a solid diagram in $\mathbb {D}$ of the form

whose image in $\mathbb {C}$ has a fixed dashed extension (i.e. including the top squares), there exists a contractible space of dashed extensions as indicated. This follows immediately from the fact that each $\mathbb {D}(-, n)\longrightarrow \mathbb {C}(-, n)$ is a left fibration of categories.

For completeness of $X(-, 0)$, recall the simplicial set $H=\Delta [3]/\sim$ in the definition of completeness from § 2, and note that a map

corresponds to a diagram in $\mathbb {D}$ of the form

in which $x_{00}\longrightarrow x_{02}$ and are degenerate and $x_{11}\longrightarrow x_{13}$ and are degenerate, and whose image in $\mathbb {C}$ consists only of degenerate arrows (since $\mathbb {C}$ is complete). We need to show that the above diagram is degenerate both in the horizontal and the vertical direction. Using that each $\mathbb {D}(-, n)\longrightarrow \mathbb {C}(-, n)$ is a left fibration, one sees that the above diagram is degenerate in the horizontal direction. Furthermore, the rightmost column is degenerate in the vertical direction, since both and are degenerate and $\mathbb {D}(0, -)$ is a category.

Proof. We verify each of the conditions, starting with the fact that $\Psi ^\perp (\mathbb {D})$ is a double category.

Segal conditions. Recall the double category $X$ from Lemma 2.18. Since each $\Psi ^\perp (\mathbb {D})(-, n)\subseteq X(-, n)$ is a full subcategory of the category $X(-, n)$, it follows that $\Psi ^\perp (\mathbb {D})(-, n)$ is a category (i.e. complete Segal space) itself. In the other direction, note that there is the following pullback square of simplicial spaces.

Since $X([m], -)$ is a category by Lemma 2.18, it suffices to verify that $\Psi ^\perp (\mathbb {D})(0, -)\hookrightarrow X(0, -)$ is a subcategory. But this can simply be identified with the inclusion of the wide subcategory of $\mathbb {D}(0, -)$ whose morphisms are cartesian morphisms.

Vertical right fibration. Next, let us verify that each $\Psi ^\perp (\mathbb {D})(m, -)\longrightarrow \mathbb {C}(m, -)$ is a right fibration. By the horizontal Segal condition, it suffices to treat the case $m=0$ and $m=1$. Unraveling the definitions, we have to verify that there are contractible spaces of diagonal lifts

with the following property: the vertical arrow in $[0, 1]$, respectively the vertical arrows in $\{00, 11\}\times [0, 1]$, are sent to cartesian vertical arrows in $\mathbb {D}$. For the left square, there is a unique such lift since $\mathbb {D}(0, -)\longrightarrow \mathbb {C}(0, -)$ is a cartesian fibration. For the right square, we have to verify that for any solid diagram

there exist unique objects $x_{00, 0}, x_{01, 0}, x_{11, 0}$ together with a dashed extension as indicated, such that the two arrows and are cartesian. To see this, first take these two arrows to be the (unique) cartesian lifts of their image in $\mathbb {C}$. Then the top square can be filled uniquely using that $\mathbb {D}\longrightarrow \mathbb {C}$ is a left fibration in the horizontal direction. The right square (living entirely in the vertical direction) can then be filled uniquely since was cartesian.

Horizontal cocartesian fibration. Note that the space $\Psi ^\perp (\mathbb {D})(1, 0)$ of horizontal arrows in $\Psi ^\perp (\mathbb {D})$ is the space of diagrams in $\mathbb {D}$ of the form

covering a horizontal arrow in $\mathbb {C}$. Let us say that a horizontal arrow in $\Psi ^\perp (\mathbb {D})$ is marked if the corresponding vertical map is an equivalence in $\mathbb {D}$. We claim that all marked horizontal arrows in $\Psi ^\perp (\mathbb {D})$ are $\Psi ^\perp (p)$-cocartesian. In other words, we have to find a unique diagonal lift for every diagram of categories

where the top map sends $0\longrightarrow 1$ to a marked arrow. Unraveling the definitions, this corresponds to a solid diagram in $\mathbb {D}$ covering a horizontal $2$-simplex in $\mathbb {C}$

where the map is an equivalence, of which one has to find a unique dashed extension as indicated. To find the desired unique extension, using that $\mathbb {D}\longrightarrow \mathbb {C}$ is a left fibration in the horizontal direction, it follows that there is a unique way to extend the square making the top triangle (living purely in the horizontal direction) commute. Since was an equivalence, the resulting vertical arrow is an equivalence as well. It then follows that there is a unique vertical map with a filling of the right triangle (purely in the vertical direction). We conclude that all marked arrows in $\Psi ^\perp (\mathbb {D})$ are cocartesian. Finally, for every horizontal arrow $f\colon c\longrightarrow c'$ in $\mathbb {C}$ and a lift $x\in \Psi ^\perp (\mathbb {D})_c$, there exist a marked lift $\tilde {f}\colon x\longrightarrow x'$, using that $\mathbb {D}(-, 0)\longrightarrow \mathbb {C}(-, 0)$ was a left fibration: indeed, one can take $\tilde {f}$ to correspond to the diagram in $\mathbb {D}$, where the vertical morphism is the identity.

Notation 3.3 Let $\mathbf {\Delta }^{\times 2}_{\downarrow }$ denote the category obtained from $\mathbf {\Delta }^{\times 2}$ by freely adding a factorization of the vertical degeneracy map $[0, 1]\longrightarrow [0, 0]$ as follows.

A presheaf $X\colon (\mathbf {\Delta }^{\times 2}_{\downarrow })^{\mathrm {op}}\longrightarrow \mathscr {S}$ is said to be a vertically marked bisimplicial space if the map $X([0, 1]^\sharp )\longrightarrow X(0, 1)$ is a subspace inclusion. We will write $\mathrm {bis}\mathscr {S}_{\downarrow }\subseteq \mathrm {Fun}(\mathbf {\Delta }^{\times 2, \mathrm {op}}_{\downarrow }, \mathscr {S})$ for the full subcategory spanned by the vertically marked bisimplicial spaces, and typically use the same symbol for a vertically marked bisimplicial space and its underlying bisimplicial space.

Notation 3.6 As usual, the forgetful functor $\mathrm {Cat}^{\otimes 2}_{\downarrow }\longrightarrow \mathrm {Cat}^{\otimes 2}$ admits a fully faithful right adjoint, sending a double category $\mathbb {C}$ to the vertically marked double category $\mathbb {C}^{\sharp }$ where all vertical arrows are marked. We will also write $\mathbb {C}^{\sharp }$ for $\mathbb {C}$ with all horizontal arrows marked; it will always be clear from the context whether we mark horizontal or vertical arrows.

Proof. By Remark 3.4, the forgetful functor $\mathrm {Fun}\big ([1], \mathrm {Cat}^{\otimes 2}_{\downarrow })\longrightarrow \mathrm {Fun}\big ([1], \mathrm {Cat}^{\otimes 2})$ induces inclusions of path components on mapping spaces, whose images consist of those commuting squares of double categories such that all maps preserve marked arrows. The result follows immediately from this.

Construction 3.10 Let us fix the following data.

1. (a) For each $m, n\geq 0$, a map of vertically marked double categories $S[m, n]\longrightarrow [m, n]^\sharp$, natural in the sense that it defines a functor fitting into the following commuting triangle.

   

2. (b) In the vertically marked double category $S[1, 0]$, an additional set of path components of marked vertical edges. We will write $S[1, 0]^+$ for the resulting vertically marked double category.

Associated to these data is the following natural map of cartesian fibrations.

(3.11)

To construct this diagram, observe that the codomain projection admits a left adjoint $L\colon \mathrm {bis}\mathscr {S}\longrightarrow \mathrm {Fun}^\sharp ([1], \mathrm {bis}\mathscr {S}_{\downarrow })$, sending $\mathbb {C}$ to $\emptyset \longrightarrow \mathbb {C}^\sharp$; the same holds in the horizontally marked case. Now consider the following commuting squares.

In the left square, $h$ is the Yoneda embedding and $\sigma$ is the functor sending

\[ \big(1, [m, n]\big)\mapsto \big(\emptyset\longrightarrow [m, n]^\sharp\big) \qquad \big(0, [m, n]\big)\mapsto \big(S[m, n]\longrightarrow [m, n]^\sharp\big). \]

Furthermore, $\sigma$ sends $(0, [1, 0]^\sharp )$ to $S[1, 0]^+\longrightarrow [1, 0]^\sharp$ and $(1, [1, 0]^\sharp )$ to $\emptyset \longrightarrow [1, 0]^\sharp$. This induces the commuting square of left adjoint functors on the right, using that $h$ and $\sigma$ induce unique colimit-preserving functors out of the corresponding presheaf categories [Reference DuggerDug01], [Reference LurieLur09a, Theorem 5.1.5.6].

Let us write $\Psi _S$ for the right adjoint to $\Phi _S$. Explicitly, let $p\colon Y\longrightarrow X^\sharp$ be a map of vertically marked bisimplicial spaces whose target is maximally marked. Then $\Psi _S(p)\colon \Psi _S(Y)\longrightarrow X^\sharp$ can be described as follows: the underlying bisimplicial space of $\Psi _S(Y)$ has $(m, n)$-simplices determined by the following.

(3.12)

Furthermore, the space of marked $[1, 0]$-simplices in $\Psi _S(Y)$ is the subspace consisting of those maps $S[1, 0]\longrightarrow Y$ that also preserve the additional marked arrows from data (b). In particular, this implies that $\Psi _S$ sends vertically marked bisimplicial spaces into horizontally marked bisimplicial spaces (i.e. marked edges form a subspace). We therefore obtain the desired commuting diagram (3.11). Furthermore, it follows immediately from (

) that $\Psi _K$ preserves cartesian morphisms: for every $Y\longrightarrow X^\sharp$ and $X'\longrightarrow X$, we have an equivalence

Construction 3.13 For any $[m, n]\in \mathbf {\Delta }^{\times 2}$, consider the following natural map of vertically marked double categories:

Here $K[m, n]$ comes equipped with the marking where for each $ii\in \mathbb {A}\mathrm {r}[m]$ (cf. Example 2.6), all vertical arrows in $\{ii\}\times [0, n]$ are marked. Furthermore, $K[1, 0]^+$ is the double category $\mathbb {A}\mathrm {r}[1]$, where, in addition, the vertical arrow is marked.

Proof. Unraveling the definition, one sees that the map of bisimplicial spaces $\Psi _K(\mathbb {D})\longrightarrow \mathbb {C}$ is precisely given by the reflection $\Psi ^\perp (p)\colon \Psi ^\perp (\mathbb {D})\longrightarrow \mathbb {C}$ from Definition 2.21; in particular, it is a (cocart, right)-fibration by Proposition 2.22.

It remains to identify the marked horizontal arrows in $\Psi _K(\mathbb {D}^{\natural })$; these correspond to diagrams in $\mathbb {D}$ with the property that is a marked arrow covering an equivalence in $\mathbb {C}$. This just means that is an equivalence, so that the marked horizontal arrows coincide with the marked horizontal arrows introduced in the proof of Proposition 2.22. We have already seen there that these marked arrows are precisely the cocartesian horizontal arrows.

Proof. Consider the natural map $K[m, n]=\mathbb {A}\mathrm {r}[m]\times [0, n]\longrightarrow \mathbb {A}\mathrm {r}[m]\stackrel {\pi _\mathrm {ver}}{\longrightarrow } [0, m]$, where the second map is the vertical projection (see Example 2.6). This map sends the marked vertical arrows in $K[m, n]$ from Construction 3.13 to degenerate edges. For each $\mathbb {D}\in \mathrm {Fib}^{\mathrm {left, cart}}$, we therefore obtain a natural transformation of double categories, given in degree $(m, n)$ by

Taking $\mathbb {D}=[0]\boxtimes \mathscr {C}$, this produces the desired map $\mathscr {C}\boxtimes [0]\longrightarrow \Psi ^\perp ([0]\boxtimes \mathscr {C})$. To see that this is an equivalence, note that the domain and codomain are both constant in the vertical direction and complete Segal spaces in the horizontal direction. In degree $(0, 0)$, the above is an equivalence since $K[0, 0]\longrightarrow [0, 0]$ is an isomorphism. In degree $(1, 0)$, it suffices to note that the map $\pi _\mathrm {ver}\colon \mathbb {A}\mathrm {r}[1]\longrightarrow [0, 1]$ induces an equivalence on mapping spaces into a double category of the form $[0]\boxtimes \mathscr {C}$.

Variant 3.16 Let us introduce the following two variants of the functor $\Psi ^\perp$.

1. (1) There is a functor $\Psi ^\top \colon \mathrm {Fib}^{\mathrm {cocart, right}}\longrightarrow \mathrm {Fib}^{\mathrm {left, cart}}$, obtained by conjugating $\Psi ^\perp$ with the functor sending $\mathbb {C}\mapsto (\mathbb {C}^\mathrm {rev})^{(1, 2)-\mathrm {op}}$. Unraveling the definitions, one sees that $\Psi ^\top =\Psi _{K'}$ is defined as in Construction 3.10 (with the roles of horizontal and vertical reversed), but this time using the maps

   
   Here we mark the horizontal arrows in $K'[m, n]$ contained in $[m, 0]\times \{ii\}$ for each $i\in [n]$. Furthermore, $K'([0, 1]^\sharp )$ is given by $\mathbb {A}\mathrm {r}[1]$ where we mark all horizontal arrows.

2. (2) There is another functor $\Psi ^{{\dagger} }\colon \mathrm {Fib}^{\mathrm {cocart, left}}\longrightarrow \mathrm {Fib}^{\mathrm {left, cocart}}$, obtained by conjugating $\Psi ^\perp$ with the functor sending $\mathbb {C}\mapsto (\mathbb {C}^{\mathrm {rev}})^{1-\mathrm {op}}$. Unraveling the definitions, one sees that $\Psi ^{\dagger} =\Psi _L$ is defined as in Construction 3.13, but the role of the vertically marked double category $K[m, n]$ is now played by

   
   In $L[m, n]$, we mark the horizontal arrows in each $[m, 0]\times \{ii\}$. Furthermore, $L([1, 0]^\sharp )$ is given by $\mathbb {T}\mathrm {w}([1]^{\mathrm {op}})$ with all horizontal arrows marked.

Construction 3.18 For each $[m, n]\in \mathbf {\Delta }^{\times 2}$, let

In $A[m, n]$, we mark all vertical arrows contained in $\mathbb {A}\mathrm {r}[m]\times \{i\}$, for all $i\in [n]$; in particular, note that this marking is different from Construction 3.13. In addition, define $A([0, 1])^+=[0, 1]^\sharp$.

Likewise, for each $[m, n]\in \mathbf {\Delta }^{\times 2}$, let us denote

In $B[m, n]=\mathbb {A}\mathrm {r}[m]\times \big ([0]\boxtimes \mathrm {Fun}([1], [n])\big )$, we mark the following vertical arrows:

• – for each object $aa\in \mathbb {A}\mathrm {r}[m]$ with $0\leq a\leq m$, we mark all edges in $\{aa\}\boxtimes \mathrm {Fun}([1], [n])$ of the form $ik\longrightarrow jk$;

• – for each $ii\in \mathrm {Fun}([1], [n])$ with $0\leq i\leq n$, we mark all vertical arrows in $\mathbb {A}\mathrm {r}[m]\times \{ii\}$.

In addition, define $B([0, 1])^+$ to be $[0]\boxtimes \mathrm {Fun}([1], [1])$, where all vertical arrows are marked.

Note that these vertically marked double categories are related by natural transformations as follows.

Here $\zeta \colon \mathbb {A}\mathrm {r}[m]\times [0, n]\longrightarrow [m, n]$ just arises from the horizontal projection $\mathbb {A}\mathrm {r}[m]\longrightarrow [m, 0]$. This sends the vertical arrows within each $\mathbb {A}\mathrm {r}[m]\times \{i\}$ to degenerate arrows and, hence, preserves markings. The map $\eta \colon \mathbb {A}\mathrm {r}[m]\times [0, n]\longrightarrow \mathbb {A}\mathrm {r}[m]\times \big ([0]\boxtimes \mathrm {Fun}([1], [n])\big )$ arises from the map $\mathrm {cst}\colon [n]\longrightarrow \mathrm {Fun}([1], [n])$ sending each $i\in [n]$ to the constant arrow $i\leq i$.

Furthermore, the maps $\zeta$ and $\eta$ induce natural maps $[0, 1]^\sharp \longleftarrow A[0, 1]^+\longrightarrow B[0, 1]^+$ preserving the additional marked vertical arrows. Using the naturality of Construction 3.10, we obtain a zigzag of natural transformations between functors $\mathrm {Fun}^\sharp ([1], \mathrm {bis}\mathscr {S}_{\downarrow })\longrightarrow \mathrm {Fun}^\sharp ([1], \mathrm {bis}\mathscr {S}_{\downarrow })$ of the form

Proof. Recall from the general description (3.12) that the space of $(m, n)$-simplices in $\Psi _A(\mathbb {D}^\natural )$ is given by the space of diagrams of vertically marked double categories

and likewise for $\Psi _B(\mathbb {D}^\natural )$. The maps $\zeta ^*$ and $\eta ^*$ simply arise by restriction along the natural maps $\zeta \colon A[m, n]\longrightarrow [m, n]$ and $\eta \colon A[m, n]\longrightarrow B[m, n]$.

$\zeta ^*$ is an equivalence. Note that $\Psi _A(\mathbb {D}^\natural )\hookrightarrow X$ is a bisimplicial subspace of the bisimplicial space $X$ from Lemma 2.18. Indeed, the space of maps $[m, n]\longrightarrow \Psi _A(\mathbb {D}^\natural )$ is the subspace of those diagrams (2.19) such that $\mathbb {A}\mathrm {r}[m]\times [0, n]\longrightarrow \mathbb {D}$ sends the vertical arrows in each $\mathbb {A}\mathrm {r}[m]\times \{i\}$ with $0\leq i\leq n$ to equivalences.

By Lemma 2.18, $X$ is double category and one sees that each simplicial subspace $\Psi _A(\mathbb {D}^\natural )(m, -)\hookrightarrow X(m, -)$ is the inclusion of a full subcategory. In the horizontal direction, the proof of Lemma 2.18 shows that the simplicial subspace $\Psi _A(\mathbb {D}^\natural )(-, n)\hookrightarrow X(-, n)$ satisfies the Segal conditions and is an equivalence on spaces of objects. This implies that $\Psi _A(\mathbb {D}^\natural )(-, n)$ is a category, so that $\Psi _A(\mathbb {D}^\natural )$ is a double category.

Let us now show that the map of marked double categories $\zeta ^*\colon \mathbb {D}\longrightarrow \Psi _A(\mathbb {D}^\sharp )$ is an equivalence. Since $\zeta$ induces an isomorphism $A[0, 1]^+\stackrel {\sim }{\longrightarrow } [0, 1]^\sharp$, $\zeta ^*$ is an equivalence on marked vertical arrows. It then remains to verify that the map on $(1,1)$-simplices is an equivalence. Unraveling the definitions, a $(1,1)$-simplex in $\Psi _A(\mathbb {D}^\natural )$ is a diagram in $\mathbb {D}$

where the right square covers a vertical arrow in $\mathbb {C}$. The map $\zeta ^*\colon \mathbb {D}(1, 1)\longrightarrow \Psi _A(\mathbb {D})(1, 1)$ is simply the inclusion of the diagrams of the above form where the right square is degenerate; this is an equivalence by completeness of $\mathbb {D}$.

$\eta ^*$ is an equivalence. Let us again consider the double category $X$ from Lemma 2.18. For each fixed $n$, consider the mapping double category $X^{[0]\boxtimes \mathrm {Fun}([1], [n])}$ and note that there is a diagram of simplicial spaces

where $P$ denotes the pullback. Since the bottom map is the inclusion of a wide subcategory (as we have seen above), $P$ is a wide subcategory of $X^{[0]\boxtimes \mathrm {Fun}([1], [n])}\big (-, 0\big )$. Unraveling the definitions, we then see that the inclusion $\Psi _B(\mathbb {D}^\natural )(-, n)\hookrightarrow P$ is the inclusion of a full subcategory: it is the inclusion of the full subcategory on those diagrams $[0]\boxtimes \mathrm {Fun}([1], [n])\longrightarrow X$ such that each vertical arrow $ik\longrightarrow jk$ in $\mathrm {Fun}([1], [n])$ is sent to a cartesian arrow in $X(0, -)\simeq \mathbb {D}(0, -)$.

It follows that $\Psi _B(\mathbb {D}^\natural )$ satisfies the (complete) Segal conditions in the horizontal direction. It therefore suffices to verify that $\eta ^*\colon \Psi _B(\mathbb {D}^\natural )(1, -)\longrightarrow \Psi _A(\mathbb {D})(1, -)$ is an equivalence of simplicial spaces. Unraveling the definitions, a $(1, n)$-simplex in $\Psi _B(\mathbb {D}^\natural )$ is given by the following diagram.

(3.20)

Here each of the slices labeled by $x, y$ and $z$ is a diagram in $\mathbb {D}(0, -)$ parametrized by $\mathrm {Fun}([1], [n])$, while the maps define a map $\mathbb {A}\mathrm {r}[1]\longrightarrow \mathbb {D}$. Furthermore, the condition that $B[1, n]=\mathbb {A}\mathrm {r}[1]\times \Big ([0]\boxtimes \mathrm {Fun}([1], [n])\Big )\longrightarrow \mathbb {D}$ preserves marked arrows translates into the following conditions.

1. (a) The maps are equivalences for all $0\leq i\leq n$, as indicated.

2. (b) All downward maps in the $x$-slice and the $z$-slice are $p$-cartesian, indicated by .

The map $\eta ^*\colon \Psi _B(\mathbb {D}^\natural )(1, n)\longrightarrow \Psi _A(\mathbb {D}^{\natural })(1, n)$ restricts the above diagram to the diagonal face, spanned by all $x_{ii}, y_{ii}$ and $z_{ii}$. We have to prove that there is a unique way to extend the above diagram (with properties (a) and (b)) from this diagonal face (covering the map $[m, n]\longrightarrow \mathbb {C}$). Let us first look at the $x$- and $z$-slices, both of which correspond to diagrams as follows.

Let $\mathrm {cst}\colon [n]\hookrightarrow \mathrm {Fun}([1], [n])$ be the inclusion of the diagonal. For every $ij\in \mathrm {Fun}([1], [n])$, the comma category $ij/\mathrm {cst}$ admits an initial object, corresponding to the map $ij\longrightarrow jj$. Using this, one sees that condition (b) is equivalent to the assertion that the two diagrams $x, z\colon \mathrm {Fun}([1], [n])\longrightarrow \mathbb {D}$ are relative right Kan extensions of their restriction to the diagonal.

Consequently, starting from the diagonal face there is a unique way to fill the $x$- and $z$-slices in (3.20) so that condition (b) holds [Reference LurieLur09a, Proposition 4.3.2.1]. Using that $\mathbb {D}\longrightarrow \mathbb {C}$ is a left fibration in the horizontal direction, there is then a unique extension to a horizontal transformation between the $x$- and $y$-slices $(x\Rightarrow y)\colon [1]\boxtimes \mathrm {Fun}([1], [n])\longrightarrow \mathbb {D}$. Finally, the natural transformation on the diagonal then extends uniquely to a natural transformation , since the target is the relative right Kan extension of its restriction to the diagonal. We conclude that $\eta ^*\colon \Psi _B(\mathbb {D}^\sharp )\longrightarrow \Psi _A(\mathbb {D}^\sharp )$ is an equivalence on the underlying bisimplicial spaces. Finally, it identifies the marking since $A[0, 1]^+\simeq B[0, 1]^+\simeq [0, 1]^\sharp$.

Construction 3.21 For each $m, n\geq 0$, let $I[m, n]$ denote the $(1, 1)$-category whose objects are quadruples of maps of linear orders

(3.22)\begin{equation} [a]\stackrel{\alpha}{\longrightarrow} [m], \quad [b]\star [a]\stackrel{\beta}{\longrightarrow} [n], \quad[t]\star [s]\stackrel{\gamma}{\longrightarrow} [a],\quad [t]\stackrel{\delta}{\longrightarrow} [b]. \end{equation}

Given two such quadruples, a map between them is given by a quadruple of maps

\[ [a]\longrightarrow [a'], \quad [b]\longrightarrow [b'], \quad [s]\longrightarrow [s'], \quad [t]\longrightarrow [t'] \]

that intertwines $(\alpha, \beta, \gamma, \delta )$ and $(\alpha ', \beta ', \gamma ', \delta ')$. Note that $I[m, n]$ depends functorially on $[m, n]\in \mathbf {\Delta }^{\times 2}$, by postcomposing $\alpha$ and $\beta$.

Now consider the natural map of bisimplicial spaces (in fact, sets)

defined as follows: to each $[s, t]\longrightarrow T[m, n]$ determined by a tuple $(\alpha, \beta, \gamma, \delta )$, we associate the map $\sigma \boxtimes \tau _0\colon [s, t]\longrightarrow [m, n]$ determined by

\[ \sigma\colon [s] \longrightarrow [t]\star [s] \stackrel{\gamma}{\longrightarrow} [a]\stackrel{\alpha}{\longrightarrow} [m], \quad \tau_0\colon [t]\stackrel{\delta}{\longrightarrow} [b]\longrightarrow [b]\star[a]\stackrel{\beta}{\longrightarrow} [n]. \]

We view $T[m, n]$ as a marked bisimplicial space by marking those $(0, 1)$-simplices corresponding to a quadruple $(\alpha, \beta, \gamma, \delta )$ (with $s=0, t=1$) such that either:

1. (a) the map $\gamma \colon [1]\star [0]\longrightarrow [a]$ factors over a $0$-simplex of $[a]$;

2. (b) the map $\beta \colon [b]\star [a]\longrightarrow [n]$ factors over a $0$-simplex of $[n]$.

For every map $[m, n]\longrightarrow [m', n']$, the induced map $T[m, n]\longrightarrow T[m', n']$ preserves such marked arrows.

Finally, let $T[0, 1]^+$ denote $T[0, 1]$, where all vertical arrows are marked.

Proof. Recall from Construction 3.10 that $\Psi _{K'}$ arises as the right adjoint to a functor $\Phi _{K'}$. This functor $\Phi _{K'}\colon \mathrm {Fun}^\sharp \big ([1], \mathrm {bis}\mathscr {S}_{\downarrow }\big )\longrightarrow \mathrm {Fun}^\sharp \big ([1], \mathrm {bis}\mathscr {S}_{\to }\big )$ is uniquely determined by the fact that it preserves colimits and sends $\emptyset \longrightarrow [m, n]^\sharp$ to $\emptyset \longrightarrow [m, n]^\sharp$ and

Here $K'[m, n]$ was described in Variant 3.16. Using Example 2.6, we can write $K'[m, n]$ as a (canonical) colimit of bisimplices

\[ K'[m, n]=\mathop {{\rm colim}}\limits_{_{\substack{\alpha\colon [a]\to [m]\\ \beta\colon [b]\star [a]\to [n]}}} [a, b]. \]

Unraveling the definitions, the marked $(1, 0)$-simplices of $K'[m, n]$ correspond precisely to tuples of $\alpha \colon [1]\longrightarrow [m]$ and $\beta \colon [0]\star [1]\longrightarrow [n]$ where $\beta$ is degenerate. Furthermore, all $(1, 0)$-simplices in $K'[0, 1]$ are marked.

We can apply the same reasoning to the left adjoint $\Phi _K\circ \Phi _{K'}$: this functor preserves colimits and sends $\emptyset \longrightarrow [m, n]^\sharp$ to $\emptyset \longrightarrow [m, n]^\sharp$. It is therefore uniquely determined by its values on $[m, n]\longrightarrow [m, n]^\sharp$ and $[0, 1]^\sharp \longrightarrow [0, 1]^\sharp$. The value on $[m, n]$ is given by

\[ \Phi_K\big(K'[m, n]\big)=\mathop {{\rm colim}}\limits_{_{\substack{\alpha\colon [a]\to [m]\\ \beta\colon [b]\star [a]\to [n]}}} K[a, b]\simeq \mathop {{\rm colim}}\limits_{_{\substack{\alpha\colon [a]\to [m]\\ \beta\colon [b]\star [a]\to [n]}}} \bigg(\mathop {{\rm colim}}\limits_{_{\substack{\gamma\colon [t]\star [s]\to [a]\\ \delta\colon [t]\to [b]}}} [s, t]\bigg). \]

In other words, it is precisely given by $T[m, n]$; unraveling the definitions shows that the marking coincides with that of Construction 3.21. Likewise, $\Phi _K\big (K'[0, 1]^+\big )$ can be identified with $T[0, 1]$ where all vertical arrows are marked. It follows that $\Phi _K\circ \Phi _{K'}\simeq \Phi _T$, so that by adjunction $\Psi _{T}\simeq \Psi _{K'}\circ \Psi _K$.

Proof (of Theorem 3.1) By Proposition 3.14 (and Variant 3.16), we have functors

\[ \Psi^\perp\colon \mathrm{Fib}^{\mathrm{left, cart}}\longrightarrow \mathrm{Fib}^{\mathrm{cocart, right}}\quad \text{and} \quad \Psi^\top\colon \mathrm{Fib}^{\mathrm{cocart, right}}\longrightarrow \mathrm{Fib}^{\mathrm{left, cart}}. \]

To see that they are mutually inverse, it suffices to prove that $\Psi ^{\top }\circ \Psi ^{\perp }\simeq \mathrm {id}$. Indeed, in this case, conjugating by the functor $\mathbb {C}\mapsto (\mathbb {C}^\mathrm {rev})^{(1, 2)-\mathrm {op}}$ provides the desired equivalence $\Psi ^\perp \circ \Psi ^\top \simeq \mathrm {id}$. We construct the desired natural equivalence as a zigzag

Here the first two equivalences follow from Proposition 3.19 and the last identification follows from Lemma 3.23. The remaining natural transformation $\theta ^*$ is induced by a natural map $\theta \colon T[m, n]\longrightarrow B[m, n]$ of vertically marked bisimplicial spaces over $[m, n]$. This map is defined as follows: for every $(s, t)$-simplex of $T[m, n]$, determined by a quadruple $(\alpha, \beta, \gamma, \delta )$ as in (3.22), consider the map

Clearly $\tau _0(i)\leq \tau _1(i)$ for all $i$, where $\tau _0$ is defined in Construction 3.21. We therefore obtain a map $(\tau _0\leq \tau _1)\colon [s, t]\longrightarrow [0]\boxtimes \mathrm {Fun}([1], [n])$. Likewise, note that $\alpha \circ \gamma$ defines a map $\alpha \circ \gamma \colon [s, t]\longrightarrow \mathbb {A}\mathrm {r}[m]$ (cf. Example 2.6). Combining these two, we obtain a natural map

for every $(s, t)$-simplex in $T[m, n]$. This defines the desired natural map of bisimplicial spaces $\theta \colon T[m, n]\longrightarrow B[m, n]$. Note that $\theta$ sends a marked $(0, 1)$-simplex in $T[m, n]$ (Construction 3.21) to a marked $(0, 1)$-simplex in $B[m, n]$ (Construction 3.18). Finally, note that in both $T[0, 1]^\sharp$ and $B[0, 1]^\sharp$, all vertical arrows are marked, so $\theta$ induces a natural map $T[0, 1]^\sharp \longrightarrow B[0, 1]^\sharp$ as well.

For any (left, cart)-fibration $p\colon \mathbb {D}^\natural \longrightarrow \mathbb {C}^\sharp$, restriction along $\theta$ therefore defines the following natural map of marked bisimplicial spaces.

Since both downwards-pointing maps are (left, cart)-fibrations (with marked arrows being the cartesian arrows), it suffices to verify that $\theta ^*$ induces equivalences between spaces of vertices and vertical arrows. On vertices, note that $T[0, 0]\simeq B[0, 0]\simeq [0, 0]$, so that the spaces of objects of $\Psi _B(\mathbb {D}^\natural )$ and $\Psi _T(\mathbb {D}^\natural )$ are both equivalent to the space of objects of $\mathbb {D}$.

Unraveling the definitions, a vertical arrow in $\Psi _T(\mathbb {D}^\natural )\simeq \Psi ^\top \big (\Psi ^\perp (\mathbb {D}^\natural )\big )$ covering a vertical arrow in $\mathbb {C}^\sharp$ is given by a diagram in $\mathbb {D}$ of the form

Here the first two maps live in the fiber over $c$ and the vertical arrow is a cartesian lift of . Since $\mathbb {D}\longrightarrow \mathbb {C}$ was a horizontal left fibration, this means that the arrow $d_0\longrightarrow d_0'$ is an equivalence.

On the other hand, a vertical arrow in $\Psi _B(\mathbb {D}^\natural )$ is a sequence of vertical maps where the first map lives in the fiber over $c$ and is cartesian. The map $\theta ^*$ can be identified with the map sending this sequence to ; it is then an equivalence by the completeness of $\mathbb {D}$.

Notation 4.3 For a category $\mathscr {I}$, let us denote by $\lambda \colon \mathbf {\Delta }\times \mathscr {I}\longrightarrow \mathbf {\Delta }_{\mathscr {I}}$ the ($(\infty, 1)$-categorical) localization of $\mathbf {\Delta }\times \mathscr {I}$ at the morphisms in $\{0\}\times \mathscr {I}$. Recall that restriction and Kan extension determine adjoint pairs

that exhibit the category of presheaves on $\mathbf {\Delta }_{\mathscr {I}}$ both as a localization and a colocalization of the category of presheaves on $\mathbf {\Delta }\times \mathscr {I}$.

Proof. For each $([m], i)\in \mathbf {\Delta }\times \mathscr {I}$, let $h_{[m], i}=\mathrm {Map}\big (-, ([m], i)\big )$ be the representable presheaf. For assertion (1), consider the following (homotopy) pushout of (space-valued) presheaves.

Here the top map is given pointwise by the inclusion of the subset of maps $([n], j)\longrightarrow ([m], i)$ such that the map $[n]\longrightarrow [m]$ factors over $[0]$. Using this, it follows immediately that the presheaf $F$ takes values in sets and sends maps in $\{0\}\times \mathscr {I}$ to isomorphisms. Consequently, $F\simeq \lambda ^*G$ for some set-valued presheaf $G$ on $\mathbf {\Delta }_{\mathscr {I}}$.

On the other hand, the left Kan extension functor $\lambda _!$ sends the left vertical map to an equivalence, so that we find $h_{\lambda ([m], i)}\simeq \lambda _!h_{[m], i}\simeq \lambda _!F$. We therefore find that the representable presheaf $h_{\lambda ([m], i)}$ is equivalent to the set-valued presheaf $\lambda _!\lambda ^*G\simeq G$, so that $\mathbf {\Delta }_{\mathscr {I}}$ is a $(1, 1)$-category. In the special case where $[m]=0$, we see that $F$ is the terminal presheaf, so that the image of $([0], i)$ defines a terminal object in $\mathbf {\Delta }_{\mathscr {I}}$.

For assertion (2), let us abbreviate $Y=X\times _{i_*i^*X} i_*\big (X(0, t)\big )$. Since $i$ is fully faithful, $i^*i_*\simeq \mathrm {id}$ and the restriction of $Y$ along $i\colon \{0\}\times \mathscr {I}\longrightarrow \mathbf {\Delta }\times \mathscr {I}$ coincides with the constant presheaf with value $X(0, t)$. This means that $Y$ is contained in the essential image of $\lambda ^*$ or, equivalently, that $Y$ is a $\lambda ^*\lambda _*$-colocal object.

It therefore remains to verify that the projection $Y\longrightarrow X$ is a colocal equivalence, i.e. that it is sent to an equivalence by $\lambda _*$. Since $Y\longrightarrow X$ is the base change of the map $i_*(X(0, t))\longrightarrow i_*i^*X$, it suffices to verify that $\lambda _*i_*(X(0, t))\longrightarrow \lambda _*i_*i^*X$ is an equivalence. By part (1), the functor $\lambda i$ decomposes as $\mathscr {I}\longrightarrow \ast \longrightarrow \mathbf {\Delta }_{\mathscr {I}}$, where the second functor is the inclusion of the terminal object. The result then follows from the fact that the natural map

is an equivalence since $t$ was the terminal object of $\mathscr {I}$.

Notation 4.5 We inductively define $\mathbf {\Delta }_d$ by $\mathbf {\Delta }_0=\ast$ and $\mathbf {\Delta }_{d}=\mathbf {\Delta }_{\mathbf {\Delta }_{d-1}}$, so that there is the following composition of localization functors of $(1, 1)$-categories:

(4.6)

By construction, restriction along $\delta _d$ determines a fully faithful inclusion

whose essential image consists of $d$-fold simplicial spaces $X$ such that each $(d-k)$-fold simplicial space $X(\vec {n}_{k-1}, 0, -, \dots, -)$ is constant. We will systematically omit $\delta _d^*$ and simply identify presheaves on $\mathbf {\Delta }_d$ with $d$-fold simplicial spaces using this inclusion. Likewise, we can identify presheaves on $\mathbf {\Delta }_d$ with simplicial object in presheaves on $\mathbf {\Delta }_{d-1}$ which are constant in simplicial degree $0$, etc.

Proof. We proceed by induction, the case $d=1$ being clear. Now let us take $d>1$ and assume that $r_{d-1}$ sends $(d-1)$-fold categories to $d$-categories. Suppose that $X\colon \mathbf {\Delta }^{\times d, \mathrm {op}}\longrightarrow \mathscr {S}$ is a $d$-fold category and factor the map $\delta _d$ as

The right Kan extension along the functor $\mathrm {id}\times \delta _{d-1}$ sends $X$ to the simplicial object $X'\colon \mathbf {\Delta }^{\mathrm {op}}\longrightarrow \mathrm {Fun}(\mathbf {\Delta }_{d-1}^{\mathrm {op}}, \mathscr {S})$ given by $X'(n)=r_{d-1}(X(n))$. Each $X'(n)$ is then a $(d-1)$-category by inductive hypothesis and the simplicial object $X'$ satisfies the complete Segal conditions since $r_{d-1}$ is a right adjoint.

Next, we compute $r_d(X)=\lambda _*(X')$ using Lemma 4.4(2). Denoting by $i\colon \{0\}\times \mathbf {\Delta }_{d-1}\hookrightarrow \mathbf {\Delta }\times \mathbf {\Delta }_{d-1}$ the inclusion, we have that

\[ r_d(X)=X'\times_{i_*i^*X'} i_*(X'(\vec{0}_d)), \]

at least after precomposing with the localization $\lambda \colon \mathbf {\Delta }\times \mathbf {\Delta }_{d-1}\longrightarrow \mathbf {\Delta }_d$. The value of this fiber product at $[m]\in \mathbf {\Delta }$ is given by the presheaf on $\mathbf {\Delta }_{d-1}$ of the form

\[ r_d(X)(m, -) = X'(m, -)\times_{X'(0, -)^{\times m+1}} X'\big(0, \vec{0}_{d-1}\big)^{\times m+1}. \]

Here the map $X'(m, -)\longrightarrow X'(0, -)^{\times m+1}$ arises from the vertex inclusions $\{k\}\hookrightarrow [m]$ for all $k\in [m]$. This implies that $r_d(X)$ satisfies the Segal condition in the first simplicial direction (as the spine inclusions induce bijections on objects). Furthermore, each $r_d(X)(m)$ is a $(d-1)$-category, since all terms in the above pullback diagram are $(d-1)$-categories.

It remains to verify the completeness condition. By Remark 4.2, it suffices to verify the completeness condition after setting the last variables equal to $\vec {0}_{d-1}$. But in that case the above formula shows that $r_d(X)(-, \vec {0}_{d-1})\simeq X'(-, \vec {0}_{d-1})$, for which we already verified that it was a complete Segal space.

Construction 4.15 The $d$-categorical model structure on $\mathrm {sPSh}(\mathbf {\Delta }_d)=\mathrm {Fun}(\mathbf {\Delta }_d^{\mathrm {op}}, \mathrm {sSet})$ is the left proper, combinatorial model structure characterized uniquely by the following properties [Reference LurieLur09b, Proposition 1.5.4].

1. (1) The cofibrations are the monomorphisms and, in particular, every object is cofibrant.

2. (2) An object $X$ is fibrant if and only if it is fibrant in the injective model structure and it is a $d$-category model.

3. (3) A weak equivalence between $d$-category models is a levelwise weak equivalence of simplicial sets (in the Kan–Quillen model structure).

We will write $\mathsf {Cat}_d$ for the full subcategory of $\mathrm {sPSh}(\mathbf {\Delta }_d)$ on the injectively fibrant $d$-category models.

Construction 4.18 Let $\mathsf {CatAlg}_d\subseteq \mathrm {sPSh}(\mathbf {\Delta }_d)\subseteq \mathrm {Fun}\big (\mathbf {\Delta }^{\mathrm {op}}, \mathrm {sPSh}(\mathbf {\Delta }_{d-1})\big )$ denote the full subcategory spanned by the functors $X\colon \mathbf {\Delta }^{\mathrm {op}}\longrightarrow \mathsf {Cat}_{d-1}$ that are $d$-categorical algebra models, i.e. explicitly:

• – each $X(n)\colon \mathbf {\Delta }_{d-1}^{\mathrm {op}}\longrightarrow \mathrm {sSet}$ is an injectively fibrant $(d-1)$-category model and $X(0)$ is constant on a Kan complex;

• – $X$ satisfies the Segal conditions in the first simplicial variable.

We will say that a map $Y\longrightarrow X$ is a Dwyer–Kan equivalence if the corresponding map of categorical algebras is a Dwyer–Kan equivalence as in Remark 4.13. Furthermore, we will say that a map $Y\longrightarrow X$ is an isofibration if:

1. (1) each $Y(n)\longrightarrow X(n)$ is a fibration in $\mathsf {Cat}_{d-1}$, i.e. an injective fibration of simplicial presheaves on $\mathbf {\Delta }_{d-1}$;

2. (2) the induced map on homotopy categories $\mathrm {ho}(Y)\longrightarrow \mathrm {ho}(X)$ is an isofibration.

With these classes of maps, $\mathsf {CatAlg}_d$ becomes a category of fibrant objects as in Appendix A. A map $Y\longrightarrow X$ is an acyclic fibration if and only if each $Y(n)\longrightarrow X(n)$ is an injective fibration, $Y(0)\longrightarrow X(0)$ is surjective on $\pi _0$ and $Y(1)\longrightarrow X(1)\times _{X(0)^\times 2} Y(0)^{\times 2}$ is a weak equivalence. To factor a map $Y\longrightarrow X$ in $\mathsf {CatAlg}_{d}$ into a weak equivalence followed by a fibration, one can simply factor it into a weak equivalence followed by a fibration with respect to the $d$-categorical model structure on $\mathrm {sPSh}(\mathbf {\Delta }_d)$; such fibrations are isofibrations by the right lifting property against $\{0\}\longrightarrow H$, where $H$ is the simplicial set from § 2.

Construction 4.19 Let us write $\mathrm {Cat}(\mathsf {Cat}_{d-1})$ for the $(1, 1)$-category of categories (strictly) enriched over the cartesian monoidal category $\mathsf {Cat}_{d-1}$. The nerve provides a fully faithful functor $\mathrm {N}\colon {{\rm Cat}}(\mathsf {Cat}_{d-1})\hookrightarrow \mathsf {CatAlg}_d$, which sends an enriched category $\mathsf {C}$ to the simplicial diagram in $\mathsf {Cat}_{d-1}$ given by

\begin{align*} \mathrm{N}(\mathsf{C})(n) = \coprod_{c_0, \dots, c_n} \mathrm{Map}_{\mathsf{C}}(c_{n-1}, c_n)\times \dots\times \mathrm{Map}_{\mathsf{C}}(c_0, c_1). \end{align*}

In particular, $\mathrm {N}(\mathsf {C})(0)$ is simply the set of objects of $\mathsf {C}$. We will say that a map of enriched categories is a Dwyer–Kan equivalence, respectively, an isofibration, if its image under the nerve functor is such. This makes $\mathrm {Cat}(\mathsf {Cat}_{d-1})$ a category of fibrant objects.

More precisely, the category of categories enriched in $\mathrm {sPSh}(\mathbf {\Delta }_{d-1})$ carries a model structure [Reference LurieLur09a, Proposition A.3.2.4], since the $(d-1)$-categorical model structure on $\mathrm {sPSh}(\mathbf {\Delta }_{d-1})$ is excellent in the sense of [Reference LurieLur09a, Definition A.3.2.16]. It follows from [Reference LurieLur09a, Theorem A.3.2.24] that ${{\rm Cat}}(\mathsf {Cat}_{d-1})$ (with the above classes of Dwyer–Kan equivalences and isofibrations) is the category of fibrant objects associated to this model category.

Proof. The right inclusion $\mathsf {Cat}_d\longrightarrow \mathsf {CatAlg}_{d}$ has a homotopy inverse $L_\mathrm {DK}$, given by taking a fibrant replacement of a $d$-categorical algebra model in the $d$-categorical model structure. The resulting objects are then related by a Dwyer–Kan equivalence.

Furthermore, the composite functor $L_\mathrm {DK}\circ \mathrm {N}\colon \mathrm {Cat}(\mathsf {Cat}_{d-1})\longrightarrow \mathsf {Cat}_d$ can be identified with the composite

where $\mathrm {Seg}_{\mathrm {sPSh}(\mathbf {\Delta }_{d-1})}$ is the category of Segal precategories from [Reference LurieLur09b, § 2]. Here the first functor $\mathrm {N}$ is a right Quillen equivalence preserving weak equivalences [Reference LurieLur09b, Theorem 2.2.16, Remark 2.2.19]. The functor $\mathrm {UnPre}$ is a left Quillen equivalence preserving weak equivalences by [Reference LurieLur09b, Propositions 2.3.1 and 2.3.9 and Lemma 2.3.14]: this uses the fact that the inclusion $\mathrm {sPSh}(\mathbf {\Delta }_{d})\hookrightarrow \mathrm {Fun}\big (\mathbf {\Delta }^{\mathrm {op}}, \mathrm {sPSh}(\mathbf {\Delta }_{d-1})\big )$ is a Quillen equivalence between the $d$-categorical model structure and the ‘complete Segal model structure’ of [Reference LurieLur09b], which detects cofibrations and weak equivalences. The last functor takes a fibrant replacement, so it follows that $L_\mathrm {DK}\circ \mathrm {N}$ induces an equivalence on localizations.