# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 4.8.1 $(n,1)$-Categories

Recall that a simplicial set $\operatorname{\mathcal{C}}$ is (isomorphic to) the nerve of a category if and only if, for every pair of integers $0 < i < m$, the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m} , \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, \operatorname{\mathcal{C}})$

is a bijection. In this section, we study a hierarchy of weaker filling conditions.

Definition 4.8.1.1. Let $n$ be a positive integer. We say that a simplicial set $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if it satisfies the following condition for every pair of integers $0 < i < m$:

$(\ast )$

Every morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $m$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$. Moreover, if $m > n$, then $\sigma$ is unique.

Remark 4.8.1.2. Let $n$ be a positive integer and let $\operatorname{\mathcal{C}}$ be a simplicial set. If $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, then it is an $\infty$-category. Conversely, if $\operatorname{\mathcal{C}}$ is an $\infty$-category, then it is an $(n,1)$-category if and only if, for every pair of integers $0 < i < m$ with $m > n$, the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, \operatorname{\mathcal{C}})$ is injective.

Example 4.8.1.3. An $\infty$-category $\operatorname{\mathcal{C}}$ is a $(1,1)$-category if and only if it is isomorphic to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$, for some category $\operatorname{\mathcal{C}}_0$. By virtue of Proposition 4.8.1.7, this is a restatement of Proposition 1.3.4.1. Note that in this case, the category $\operatorname{\mathcal{C}}_0$ is well-defined up to unique isomorphism (Proposition 1.3.3.1).

Example 4.8.1.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category, and suppose that every $2$-morphism in $\operatorname{\mathcal{C}}$ is an isomorphism. Then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category, in the sense of Definition 4.8.1.8. This follows by combining Propositions 4.8.1.7 and 2.3.3.1.

Warning 4.8.1.5. We have now given several a priori different definitions for the notion of $(2,1)$-category:

$(1)$

According to Definition 2.2.8.5, a $(2,1)$-category is a $2$-category $\operatorname{\mathcal{C}}$ in which every $2$-morphism is an isomorphism.

$(2)$

According to Definition 4.8.1.8 (or Definition 4.8.1.1), a $(2,1)$-category is a simplicial set which satisfies some additional conditions.

However, these definitions are compatible with one another. If $\operatorname{\mathcal{C}}$ is a $(2,1)$-category in the sense of $(1)$, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category in the sense of $(2)$ (Example 4.8.1.4). We will later see that the converse is also true: every $(2,1)$-category in the sense $(2)$ is isomorphic to the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a $2$-category in which every $2$-morphism is an isomorphism (in this case, Theorem 2.3.4.1 guarantees that $\operatorname{\mathcal{C}}$ is unique up to non-strict isomorphism). See Proposition .

Exercise 4.8.1.6. Let $n$ be a positive integer and let $\operatorname{\mathcal{C}}$ be a differential graded category which satisfies the following condition:

• For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ is concentrated in degrees $< n$: that is, the abelian groups $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{m}$ vanish for $m \geq n$.

Show that the differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is an $(n,1)$-category (see the proof of Theorem 2.5.3.10).

It will be useful to work with a reformulation of Definition 4.8.1.1. Recall that:

• A simplicial set $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal if the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ m, \operatorname{\mathcal{C}})$

is a bijection for $m \geq n+2$ and an injection for $m = n+1$ (Definition 3.5.4.1).

• An $\infty$-category $\operatorname{\mathcal{C}}$ is minimal in dimension $n$ if, whenever $\sigma$ and $\tau$ are $n$-simplices of $\operatorname{\mathcal{C}}$ which are isomorphic relative to $\operatorname{\partial \Delta }^ n$, then $\sigma = \tau$ (Definition 4.7.6.4).

Proposition 4.8.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be a positive integer. Then $\operatorname{\mathcal{C}}$ is an $(n,1)$-category (in the sense of Definition 4.8.1.1) if and only if it is weakly $n$-coskeletal and minimal in dimension $n$.

Proof. We proceed as in the proof of Proposition 3.5.5.12. Assume first that $\operatorname{\mathcal{C}}$ is an $(n,1)$-category. Then, for any integer $m > n$, the composition of the restriction maps

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, \operatorname{\mathcal{C}}) \xrightarrow {\theta _ m} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{1}, \operatorname{\mathcal{C}})$

is a bijection. In particular, $\theta _ m$ is an injection. To show that $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal, it will suffice to show that $\theta _{m}$ is surjective for $m \geq n+2$. Fix a morphism $\sigma _0: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, there is a unique $m$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$ satisfying $\sigma |_{ \Lambda ^{m}_{1} } = \sigma _0|_{ \Lambda ^{m}_{1} }$. We complete the argument by observing that $\sigma |_{ \operatorname{\partial \Delta }^{m} } = \sigma _0$, by virtue of the injectivity of the map $\theta _{m-1}$.

We next show that if $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, then it is minimal in dimension $n$. Let $\sigma _0, \sigma _1: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be $n$-simplices of $\operatorname{\mathcal{C}}$ and let $h: \Delta ^1 \times \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be a natural isomorphism from $\sigma _0 = h|_{ \{ 0\} \times \Delta ^ n}$ to $\sigma _1 = h|_{ \{ 1\} \times \Delta ^ n }$ whose restriction to $\Delta ^1 \times \operatorname{\partial \Delta }^ n$ factors through $\operatorname{\partial \Delta }^ n$; we wish to show that $\sigma _0 = \sigma _1$. For $0 \leq i \leq n$, let $\alpha _{i}: [n+1] \rightarrow [1] \times [n]$ denote the nondecreasing function given by the formula

$\alpha _{i}(j) = \begin{cases} (0, j) & \text{ if } j \leq i \\ (1, j-1) & \text{ if } j > i, \end{cases}$

and let $\tau _{i}$ denote the $(n+1)$-simplex of $\operatorname{\mathcal{C}}$ given by the composition

$\Delta ^{n+1} \xrightarrow { \alpha _{i} } \Delta ^1 \times \Delta ^ n \xrightarrow {h} \operatorname{\mathcal{C}}.$

Let $\rho _{i}, \rho '_{i}: \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ be the $n$-simplices of $X$ given by $\rho _{i} = d^{n+1}_{i}( \tau _ i )$ and $\rho '_{i} = d^{n+1}_{i+1}( \tau _ i )$; by construction, we have

$\sigma _0 = \rho '_{n} \quad \quad \rho _{n} = \rho '_{n-1} \quad \quad \cdots \quad \quad \rho _{1} = \rho '_{0} \quad \quad \rho _0 = \sigma _1.$

We will complete the proof by showing that $\rho _{i} = \rho '_{i}$ for $0 \leq i \leq n$. We will treat the case $i > 0$ (the case $i < n$ follows by a similar argument). Using our assumption that $h$ is constant along the boundary $\operatorname{\partial \Delta }^{n}$, we see that the degenerate $(n+1)$-simplex $s^{n}_{i}( \rho _{i} )$ coincides with $\tau _ i$ on the horn $\Lambda ^{n+1}_{i} \subset \Delta ^{n+1}$. Since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, it follows that $\tau _ i = s^{n}_{i}( \rho _{i} )$. Applying the face operator $d^{n+1}_{i+1}$, we obtain $\rho _{i} = \rho '_{i}$.

We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ is a weakly $n$-coskeletal $\infty$-category which is minimal in dimension $n$; we will show that it is an $(n,1)$-category. Fix a pair of integers $0 < i < m$ with $m > n$ and a pair of $m$-simplices $\tau _0, \tau _1: \Delta ^{m} \rightarrow X$ which coincide on the horn $\Lambda ^{m}_{i} \subset \Delta ^{m}$; we wish to show that $\tau _0 = \tau _1$. Since $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal, it will suffice to prove that $\tau _0$ and $\tau _1$ coincide on the boundary $\operatorname{\partial \Delta }^{m}$: that is, to show that the $(m-1)$-simplices $\sigma _0 = d^{m}_{i}(\tau _0)$ and $\sigma _1 = d^{m}_{i}(\tau _1)$ coincide. Note that $\sigma _0$ and $\sigma _1$ have the same restriction to the boundary $\operatorname{\partial \Delta }^{m-1}$. Consequently, if $m \geq n+2$, the desired result follows from our assumption that $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal. We may therefore assume that $m = n+1$. Since $\operatorname{\mathcal{C}}$ is minimal in dimension $n$, it will suffice to show that there is an isomorphism from $\sigma _0$ to $\sigma _1$ (in the $\infty$-category $\operatorname{Fun}(\Delta ^ n, \operatorname{\mathcal{C}})$) whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})$ is an identity morphism. In fact, we will prove a stronger claim: there is an isomorphism from $\tau _0$ to $\tau _1$ in the $\infty$-category $\operatorname{Fun}( \Delta ^{m}, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}( \Lambda ^{m}_{i}, \operatorname{\mathcal{C}})$ is an identity morphism. This follows from the observation that the restriction map $\operatorname{Fun}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Lambda ^{m}_{i}, \operatorname{\mathcal{C}})$ is a trivial Kan fibration; see Proposition 1.5.7.6. $\square$

Motivated by Proposition 4.8.1.7, we introduce a generalization of Definition 4.8.1.1.

Definition 4.8.1.8. Let $n$ be an integer. We say that a simplicial set $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if it is an $\infty$-category which is weakly $n$-coskeletal and (if $n \geq 0$) minimal in dimension $n$.

For $n > 0$, Definitions 4.8.1.8 and 4.8.1.1 are equivalent: this is the content of Proposition 4.8.1.7. The advantage of Definition 4.8.1.8 is that it also makes sense for $n \leq 0$. However, for $n < 0$ it is rather trivial:

Example 4.8.1.9. A simplicial set $\operatorname{\mathcal{C}}$ is a $(-1,1)$-category if and only if it is either empty or isomorphic to $\Delta ^0$. See Example 3.5.4.4.

Example 4.8.1.10. For $n \leq -2$, a simplicial set $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only it is isomorphic to $\Delta ^0$. See Example 3.5.4.3.

Example 4.8.1.11. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. Then $X$ is an $n$-groupoid (in the sense of Definition 3.5.5.1) if and only if it is an $(n,1)$-category (in the sense of Definition 4.8.1.8). This is a reformulation of Proposition 3.5.5.12.

Remark 4.8.1.12 (Monotonicity). Let $\operatorname{\mathcal{C}}$ be an $(m,1)$-category for some integer $m$. Then $\operatorname{\mathcal{C}}$ is an $(n,1)$-category for every integer $n \geq m$.

Remark 4.8.1.13 (Symmetry). Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is also an $(n,1)$-category.

Remark 4.8.1.14. Let $n$ be an integer and let $\{ \operatorname{\mathcal{C}}_ j \} _{j \in J}$ be a collection of $(n,1)$-categories. Then the product $\operatorname{\mathcal{C}}= \prod _{j \in J} \operatorname{\mathcal{C}}_{j}$ is also an $(n,1)$-category.

Proposition 4.8.1.15. A simplicial set $\operatorname{\mathcal{C}}$ is a $(0,1)$-category if and only if there exists an isomorphism $\operatorname{\mathcal{C}}\simeq \operatorname{N}_{\bullet }(Q)$, for some partially ordered set $(Q, \leq )$.

Proof. By virtue of Remark 4.8.1.12, we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is a $(1,1)$-category: that is, it is isomorphic to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$, for some category $\operatorname{\mathcal{C}}_0$ (see Example 4.8.1.3). In this case, $\operatorname{\mathcal{C}}$ is a $(0,1)$-category if and only if it satisfies the following additional conditions:

$(0)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is minimal in dimension $0$: that is, if $X$ and $Y$ are isomorphic objects of $\operatorname{\mathcal{C}}_0$, then $X = Y$.

$(1)$

The restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$ is injective: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}_0$, there is at most one morphism from $X$ to $Y$.

$(2)$

The restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^2, \operatorname{\mathcal{C}})$ is bijective: that is, every diagram

$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z }$

in the category $\operatorname{\mathcal{C}}_0$ is automatically commutative.

Note that conditions $(1)$ and $(2)$ are equivalent to one another: they both assert that $\operatorname{\mathcal{C}}_0$ can be recovered from the set of objects $Q = \operatorname{Ob}(\operatorname{\mathcal{C}})$, endowed with the reflexive and transitive relation $\leq _{Q}$ defined by

$( X \leq _{Q} Y ) \Leftrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}(X,Y) \neq \emptyset .$

In this case, condition $(0)$ is satisfied if and only if the relation $\leq _{Q}$ is also antisymmetric: that is, the relation $\leq _{Q}$ is a partial ordering of $Q$. $\square$

Remark 4.8.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset which is also an $\infty$-category. If $\operatorname{\mathcal{C}}$ is an $(n,1)$-category for some integer $n \geq -1$, then $\operatorname{\mathcal{C}}_0$ is also an $(n,1)$-category. For $n \geq 1$, this follows from Remark 4.8.1.2. The case $n = 0$ follows from Proposition 4.8.1.15 (since any subcategory of a partially ordered set is also a partially ordered set), and the case $n = -1$ is trivial (see Example 4.8.1.9).

Example 4.8.1.17. Let $n \geq 0$ and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is an $n$-groupoid. This follows by combining Remark 4.8.1.16 with Example 4.8.1.11.

Remark 4.8.1.18. Let $\{ \operatorname{\mathcal{C}}_{j} \} _{j \in \operatorname{\mathcal{J}}}$ be a diagram of simplicial sets having limit $\operatorname{\mathcal{C}}= \varprojlim _{j \in \operatorname{\mathcal{J}}} \operatorname{\mathcal{C}}_ j$ and let $n$ be an integer. If each $\operatorname{\mathcal{C}}_{j}$ is an $(n,1)$-category and $\operatorname{\mathcal{C}}$ is an $\infty$-category, then $\operatorname{\mathcal{C}}$ is also an $(n,1)$-category. For $n \leq -2$, this is trivial (Example 4.8.1.10). The case $n \geq -1$ follows from Remarks 4.8.1.14 and 4.8.1.16, since $\operatorname{\mathcal{C}}$ can be identified with a simplicial subset of the product $\prod _{j \in J} \operatorname{\mathcal{C}}_{j}$.

Proposition 4.8.1.19. Let $n$ be a positive integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ are $(n-1)$-groupoids.

Proof. We will show that the right-pinched morphism space $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ is an $(n-1)$-groupoid; the analogous statement for the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ follows by a similar argument. If $n = 1$, then $\operatorname{\mathcal{C}}$ can be identified with the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$ (Example 4.8.1.3) and the desired result follows from Example 4.6.5.12. We may therefore assume that $n > 1$. Since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is a Kan complex, it will suffice to show that it is an $(n-1,1)$-category (Example 4.8.1.11). Let $m \geq n$ and let $\sigma _0$ and $\sigma _1$ be $m$-simplices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ which satisfy $\sigma _0|_{ \Lambda ^{m}_{i} } = \sigma _1|_{ \Lambda ^{m}_{i} }$ for some $0 < i < m$; we wish to show that $\sigma _0 = \sigma _1$. Let us identify $\sigma _0$ and $\sigma _1$ with morphisms $\tau _0, \tau _1: \Delta ^{m+1} \rightarrow \operatorname{\mathcal{C}}$ which carry the simplicial subset $\Delta ^{m} \subset \Delta ^{m+1}$ to the object $X$ and the final vertex of $\Delta ^{m+1}$ to the object $Y$. Our assumptions then guarantee that $\tau _0$ and $\tau _1$ have the same restriction to $\Lambda ^{m+1}_{i}$. Since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category, it follows that $\tau _0 = \tau _1$. $\square$

Corollary 4.8.1.20. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated.

Proof. For $n \leq -1$, there is nothing to prove (see Examples 4.8.1.10 and 4.8.1.9). The case $n = 0$ follows from Proposition 4.8.1.15. We may therefore assume $n > 0$. By virtue of Proposition 4.6.5.10, it will suffice to show that the pinched morphism space $K = \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated. This follows from Example 3.5.7.2, since $K$ is an $(n-1)$-groupoid (Proposition 4.8.1.19). $\square$

Warning 4.8.1.21. Let $n \geq 1$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Corollary 4.8.1.20 guarantees that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated, and is therefore homotopy equivalent to an $(n-1)$-groupoid (for example, it is homotopy equivalent to the pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$). Beware that, for $n \geq 2$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ itself is generally not an $(n-1)$-groupoid. For example, this usually fails in the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}_0)$ arises as the Duskin nerve of a $2$-category $\operatorname{\mathcal{C}}_0$: see Remark 8.1.8.8. However, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is always weakly $(n-1)$-coskeletal: see Corollary 4.8.3.5.

Proof. It follows from Theorem 1.5.3.7 that $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is an $\infty$-category. Since $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal, the $\infty$-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is also weakly $n$-coskeletal (Corollary 3.5.4.13). To complete the proof, it will suffice to show that if $n \geq 0$, then $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is minimal in dimension $n$ (Proposition 4.8.1.7). Suppose we are given a pair of $n$-simplices $\sigma _0, \sigma _1: \Delta ^ n \rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ and an isomorphism $\sigma _0 \xrightarrow {\sim } \sigma _1$ whose restriction to $\operatorname{\partial \Delta }^ n$ is an identity morphism; we wish to show that $\sigma _0 = \sigma _1$. Let us identify $\sigma _0$ and $\sigma _1$ with diagrams $f_0, f_1: \Delta ^ n \times K \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal, it will suffice to show that $f_0$ and $f_1$ coincide on $m$-simplices $\tau = (\tau ', \tau '')$ of $\Delta ^ n \times K$ for $m \leq n$. If $\tau '$ factors through the boundary $\operatorname{\partial \Delta }^ n$, this follows immediately from the equality $\sigma _0|_{\operatorname{\partial \Delta }^ n} = \sigma _1|_{ \operatorname{\partial \Delta }^ n}$. We may therefore assume without loss of generality that $m = n$ and that $\tau ': \Delta ^ m \rightarrow \Delta ^ n$ is the identity map. In this case, our assumption guarantees that there is an isomorphism of $f_0( \tau )$ with $f_1(\tau )$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$ is an identity morphism. The equality $f_0( \tau ) = f_1(\tau )$ now follows from the fact that $\operatorname{\mathcal{C}}$ is minimal in dimension $n$ (Proposition 4.8.1.7). $\square$

Corollary 4.8.1.23. Let $n$ be an integer and let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $(n,1)$-categories. Then the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ and the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ are also $(n,1)$-categories.

Proof. By definition, the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ can be realized as an iterated fiber product

$\operatorname{\mathcal{C}}_0 \times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}_1,$

which is an $(n,1)$-category by virtue of Propositions 4.8.1.22 and Remark 4.8.1.18. The homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a full subcategory of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, which coincides with $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ for $n \leq -1$. Applying Remark 4.8.1.16, we see that it is also an $(n,1)$-category. $\square$