Kerodon

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

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

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

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

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. $\square$

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

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 $Q$. $\square$

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 $\tau _0 = \tau _1$. $\square$

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). $\square$

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). $\square$

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

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. $\square$