# Kerodon

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### 4.8.3 Minimality Conditions

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be an integer. We then have the following implications (see Proposition 4.8.2.8):

$\xymatrix@R =50pt@C=50pt{ \textnormal{\operatorname{\mathcal{C}} is an (n,1)-category} \ar@ {=>}[d] \\ \textnormal{\operatorname{\mathcal{C}} is weakly n-coskeletal} \ar@ {=>}[d] \\ \textnormal{\operatorname{\mathcal{C}} is (n+1)-coskeletal} \ar@ {=>}[d] \\ \textnormal{\operatorname{\mathcal{C}} is locally (n-1)-truncated}. }$

Beware that, in general, none of these implications is reversible. However, the failure of reversibility can be measured using the minimality conditions introduced in ยง4.7.6.

Proposition 4.8.3.1. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Assume that, if $n \leq -2$, then $\operatorname{\mathcal{C}}$ is nonempty. Then:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is $(n+1)$-coskeletal if and only if it is locally $(n-1)$-truncated and minimal in dimensions $\geq n+2$ (see Definition 4.7.6.4).

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal if and only if it is locally $(n-1)$-truncated and minimal in dimensions $\geq n+1$.

$(3)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if it is locally $(n-1)$-truncated and minimal in dimensions $\geq n$.

We will give the proof of Proposition 4.8.3.1 at the end of this section. First, let us collect some consequences.

Corollary 4.8.3.2. Let $\operatorname{\mathcal{C}}$ be a minimal $\infty$-category and let $n \geq -1$ be an integer. Then $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if it is locally $(n-1)$-truncated.

Corollary 4.8.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n \geq -1$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated if and only if it is equivalent to an $(n,1)$-category.

Corollary 4.8.3.4. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is weakly $(n-1)$-coskeletal. Then $\operatorname{\mathcal{C}}$ is an $(n,1)$-category.

Corollary 4.8.3.5. 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 morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is weakly $(n-1)$-coskeletal.

Proof. For $n < 0$, the result is trivial (see Example 4.8.1.9). We will therefore assume that $n \geq 0$. It follows from Corollary 4.8.1.23 that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\}$ is also $(n,1)$-category; in particular, it is minimal in dimensions $\geq n$. Since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated (Corollary 4.8.1.20), it is locally $(n-2)$-truncated (Example 4.8.2.4). Applying Proposition 4.8.3.1, we conclude that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is weakly $(n-1)$-coskeletal. $\square$

Our proof of Proposition 4.8.3.1 will make use of some auxiliary results of independent interest. Recall that, if $\operatorname{\mathcal{C}}$ is a simplicial set, then the weak $n$-coskeleton $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is the simplicial subset of $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ given by the image of the tautological map $\operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ (see Notation 3.5.4.19).

Proposition 4.8.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $n$ be an integer, and let $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ denote the weak $n$-coskeleton of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The simplicial set $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is an $\infty$-category.

$(2)$

The tautological map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is an inner fibration of $\infty$-categories.

$(3)$

If $n \geq -1$, the functor $F$ exhibits exhibits $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ as a local $(n-1)$-truncation of $\operatorname{\mathcal{C}}$.

$(4)$

If $n \neq 0$, then $F$ is an isofibration of $\infty$-categories.

Proof. For $n < 0$, the weak coskeleton $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is either empty (if $n = -1$ and $\operatorname{\mathcal{C}}$ is empty) or isomorphic to $\Delta ^0$; in either case, assertions $(1)$ through $(4)$ are clear. We may therefore assume that $n \geq 0$. The map $F$ factors as a composition

$\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}}) \xrightarrow {F''} \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}),$

where $F''$ is a trivial Kan fibration (Proposition 3.5.4.22). Since $\operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}})$ is an $\infty$-category (Proposition 4.8.2.13), assertion $(1)$ follows from Proposition 1.5.5.11.

To prove $(2)$, we proceed as in Proposition 3.5.4.26. Suppose we are given a pair of integers $0 < i < m$; we wish to show that every lifting problem

4.80
$$\begin{gathered}\label{equation:fibration-to-strong-coskeleton5} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}}) } \end{gathered}$$

admits a solution. We consider two cases:

• If $m \leq n+1$, then we can choose an $m$-simplex $\sigma$ of $\operatorname{\mathcal{C}}$ satisfying $F(\sigma ) = \overline{\sigma }$. Since $F$ is bijective on simplices of dimension $\leq n$, the commutativity of the diagram (4.80) guarantees that $\sigma |_{ \Lambda ^{m}_{i} } = \sigma _0$.

• If $m \geq n+2$, then our assumption that $\operatorname{\mathcal{C}}$ is an $\infty$-category guarantees that $\sigma _0$ can be extended to an $m$-simplex $\sigma$ of $X$. The commutativity of the diagram (4.80) then guarantees that $F(\sigma )$ and $\overline{\sigma }$ have the same restriction to the horn $\Lambda ^{m}_{i} \subset \Delta ^{m}$. In particular, they have the same restriction to the $n$-skeleton of $\Delta ^{m}$, so $F(\sigma ) = \overline{\sigma }$.

Since $F''$ is an equivalence of $\infty$-categories, assertion $(3)$ follows by combining Proposition 4.8.2.13 with Remark 4.8.2.12. It remains to prove $(4)$. Let $Y$ be an object of $\operatorname{\mathcal{C}}$ and suppose we are given an isomorphism morphism $\overline{u}: \overline{X} \rightarrow Y$ in the $\infty$-category $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$. If $n \geq 1$, then $F$ is bijective on vertices and edges; it follows that we can write $\overline{u} = F(u)$ for a unique morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $u$ is an isomorphism. Equivalently, we wish to show that the homotopy class $[u]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. This is clear: the nerve $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is weakly $1$-coskeletal (Example 3.5.4.6), so the tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ factors (uniquely) through $\operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}})$ (Proposition 3.5.4.18). $\square$

Warning 4.8.3.7.. The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is generally not an isofibration in the case $n = 0$.

Warning 4.8.3.8. In the situation of Proposition 4.8.3.6, the map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}})$ is generally not an inner fibration.

Corollary 4.8.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n \geq -2$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if the tautological map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n+1}^{\circ }(\operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Proof. It follows from Proposition 4.8.3.6 that $F$ exhibits $\operatorname{cosk}^{\circ }_{n+1}(\operatorname{\mathcal{C}})$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$. Applying Proposition 4.8.2.14, we see that $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if $F$ is an equivalence of $\infty$-categories. We wish to show that if this condition is satisfied, then $F$ is a trivial Kan fibration. By virtue of Proposition 4.5.5.20, it will suffice to show that $F$ is an isofibration. For $n \neq -1$, this is automatic (Proposition 4.8.3.6). We will therefore assume that $n = -1$. Using Proposition 4.8.3.6, we see that $F$ is an inner fibration. Fix a morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in the $\infty$-category $\operatorname{cosk}_{0}^{\circ }(\operatorname{\mathcal{C}})$. Then there are unique objects $X,Y \in \operatorname{\mathcal{C}}$ satisfying $\overline{X} = F(X)$ and $\overline{Y} = F(Y)$. Choose a morphism $u: X \rightarrow Y$ satisfying $F(u) = \overline{u}$. To complete the proof, it will suffice to show that if $\overline{u}$ is an isomorphism in $\operatorname{cosk}_{0}^{\circ }(\operatorname{\mathcal{C}})$, then $u$ is an isomorphism in $\operatorname{\mathcal{C}}$. Let $\overline{v}: \overline{Y} \rightarrow \overline{X}$ be a homotopy inverse to $\overline{u}$. Then we can write $\overline{v} = F(v)$ for some morphism $v: Y \rightarrow X$ of $\operatorname{\mathcal{C}}$. Since the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ is either empty or contractible, the composition $v \circ u$ is automatically homotopic to $\operatorname{id}_{X}$: that is $v$ is a left homotopy inverse to $u$. A similar argument shows that $v$ is right homotopy inverse to $u$, so that $u$ is an isomorphism as desired. $\square$

Corollary 4.8.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n \geq 0$ be an integer. The following conditions are equivalent:

$(1)$

For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the set $\pi _{n}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), f)$ consists of a single element.

$(2)$

Every diagram $\operatorname{\partial \Delta }^{n+2} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$.

Proof. By virtue of Proposition 4.8.2.13, we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where the $\infty$-category $\operatorname{\mathcal{C}}$ is $(n+2)$-coskeletal. In this case, the $\infty$-category $\operatorname{\mathcal{C}}$ is locally $n$-truncated (Proposition 4.8.2.8), and satisfies condition $(1)$ if and only if it is locally $(n-1)$-truncated. Applying Corollary 4.8.3.9, we see that $(1)$ is equivalent to the following:

$(1')$

The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is a trivial Kan fibration.

The equivalence of $(1')$ and $(2)$ now follows from Corollary 3.5.4.24. $\square$

Corollary 4.8.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n \geq -2$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only 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 surjective for every integer $m \geq n+3$.

Proposition 4.8.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $m > 0$ be an integer. Then the restriction map

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

is injective if and only if the following conditions are satisfied:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is minimal in dimension $m$.

$(2)$

The restriction map $\theta _{m+1}$ is surjective.

Remark 4.8.3.13. In the case $m = 0$, the formulation of Proposition 4.8.3.12 requires a slight modification. The restriction map $\theta _0$ is injective if and only if $\operatorname{\mathcal{C}}$ satisfies the following pair of conditions:

$(1)$

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

$(2')$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an isomorphism from $X$ to $Y$.

Note that condition $(2')$ is stronger than condition $(2)$ of Proposition 4.8.3.12, which demands only that there exists a morphism from $X$ to $Y$.

Proof of Proposition 4.8.3.12. Let $\operatorname{\mathcal{C}}$ be an integer and let $m > 0$ be an integer. It follows immediately from the definitions that, if the restriction map

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

is injective, then $\operatorname{\mathcal{C}}$ is minimal in dimension $m$. We claim that, if this condition is satisfied, then $\theta _{m+1}$ is surjective: that is, every morphism $\tau _0: \operatorname{\partial \Delta }^{m+1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(m+1)$-simplex of $\operatorname{\mathcal{C}}$. Fix an integer $0 < i < m+1$. Our assumption that $\operatorname{\mathcal{C}}$ is an $\infty$-category guarantees that we can choose an $(m+1)$-simplex $\tau$ of $\operatorname{\mathcal{C}}$ satisfying $\tau |_{ \Lambda ^{m+1}_{i} } = \tau _0 |_{ \Lambda ^{m+1}_{i} }$. In particular, $\tau$ and $\tau _0$ have the same restriction to the $(m-1)$-skeleton of $\Delta ^{m+1}$. Invoking the injectivity of $\theta _{m}$, we conclude that $\tau |_{ \operatorname{\partial \Delta }^{m+1} } = \tau _0$.

We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ is minimal in dimension $m$ and that $\theta _{m+1}$ is surjective; we wish to show that $\theta _{m}$ is injective. Let $\sigma _0$ and $\sigma _1$ be $m$-simplices of $\operatorname{\mathcal{C}}$ which have the same restriction to $\operatorname{\partial \Delta }^{m}$; we wish to show that $\sigma _0 = \sigma _1$. Let

$X(0) \subset X(1) \subset \cdots \subset X(m) \subset X(m+1) = \Delta ^1 \times \Delta ^ m$

be the filtration of Lemma 3.1.2.12, so that $X(0) = (\Delta ^1 \times \operatorname{\partial \Delta }^ m) \cup ( \{ 1\} \times \Delta ^ m)$ and the inclusion map $X(i) \hookrightarrow X(i+1)$ is inner anodyne for $0 \leq i < m$. There is a unique morphism of simplicial sets $h_0: X(0) \rightarrow \operatorname{\mathcal{C}}$ such that $h_0 |_{ \{ 1\} \times \Delta ^ m }$ coincides with $\sigma _1$, and $h_0|_{ \Delta ^1 \times \operatorname{\partial \Delta }^{m} }$ factors through the projection map $\Delta ^1 \times \operatorname{\partial \Delta }^{m} \twoheadrightarrow \operatorname{\partial \Delta }^ m$. Since $\operatorname{\mathcal{C}}$ is an $\infty$-category, we can extend $h_0$ to a diagram $h_{m}: X(m) \rightarrow \operatorname{\mathcal{C}}$. Invoking the surjectivity of $\theta _{m+1}$, we see that $h_{m}$ can be extended to a morphism $h: \Delta ^1 \times \Delta ^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $h|_{ \{ 0\} \times \Delta ^ m } = \sigma _0$. By construction, $h$ is an isomorphism from $\sigma _0$ to $\sigma _1$ in the $\infty$-category $\operatorname{Fun}( \Delta ^ m, \operatorname{\mathcal{C}})$ whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ m, \operatorname{\mathcal{C}})$ is an identity morphism. Since $\operatorname{\mathcal{C}}$ is minimal in dimension $m$, it follows that $\sigma _0 = \sigma _1$. $\square$

Proof of Proposition 4.8.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $n$ be an integer. For every integer $m \geq 0$, we let $\theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}})$ denote the restriction map. By virtue of Corollary 4.8.3.11, the map $\theta _{m}$ is surjective for $m \geq n+2$ if and only if $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated (and nonempty if $n \leq -2$). Assume that these equivalent conditions are satisfied. Then:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is $(n+1)$-coskeletal if and only if $\theta _{m}$ is injective for $m \geq n+2$. By virtue of Proposition 4.8.3.12 (and Remark 4.8.3.13), this is equivalent to the requirement that $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n+2$.

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is weakly $n$-coskeletal if and only if $\theta _{m}$ is injective for $m \geq n+1$. By virtue of Proposition 4.8.3.12 (and Remark 4.8.3.13), this is equivalent to the requirement that $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n+1$.

$(3)$

The $\infty$-category $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if $\operatorname{\mathcal{C}}$ is minimal in dimensions $\geq n$. This follows immediately from $(2)$ (see Definition 4.8.1.8).

$\square$

We conclude this section by recording another consequence of Proposition 4.8.3.12.

Corollary 4.8.3.14. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. The following conditions are equivalent:

$(a)$

The Kan complex $X$ is $n$-reduced: that is, it has a single $m$-simplex for each $0 \leq m \leq n$ (Definition 3.5.2.8).

$(b)$

The Kan complex $X$ is $(n+1)$-connective and minimal in dimensions $\leq n$.

Proof. Without loss of generality, we may assume that $X$ is nonempty (otherwise, neither $(a)$ nor $(b)$ is satisfied). In this case, $X$ is $n$-reduced if and only if the restriction map $\theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}})$ is injective for each $m \leq n$. Corollary 4.8.3.14 now follows by combining Proposition 4.8.3.12 (and Remark 4.8.3.13) with the criterion of Proposition 3.5.1.12. $\square$

Corollary 4.8.3.15. Let $X$ be a minimal Kan complex and let $n \geq 0$ be an integer. Then $X$ is $n$-reduced if and only if it is $(n+1)$-connective.