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4.8.9 Categorically Connective Morphisms of Simplicial Sets

Using Theorem 4.8.8.3, we can give an alternative characterization of categorical connectivity.

Proposition 4.8.9.1. Let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ be a functor of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The functor $F$ is categorically $n$-connective (Definition 4.8.7.1).

$(2)$

For every essentially $(n-1)$-categorical functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \ar [r]^-{\circ F} \ar [d]^{U \circ } & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \ar [d]^{ U \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \]

is a categorical pullback square.

$(3)$

For every $(n-1)$-categorical isofibration $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$, precomposition with $F$ induces an equivalence of $\infty $-categories

\[ \theta _{\operatorname{\mathcal{C}}}: \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}). \]

Proof. The implication $(1) \Rightarrow (2)$ is a restatement of Corollary 4.8.7.18, and the implication $(2) \Rightarrow (3)$ follows from Corollary 4.5.2.32. To show that $(3)$ implies $(1)$, we may assume without loss of generality that $F$ is an isofibration. Then the comparison map $G: \mathrm{h}_{\mathit{\leq n-1}}\mathit{(()}\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}}) \rightarrow \operatorname{\mathcal{B}}$ $(n-1)$-categorical isofibration (Propositions 4.8.8.14 and 4.8.8.21). If $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ is another $(n-1)$-categorical isofibration, then we can use Proposition 4.8.8.16 to identify $\theta _{\operatorname{\mathcal{C}}}$ with the map with the functor $\operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{B}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \mathrm{h}_{\mathit{\leq n-1}}\mathit{(\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}})}, \operatorname{\mathcal{C}})$ given by precomposition with $G$. If condition $(3)$ is satisfied, then $G$ is an equivalence of $\infty $-categories, so that $F$ is categorically $n$-connective by virtue of Corollary 4.8.8.25. $\square$

Motivated by Proposition 4.8.9.1, we introduce a generalization of Definition 4.8.7.1.

Definition 4.8.9.2. Let $f: A \rightarrow B$ be a morphism of simplicial sets and let $n$ be an integer. We say that $f$ is categorically $n$-connective if, for every essentially $(n-1)$-categorical functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [r]^-{\circ f} \ar [d]^{U \circ } & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [d]^{ U \circ } \\ \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \ar [r]^-{\circ f} & \operatorname{Fun}( A, \operatorname{\mathcal{D}}) } \]

is a categorical pullback square.

Remark 4.8.9.3. In the situation of Definition 4.8.9.2, we can assume without loss of generality that the functor $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration (see Corollary 4.5.2.23). Replacing $\operatorname{\mathcal{C}}$ by the simplicial set $\mathrm{h}_{\mathit{\leq n-1}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$, we can further arrange that the isofibration $U$ is $(n-1)$-categorical (Proposition 4.8.8.22).

Remark 4.8.9.4. Let $n$ be an integer. The notion of categorical $n$-connectivity is completely determined by the following two properties:

$(1)$

If $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ is a functor of $\infty $-categories, then it is categorically $n$-connective in the sense of Definition 4.8.9.2 if and only if it is categorically $n$-connective in the sense of Definition 4.8.7.1: that is, $F$ is $m$-full for every nonnegative integer $m \leq n$ (see Proposition 4.8.9.1).

$(2)$

Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r] \ar [d]^{f} & A' \ar [d]^{f'} \\ B \ar [r] & B', } \]

where the horizontal maps are categorical equivalences. Then $f$ is categorically $n$-connective if and only if $f'$ is categorically $n$-connective. See Proposition 4.5.2.19.

If $f: A \rightarrow B$ is any morphism of simplicial sets, then we can use Proposition 4.1.3.2 to choose a commutative diagram

\[ \xymatrix { A \ar [d]^{f} \ar [r] & \operatorname{\mathcal{A}}\ar [d]^{F} \\ B \ar [r] & \operatorname{\mathcal{B}}} \]

where the horizontal maps are categorical equivalences and $F$ is a functor of $\infty $-categories. Combining $(1)$ and $(2)$, we see that $f$ is categorically $n$-connective if and only if the functor $F$ is $m$-full for $m \leq n$.

Remark 4.8.9.5. Let $f: A \rightarrow B$ be a morphism of simplicial sets. If $f$ is categorically $n$-connective, then it is $n$-connective. This follows from Remark 4.8.9.4 and Corollary 4.8.7.17. Beware that the converse is false in general (Warning 4.8.7.4).

Remark 4.8.9.6 (Transitivity). Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be morphisms of simplicial sets and let $n$ be an integer.

$(1)$

Suppose that $f$ and $g$ are categorically $n$-connective. Then $g \circ f$ is categorically $n$-connective.

$(2)$

Suppose that $g \circ f$ is categorically $n$-connective, $g$ is categorically $(n+1)$-connective, and $n \geq 1$. Then $f$ is categorically $n$-connective.

$(3)$

Suppose that $g \circ f$ is categorically $n$-connective and that $f$ is categorically $(n-1)$-connective. Then $g$ is categorically $n$-connective.

To prove these assertions, we can use Remark 4.8.9.4 to reduce to the case where $A$, $B$, and $C$ are $\infty $-categories, in which case the result follows from Proposition 4.8.7.12

Proposition 4.8.9.7. Suppose we are given a categorical pushout square of simplicial sets

4.89
\begin{equation} \begin{gathered}\label{equation:pushout-of-categorically-connective} \xymatrix@C =50pt@R=50pt{ A \ar [d]^{f} \ar [r] & A' \ar [d]^{f'} \\ B \ar [r] & B', } \end{gathered} \end{equation}

where $f$ is categorically $n$-connective. Then $f'$ is also categorically $n$-connective.

Proof. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an essentialy $(n-1)$-categorical functor of $\infty $-categories, and consider the cubical diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}(B', \operatorname{\mathcal{C}}) \ar [dr] \ar [rr] \ar [dd] & & \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [dr] \ar [dd] & \\ & \operatorname{Fun}(B', \operatorname{\mathcal{D}}) \ar [rr] \ar [dd] & & \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \ar [dd] \\ \operatorname{Fun}(A', \operatorname{\mathcal{C}}) \ar [rr] \ar [dr] & & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \ar [dr] & \\ & \operatorname{Fun}(A', \operatorname{\mathcal{D}}) \ar [rr] & & \operatorname{Fun}(A, \operatorname{\mathcal{D}}). } \]

Our assumption that $f$ is categorically $n$-connective guarantees that the right face is a categorical pullback square, and our assumption on (4.89) guarantees that the front and back faces are categorical pullback squares. Applying Proposition 4.5.2.18, we conclude that the left face is also a categorical pullback square. $\square$

Proposition 4.8.9.8. Let $f: A \hookrightarrow B$ be a monomorphism of simplicial sets and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The morphism $f$ is categorically $n$-connective.

$(2)$

For every essentially $(n-1)$-categorical functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the restriction map

\[ V: \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories.

$(3)$

For every essentially $(n-1)$-categorical isofibration of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the functor $V$ is a trivial Kan fibration.

$(3)$

Every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]

admits a solution, provided that $U$ is an essentially $(n-1)$-categorical isofibration of $\infty $-categories.

Proof. The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ follow from Remarks 4.8.7.19 and 4.8.9.3, and the implication $(3) \Rightarrow (4)$. is immediate. We will complete the proof by showing that $(4)$ implies $(3)$. Assume that condition $(4)$ is satisfied, and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an essentially $(n-1)$-categorical isofibration of $\infty $-categories. We wish to show that, for every simplicial set $B'$ and every simplicial subset $A' \subseteq B'$, every lifting problem

4.90
\begin{equation} \begin{gathered}\label{equation:categorically-connective-monomorphism} \xymatrix@C =50pt@R=50pt{ A' \ar [r] \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [d]^{V} \\ B' \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

admits a solution. Unwinding the definitions, we can rewrite (4.90) as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d]^{f} & \operatorname{Fun}(B', \operatorname{\mathcal{C}}) \ar [d]^{V'} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A', \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A', \operatorname{\mathcal{D}}) } \operatorname{Fun}(B', \operatorname{\mathcal{D}}). } \]

The existence of a solution follows from $(4)$, since $V'$ is also an essentially $(n-1)$-categorical isofibration of $\infty $-categories (Corollary 4.8.6.20 and Proposition 4.4.5.1). $\square$

Example 4.8.9.9. Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset which contains the $n$-skeleton of $B$. Then the inclusion map $A \hookrightarrow B$ is categorically $n$-connective. In particular, for every simplicial set $B$, the inclusion map $\operatorname{sk}_{n}(B) \hookrightarrow B$ is categorically $n$-connective.

Proposition 4.8.9.10. Let $n \geq 0$ be an integer and let $f: A \rightarrow B$ be a morphism of simplicial sets which is bijective on simplices of dimension $< n$ and surjective on $n$-simplices. Then $f$ is categorically $n$-connective.

Proof. Using Proposition 1.1.4.12, we can choose a simplicial subset $A' \subseteq \operatorname{sk}_{n}(A)$ which contains the $(n-1)$-skeleton of $A$, such that $f$ restricts to an isomorphism of $A'$ with the $n$-skeleton of $B$. It follows from Example 4.8.9.9 $f|_{A'}$ is categorically $n$-connective, and that the inclusion map $A' \hookrightarrow A$ is categorically $(n-1)$-connective. Applying Remark 4.8.9.6, we deduce that $f$ is categorically $n$-connective. $\square$