Definition 4.8.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ is categorically $n$-connective if it is $m$-full for every integer $0 \leq m \leq n$ (see Definition 4.8.2.7).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8.5 Categorically Connective Functors
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Recall that $F$ is $n$-faithful if it is $m$-full for every nonnegative integer $m > n$. In this section, we study a dual version of this condition.
Example 4.8.5.2. For small values of $n$, we can make Definition 4.8.5.1 more concrete:
For $n < 0$, every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $n$-connective.
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $0$-connective if and only if it is essentially surjective.
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $1$-connective if and only if it is full and essentially surjective.
Example 4.8.5.3. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n$ be an integer. Then $f$ is categorically $n$-connective (in the sense of Definition 4.8.5.1) if and only if it is $n$-connective (in the sense of Definition 3.5.1.13). See Proposition 4.8.2.20.
Warning 4.8.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. If $F$ is categorically $n$-connective, then it is an $n$-connective morphism of simplicial sets (Corollary 4.8.5.22). Beware that the converse is false in general. For example, the projection map $\Delta ^1 \twoheadrightarrow \Delta ^0$ is a homotopy equivalence (and therefore $n$-connective for every integer $n$) which is not categorically $2$-connective.
Remark 4.8.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is categorically $n$-connective if and only if it satisfies the following pair of conditions:
The functor $F$ is locally $(n-1)$-connective. That is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
is $(n-1)$-connective.
If $n \geq 0$, then $F$ is essentially surjective.
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
See Corollary 4.8.2.22.
Remark 4.8.5.6 (Symmetry). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is categorically $n$-connective if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is categorically $n$-connective.
Remark 4.8.5.7 (Monotonicity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $m \leq n$ be integers. If $F$ is categorically $n$-connective, then it is categorically $m$-connective.
Remark 4.8.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if it is categorically $n$-connective for every integer $n$. This is a restatement of Theorem 4.8.4.1.
Remark 4.8.5.9 (Products). Let $n$ be an integer, let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of functors of $\infty $-categories, and let $F: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \prod _{i \in I} \operatorname{\mathcal{D}}_ i$ be their product. If each of the functors $F_ i$ is categorically $n$-connective, then $F$ is categorically $n$-connective. The converse holds if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is nonempty. See Remark 4.8.2.10.
Remark 4.8.5.10 (Coproducts). Let $n$ be an integer, let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of functors of $\infty $-categories, and let $F: \coprod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \coprod _{i \in I} \operatorname{\mathcal{D}}_ i$ be their coproduct. Then $F$ is categorically $n$-connective if and only if each $F_{i}$ is categorically $n$-connective. See Remark 4.8.2.11.
Remark 4.8.5.11. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which can be realized as the colimit of a filtered diagram $\{ F_{\alpha }: \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}_{\alpha } \} $ in the category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$. If each $F_{\alpha }$ is a categorically $n$-connective functor of $\infty $-categories, then $F$ is a categorically $n$-connective functor of $\infty $-categories. See Remark 4.8.2.12.
Remark 4.8.5.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Remark 4.8.5.5 (and Theorem 4.8.4.1) that if $F$ is categorically $(n+1)$-connective, then the induced map of homotopy $n$-categories $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}})}} \rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{D}})}}$ is an equivalence. In particular, if $F$ is categorically $2$-connective, then it induces an equivalence of homotopy categories $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$.
Remark 4.8.5.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
If $F$ is categorically $1$-connective and $\operatorname{\mathcal{C}}$ is a Kan complex, then $\operatorname{\mathcal{D}}$ is also a Kan complex.
If $F$ is categorically $2$-connective, then $\operatorname{\mathcal{C}}$ is a Kan complex if and only if $\operatorname{\mathcal{D}}$ is a Kan complex.
Remark 4.8.5.14. Let $n$ be an integer, and suppose we are given a categorical pullback diagram of $\infty $-categories If $F$ is categorically $n$-connective, then $F'$ is categorically $n$-connective. The converse holds if $G$ is full and essentially surjective. See Corollary 4.8.2.27.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \]
Proposition 4.8.5.15 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. Then:
If $F$ and $G$ are categorically $n$-connective, then the composite functor $G \circ F$ is categorically $n$-connective.
If $G \circ F$ is categorically $n$-connective, $G$ is categorically $(n+1)$-connective, and $n \geq 1$, then $F$ is categorically $n$-connective.
If $G \circ F$ is categorically $n$-connective and $F$ is categorically $(n-1)$-connective, then $G$ is categorically $n$-connective.
Proof. Assertions $(1)$ and $(3)$ follow by combining Remark 4.8.2.14 with Proposition 4.8.2.30 (supplemented by Remark 4.8.2.31), respectively. It will therefore suffice to prove $(2)$. Assume that $n \geq 1$, that $G \circ F$ is categorically $n$-connective, and that $G$ is categorically $(n+1)$-connective; we wish to prove that $F$ is categorically $n$-connective: that is, that $F$ is $m$-full for $m \leq n$. If $m > 0$, this follows from Proposition 4.8.2.28. It will therefore suffice to treat the case $m = 0$: that is, to show that $F$ is essentially surjective. This follows from the essential surjectivity of $G \circ F$, since $G$ induces an equivalence of homotopy categories (Remark 4.8.5.12). $\square$
Warning 4.8.5.16. In the case $n = 0$, assertion $(2)$ of Proposition 4.8.5.15 is not necessarily true. For example, let $\operatorname{\mathcal{D}}$ be an $\infty $-category with the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{D}}$, there exists a morphism from $X$ to $Y$. Then the projection map $G: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ is categorically $1$-connective. Any choice of object $D \in \operatorname{\mathcal{D}}$ determines a functor $F: \{ D\} \hookrightarrow \operatorname{\mathcal{D}}$ with the property that $G \circ F$ is an isomorphism (and therefore categorically $0$-connective). However, the functor $F$ need not be essentially surjective.
In the case where $F$ is an isofibration, Definition 4.8.5.1 can be reformulated as a lifting property.
Proposition 4.8.5.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The functor $F$ is categorically $n$-connective.
For every integer $0 \leq m \leq n$, every lifting problem
admits a solution.
For every simplicial set $B$ of dimension $\leq n$ and every simplicial subset $A \subseteq B$, every lifting problem
admits a solution.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B\ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.8.4.5. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 1.1.4.12. $\square$
Proposition 4.8.5.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Suppose that $F$ is bijective on simplices of dimension $< n$ and surjective on simplices of dimension $n$. Then $F$ is categorically $n$-connective.
Proof. Note that $F$ is automatically essentially surjective (since it is surjective on objects). By virtue of Remark 4.8.5.5, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is $(n-1)$-connective. This follows from Corollary 3.5.2.2, since $F_{X,Y}$ is bijective on simplices of dimension $< n-1$ and surjective on simplices of dimension $n$. $\square$
Example 4.8.5.19. Let $n \geq -2$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ (see Definition 4.7.6.13). Then $F$ is categorically $(n+2)$-connective. To prove this, we may assume without loss of generality that $F$ exhibits $\operatorname{\mathcal{D}}$ as the $(n+2)$-coskeleton of $\operatorname{\mathcal{C}}$ (Proposition 4.7.6.19). In this case, $F$ is bijective on $m$-simplices for $m \leq n+2$, so the desired result follows from Proposition 4.8.5.18.
We now establish a partial converse to Proposition 4.8.5.18:
Proposition 4.8.5.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The functor $F$ is categorically $n$-connective.
The functor $F$ factors as a composition
where $F'$ is a monomorphism which is bijective on $m$-simplices for $m \leq n$, and $U$ is a trivial Kan fibration.
The functor $F$ factors as a composition
where $F'$ is bijective on $m$-simplices for $m < n$, surjective on $n$-simplices, and $U$ is categorically $n$-connective.
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}, \]
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}, \]
Proof. We proceed as in the proof of Corollary 3.5.2.4. The implication $(2) \Rightarrow (3)$ is clear, and the implication $(3) \Rightarrow (1)$ follows from Propositions 4.8.5.15 and 4.8.5.18. We will complete the proof by showing that $(1)$ implies $(2)$. Assume that $F$ is categorically $n$-connective. Using a variant of Exercise 3.1.8.11, we can choose a factorization of $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}$ with the following properties:
- $(a)$
For every integer $m > n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [d]^{U} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]admits a solution.
- $(b)$
The morphism $F'$ can be realized as a transfinite pushout of inclusion maps $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^ m$ for $m > n$.
It follows immediately from $(b)$ that $F'$ is a monomorphism which is bijective on simplices of dimension $\leq n$. We will complete the proof by showing that $U$ is a trivial Kan fibration: that is, every lifting problem
4.87
\begin{equation} \begin{gathered}\label{equation:connective-Kan-complex-factorization-categorical} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [d]^{U} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}
admits a solution. For $m > n$, this follows from $(b)$. For $m \leq n$, we can identify (4.87) with a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}, } \]
which admits a solution by virtue of our assumption that $F$ is a categorically $n$-connective isofibration (Proposition 4.8.5.17). $\square$
Corollary 4.8.5.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is categorically $n$-connective if and only if it factors as a composition where $E$ and $G$ are equivalences of $\infty $-categories and $F'$ is bijective on simplices of dimension $\leq n$. Moreover, we can arrange that $E$ and $F'$ are monomorphisms and that $G$ is a trivial Kan fibration.
\[ \operatorname{\mathcal{C}}\xrightarrow {E} \operatorname{\mathcal{C}}' \xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}} \]
Proof. Combine Proposition 4.8.5.20 with Corollary 4.5.3.24. $\square$
Corollary 4.8.5.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is categorically $n$-connective (in the sense of Definition 4.8.5.1), then it is $n$-connective (in the sense of Variant 3.5.1.15).
Proof. Using Corollary 4.8.5.21 (and Remark 4.5.4.4), we can reduce to the case where $F$ is bijective on simplices of dimension $\leq n$. In this case, the desired result follows from Corollary 3.5.2.2. $\square$
Corollary 4.8.5.23. Let $n$ be an integer and let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ and $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories. Suppose that $F$ is categorically $n$-connective and that $G$ is $n$-faithful. Then the diagram of $\infty $-categories is a categorical pullback square.
4.88
\begin{equation} \begin{gathered}\label{equation:orthogonality-categoricity-connectivity} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \ar [r]^-{\circ F} \ar [d]^{G \circ } & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \ar [d]^{ G \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}
For a partial converse, see Proposition 4.8.8.1.
Remark 4.8.5.24. In the situation of Corollary 4.8.5.23, suppose that one of the following additional conditions is satisfied:
The functor $F$ is a monomorphism of simplicial sets.
The functor $G$ is an isofibration.
Condition $(a)$ guarantees that the horizontal maps in the diagram (4.88) are isofibrations (Corollary 4.4.5.3) and condition $(b)$ guarantees that the vertical maps are isofibrations (Corollary 4.4.5.6). In either case, the conclusion of Corollary 4.8.5.23 is equivalent to the requirement that the functor
\[ V: \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories (Proposition 4.5.3.27). If conditions $(a)$ and $(b)$ are both satisfied, then $V$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). In this case, the conclusion of Corollary 4.8.5.23 is equivalent to the requirement that $G$ is a trivial Kan fibration (Proposition 4.5.6.20).
Proof of Corollary 4.8.5.23. Using Corollaries 4.8.5.21, we can reduce to the case where $F$ is a monomorphism which is bijective on simplices of dimension $\leq n$. In this case, the desired result follows from Corollary 4.8.4.14 (and Remark 4.8.5.24). $\square$
Proposition 4.8.5.25. Let $m$ and $n$ be nonnegative integers and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a categorically $(m+n)$-connective functor of $\infty $-categories. Let $B$ be a simplicial set of dimension $\leq m$, and let $A \subseteq B$ be a simplicial subset. Then the induced functor is categorically $n$-connective.
\[ G: \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( A, \operatorname{\mathcal{D}}) } \operatorname{Fun}( B, \operatorname{\mathcal{D}}) \]
Proof. We proceed as in the proof of Proposition 3.5.2.11. Using Corollary 4.5.3.24 (and Corollary 4.5.3.34), we can reduce to the case where $F$ is an isofibration. In this case, $G$ is also an isofibration (Proposition 4.4.5.1). By virtue of Proposition 4.8.5.17, it will suffice to show that if $B'$ is a simplicial set of dimension $\leq n$ and $A' \subseteq B'$ is a simplicial subset, then every lifting problem
4.89
\begin{equation} \begin{gathered}\label{equation:exponentiation-for-connectivity-categorical} \xymatrix@R =50pt@C=50pt{ A' \ar [r] \ar [d] & \operatorname{Fun}( B, \operatorname{\mathcal{C}}) \ar [d]^{G} \\ B' \ar [r] \ar@ {-->}[ur] & \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 (4.89) as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (A \times B') \coprod _{ (A \times A' ) } (B \times A') \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \times B' \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \]
Since the simplicial set $B \times B'$ has dimension $\leq m+n$ (Proposition 1.1.3.6), the existence of a solution follows from our assumption that $F$ is categorically $(m+n)$-connective (Proposition 4.8.5.17). $\square$
Corollary 4.8.5.26. Let $m$ and $n$ be nonnegative integers, let $B$ be a simplicial set of dimension $\leq m$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which is categorically $(m+n)$-connective. Then the induced map $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is categorically $n$-connective.
Proof. Apply Proposition 4.8.5.25 in the special case $A = \emptyset $. $\square$