Definition 4.8.7.1. 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 nonnegative integer $m \leq n$ (see Definition 4.8.5.10).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8.7 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 essentially $(n-1)$-categorical if it is $m$-full for every nonnegative integer $m > n$. In this section, we study a dual version of this condition.
Example 4.8.7.2. For small values of $n$, we can make Definition 4.8.7.1 more concrete:
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $1$-connective if and only if it is full and essentially surjective.
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $0$-connective if and only if it is essentially surjective.
For $n < 0$, every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $n$-connective.
Example 4.8.7.3. 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 4.8.7.1) if and only if it is $n$-connective (in the sense of Definition 3.5.1.13). See Corollary 4.8.5.24.
Warning 4.8.7.4. 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 4.8.7.17). 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 4.8.7.5. 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 4.8.5.22.
Remark 4.8.7.6 (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 4.8.7.7 (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 4.8.7.8. 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 $n$. See Remark 4.8.5.11.
Remark 4.8.7.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Remark 4.8.7.5 that if $F$ is categorically $(n+1)$-connective, then the induced map of homotopy $n$-categories $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}})}$ is an equivalence. In particular, if $F$ is categorically $2$-connective, then it induces an equivalence of homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$.
Remark 4.8.7.10. 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 4.8.7.11. 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 4.8.5.29.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \]
Proposition 4.8.7.12 (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 4.8.5.15 with Proposition 4.8.5.33 (supplemented by Remark 4.8.5.34), 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 4.8.5.31. 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 4.8.7.9). $\square$
Proposition 4.8.7.13. 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 4.8.7.5, 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 $n$. $\square$
In the case where $F$ is an isofibration, Definition 4.8.7.1 can be reformulated as a lifting property.
Proposition 4.8.7.14. 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 4.8.5.30. The implication $(3) \Rightarrow (2)$ is immediate, and the reverse implication follows from Proposition 1.1.4.12. $\square$
We now prove a partial converse to Proposition 4.8.7.13:
Proposition 4.8.7.15. 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 $U$ 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 4.8.7.12 and 4.8.7.13. We will complete the proof by showing that $(1)$ implies $(2)$. Assume that $F$ is categorically $n$-connective. Using a variant of Exercise 3.1.7.11, 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
4.85
\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 (4.85) 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 4.8.7.14). $\square$
Corollary 4.8.7.16. 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 4.8.7.15 with Corollary 4.5.2.23. $\square$
Corollary 4.8.7.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is categorically $n$-connective, then it is $n$-connective.
Proof. Using Corollary 4.8.7.16 (and Remark 4.5.3.4), 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 4.8.7.18. 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 essentially $(n-1)$-categorical. Then the diagram of $\infty $-categories is a categorical pullback square.
4.86
\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 4.8.9.1.
Remark 4.8.7.19. In the situation of Corollary 4.8.7.18, 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 (4.86) 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 4.8.7.18 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 4.5.2.26). If conditions $(a)$ and $(b)$ are both satisfied, then $G$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). In this case, the conclusion of Corollary 4.8.7.18 is equivalent to the requirement that $G$ is a trivial Kan fibration (Proposition 4.5.5.20).
Proof of Corollary 4.8.7.18. Using Corollary 4.8.7.16 we can reduce to the case where $F$ is a monomorphism which is bijective on simplices of dimension $\leq n$. The desired result now follows from Corollary 4.8.6.21 (and Remark 4.8.7.19). $\square$
Proposition 4.8.7.20. 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 4.5.2.23 (and Corollary 4.5.2.30), 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 4.8.7.14, 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
4.87
\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 (4.87) 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 4.8.7.14). $\square$
Corollary 4.8.7.21. 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 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. Applying Proposition 4.8.7.20 in the special case $A = \emptyset $. $\square$