Question 4.8.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Under what conditions does there exists an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $(n,1)$-category?
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.8 Truncations in Higher Category Theory
Recall that a simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category if, for every pair of integers $0 < i < m$, every inner horn $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$ (Definition 1.4.0.1). In this case, $\operatorname{\mathcal{C}}$ is (isomorphic to the nerve of) an ordinary category if and only if the extension $\sigma $ is always unique (Proposition 1.3.4.1). More generally, we say that $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if the extension $\sigma $ is unique whenever $m > n$ (Definition 4.8.1.1). In §4.8.1, we summarize the formal properties of this definition and give some basic examples.
Beware that the notion of $(n,1)$-category is not homotopy-invariant: that is, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent $\infty $-categories and $\operatorname{\mathcal{D}}$ is an $(n,1)$-category, then $\operatorname{\mathcal{C}}$ need not be an $(n,1)$-category. We can therefore ask the following:
We partially address Question 4.8.0.1 in §4.8.2 by introducing the notion of a locally truncated $\infty $-category. If $m$ is an integer, we say that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $m$-truncated if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $m$-truncated (Definition 4.8.2.1). It is easy to see that every $(n,1)$-category is locally $(n-1)$-truncated (Example 4.8.2.2). Conversely, if $\operatorname{\mathcal{C}}$ is a locally $(n-1)$-truncated $\infty $ category, then there exists an equivalence $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. We will give two proofs of this result:
In §4.8.3, we show that a locally $(n-1)$-truncated $\infty $-category $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if it is minimal in dimensions $\geq n$ (Proposition 4.8.3.1). In particular, if $\operatorname{\mathcal{D}}$ is a minimal model of $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{D}}$ is an $(n,1)$-category.
In §4.8.4, we associate to every $\infty $-category $\operatorname{\mathcal{C}}$ an $(n,1)$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$, which we call the homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Construction 4.8.4.9). The homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is equipped with a comparison functor $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$, which is an equivalence if and only if $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated (Corollary 4.8.4.16).
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. Recall that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of categories if and only if it satisfies the following three conditions:
The functor $F$ is essentially surjective: that is, every object of $\operatorname{\mathcal{D}}$ is isomorphic to $F(X)$, for some objects $X \in \operatorname{\mathcal{C}}$.
The functor $F$ is full: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the function $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is surjective.
The functor $F$ is faithful: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the function $F_{X,Y}$ is injective.
Exercise 4.8.0.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Show that $F$ is faithful if and only if, for every diagram $\sigma :$ in the category $\operatorname{\mathcal{C}}$, if $F(\sigma )$ is a commutative diagram in $\operatorname{\mathcal{D}}$, then $\sigma $ is also commutative.
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
To emphasize the parallels between the preceding conditions, it is convenient to reformulate them using the language of simplicial sets. To simplify the discussion, let us assume that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration (Definition 4.4.1.1). In this case:
- $(0)$
The functor $F$ is essentially surjective if and only if it is surjective on objects: that is, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{0} \ar [r] \ar [d] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{N}_{\bullet }(F) } \\ \Delta ^0 \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \]admits a solution.
- $(1)$
The functor $F$ is full if and only if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{1} \ar [r] \ar [d] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{N}_{\bullet }(F) } \\ \Delta ^1 \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \]admits a solution.
- $(2)$
The functor $F$ is faithful if and only if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{2} \ar [r] \ar [d] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{N}_{\bullet }(F) } \\ \Delta ^2 \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}). } \]
These conditions have a counterpart in the setting of $\infty $-categories:
Definition 4.8.0.3 (Preliminary). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration $\infty $-categories and let $m \geq 0$ be an integer. We say that $F$ is $m$-full if every lifting problem admits a solution.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^ m \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
In §4.8.5, we extend Definition 4.8.0.3 to the case where $F$ is an arbitrary functor of $\infty $-categories (see Definition 4.8.5.10 for a homotopy-invariant formulation, and Proposition 4.8.5.30 for a comparison with Definition 4.8.0.3).
Example 4.8.0.4. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ be a functor of ordinary categories and let $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}_0)$ denote the induced of $\infty $-categories. Then:
The functor $F$ is $0$-full if and only if $F_0$ is essentially surjective.
The functor $F$ is $1$-full if and only if $F_0$ is full.
The functor $F$ is $2$-full if and only if $F_0$ is faithful.
For $m \geq 3$, the functor $F$ is automatically $m$-full (see Exercise 1.3.1.5).
Fix an integer $n$. We will say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially $n$-categorical if it is $m$-full for every nonnegative integer $m \geq n+2$ (Definition 4.8.6.1). In §4.8.6, we will see that this condition has many familiar specializations:
A functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially $(-1)$-categorical if and only if it is fully faithful, and essentially $(-2)$-categorical if and only if it is an equivalence of $\infty $-categories. See Remark 4.8.5.11.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -1$. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is essentially $n$-categorical if and only if $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated: that is, if and only if $\operatorname{\mathcal{C}}$ is equivalent to an $(n,1)$-category. See Example 4.8.6.4.
If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are Kan complexes, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially $n$-categorical if and only if it is $n$-truncated, in the sense of Definition 3.5.9.1. See Example 4.8.6.3.
In §4.8.7, we introduce a dual version of this condition. We will say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is categorically $n$-connective if it is $m$-full for every integer $0 \leq m \leq n$ (Definition 4.8.7.1). Roughly speaking, this condition asserts that, up to equivalence, the $\infty $-category $\operatorname{\mathcal{D}}$ can be be built from $\operatorname{\mathcal{C}}$ using only simplices of dimension strictly larger than $n$ (see Corollary 4.8.7.16 for a precise formulation). In the special case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are Kan complexes, this recovers the theory of relative connectivity developed in §3.5.1 (Example 4.8.7.3). As with the usual notion of connectivity, it can sometimes be useful to extend the notion of categorical connectivity to morphisms of between simplicial sets which are not $\infty $-categories; we consider this extension in §4.8.9 (Definition 4.8.9.2).
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. In §4.8.8, we show that $F$ admits a factorization
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}, \]
where $F'$ is categorically $(n+1)$-connective and $G$ is essentially $n$-categorical (Theorem 4.8.8.3). Moreover, this factorization is unique up to equivalence (Remark 4.8.8.8). In the special case where $\operatorname{\mathcal{D}}= \Delta ^0$, this can be achieved by taking $\operatorname{\mathcal{D}}'$ to be the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ constructed in §4.8.4 (Example 4.8.8.7). More generally, if $F$ is an inner fibration of $\infty $-categories, we will show that the system of $\infty $-categories $\{ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}_{D})} \} _{D \in \operatorname{\mathcal{D}}}$ can be realized as the fibers of an inner fibration $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ which realizes the desired factorization (Construction 4.8.8.10).
Structure
- Subsection 4.8.1: $(n,1)$-Categories
- Subsection 4.8.2: Locally Truncated $\infty $-Categories
- Subsection 4.8.3: Minimality Conditions
- Subsection 4.8.4: Higher Homotopy Categories
- Subsection 4.8.5: Full and Faithful Functors
- Subsection 4.8.6: Essentially Categorical Functors
- Subsection 4.8.7: Categorically Connective Functors
- Subsection 4.8.8: Relative Higher Homotopy Categories
- Subsection 4.8.9: Categorically Connective Morphisms of Simplicial Sets