Section 4.8 (0578): Full, Faithful, and Essentially Surjective Functors

4.8 Full, Faithful, and Essentially Surjective Functors

Our starting point is the following basic result of category theory:

Theorem 4.8.0.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then $F$ is an equivalence if and only if it satisfies the following conditions:

$(0)$: The functor $F$ is essentially surjective: that is, every object of $\operatorname{\mathcal{D}}$ is isomorphic to $F(X)$ for some object $X \in \operatorname{\mathcal{C}}$.
$(1)$: The functor $F$ is full: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is surjective.
$(2)$: The functor $F$ is faithful: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ 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 $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]

in the category $\operatorname{\mathcal{C}}$, if $F(\sigma )$ is a commutative diagram in $\operatorname{\mathcal{D}}$, then $\sigma $ is also commutative.

Remark 4.8.0.3. To emphasize the parallels between the conditions appearing in Theorem 4.8.0.1, it is convenient to reformulate them using the language of simplicial sets. To simplify the discussion, let us assume that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration of categories (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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{2} \ar [r] \ar [d] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{N}_{\bullet }(F) } \\ \Delta ^2 \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \]

admits a solution (Exercise 4.8.0.2).

The main goal of this section is to formulate a counterpart of Theorem 4.8.0.1 in the setting of $\infty $-categories. Our first step is to formulate appropriate analogues of conditions $(0)$, $(1)$, and $(2)$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$ be the induced functor of homotopy categories. We will say that $F$ is essentially surjective if the functor $\operatorname {h}\! \mathit{F}$ is essentially surjective (Definition 4.8.1.1 and Remark 4.8.1.3), and we say that $F$ is full if the functor $\operatorname {h}\! \mathit{F}$ is full (Definition 4.8.2.1 and Remark 4.8.2.3). The analogue of faithfulness is more subtle, and uses the theory of morphism spaces developed in §4.6. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a morphism of Kan complexes

\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ). \]

We will say that $F$ is faithful if each of the functors $F_{X,Y}$ induces a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ (Definition 4.8.3.8). We say that $F$ is fully faithful if it is both full and faithful: that is, each of the maps $F_{X,Y}$ is a homotopy equivalence of Kan complexes (Definition 4.8.3.1 and Remark 4.8.3.9). We study the properties of essentially surjective, full, and faithful functors in §4.8.1, §4.8.2, and §4.8.3, respectively.

In §4.8.4, we show that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories if and only if it is essentially surjective, full, and faithful (Theorem 4.8.4.1). To prove this, it will be convenient to package the hypotheses in a different way, which is motivated by Remark 4.8.0.3.

Definition 4.8.0.4 (Preliminary). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories and let $n \geq 0$ be an integer. We say that $F$ is $n$-full if every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

admits a solution.

Stated more informally, an isofibration of $\infty $-categories is $n$-full if it is surjective at the level of $n$-morphisms (provided that the source and target have been fixed). In §4.8.2, we extend Definition 4.8.0.4 to general functors (see Definition 4.8.2.7 for a homotopy-invariant formulation, and Proposition 4.8.4.5 for a comparison with Definition 4.8.0.4). This extension has the following properties:

A functor of $\infty $-categories is essentially surjective if and only if it is $0$-full.
A functor of $\infty $-categories is full if and only if it is $1$-full.
A functor of $\infty $-categories is faithful if and only if it is $n$-full for every integer $n > 1$ (Remark 4.8.3.16).

From this perspective, our main result asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories if and only if it is $n$-full for every integer $n \geq 0$. In the case where $F$ is an isofibration, this reduces to characterization of Proposition 4.5.6.20: an isofibration of $\infty $-categories is an equivalence of $\infty $-categories if and only if it is a trivial Kan fibration.

Fix an integer $n$. We will say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-faithful if it is $m$-full for every nonnegative integer $m > n$ (Definition 4.8.3.12). This condition has many familiar specializations:

A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is faithful if and only if it is $1$-faithful, and fully faithful if and only if it is $0$-faithful.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be a nonnegative integer. Then the projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is $n$-faithful if and only if $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated: that is, if and only if $\operatorname{\mathcal{C}}$ is equivalent to an $(n-1)$-category. See Example 4.8.3.14.
If $f: X \rightarrow Y$ is a morphism of Kan complexes, then it is $n$-faithful (when regarded as a functor of $\infty $-categories) if and only if it is $(n-1)$-truncated (in the sense of Definition 3.5.9.1). See Example 4.8.3.14.

In §4.8.5, 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 $0 \leq m \leq n$. Roughly speaking, this condition asserts that (up to equivalence) the $\infty $-category $\operatorname{\mathcal{D}}$ can be built from $\operatorname{\mathcal{C}}$ using only simplices of dimension strictly larger than $n$ (see Corollary 4.8.5.21 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 4.8.5.3). As with the usual notion of connectivity, it can sometimes be useful to extend the notion of categorical connectivity to morphisms between simplicial sets which are not $\infty $-categories; we consider this extension in §4.8.8 (Definition 4.8.8.2).

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. In §4.8.7, we show that $F$ admits a factorization

\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}, \]

where $F'$ is categorically $n$-connective and $G$ is $n$-faithful (Theorem 4.8.7.3). Moreover, this factorization is unique up to equivalence (Remark 4.8.7.9). In the special case where $\operatorname{\mathcal{D}}= \Delta ^0$, this can be achieved by taking $\operatorname{\mathcal{D}}'$ to be the homotopy $(n-1)$-category $\operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{C}})}}$ constructed in §4.7.9 (Example 4.8.7.7). The general case depends on a relative version of the higher homotopy category construction, which we describe in §4.8.6 (see Construction 4.8.6.13).

Structure

Subsection 4.8.1: Essentially Surjective Functors
Subsection 4.8.2: Full Functors
Subsection 4.8.3: Faithful Functors
Subsection 4.8.4: Detecting Equivalences of $\infty $-Categories
Subsection 4.8.5: Categorically Connective Functors
Subsection 4.8.6: Digression: $n$-Categorical Fibrations
Subsection 4.8.7: Canonical Factorizations
Subsection 4.8.8: Categorically Connective Morphisms of Simplicial Sets