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5.2.2 Fully Faithful and Essentially Surjective Functors

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be 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 pair of conditions:

$(1)$

The functor $F$ is fully faithful: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is bijective.

$(2)$

The functor $F$ is essentially surjective: that is, for every object $X \in \operatorname{\mathcal{D}}$, there exists an object $Y \in \operatorname{\mathcal{C}}$ and an isomorphism $X \simeq F(Y)$ in the category $\operatorname{\mathcal{D}}$.

Our goal in this section is to give an analogous characterization of equivalences in the setting of $\infty $-categories (Theorem 5.2.2.15). We begin by formulating $\infty $-categorical analogues of conditions $(1)$ and $(2)$.

Definition 5.2.2.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is fully faithful if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes.

Example 5.2.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory (Definition 4.1.2.15). Then the inclusion map $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is fully faithful. In fact, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}'$, the inclusion $\iota $ induces an isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.

Example 5.2.2.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is fully faithful if and only if the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is fully faithful (in the sense of Definition 5.2.2.1). Consequently, we can regard Definition 5.2.2.1 as a generalization of the classical notion of fully faithful functor.

Remark 5.2.2.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories, so that $F$ induces a functor of homotopy categories $f: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$. If $F$ is fully faithful, then $f$ is also fully faithful (see Remark 5.2.1.9). Beware that the converse is generally false.

Remark 5.2.2.5 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $G$ is fully faithful. Then $F$ is fully faithful if and only if $G \circ F$ is fully faithful. In particular, the collection of fully faithful functors is closed under composition.

Proposition 5.2.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then $F$ is conservative (Definition 4.4.2.6). That is, if $u: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{C}}$ for which $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$, then $u$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. Let $\overline{v}: F(Y) \rightarrow F(X)$ be a homotopy inverse to $F(u)$. Since $F$ is fully faithful, the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(X))$ is a homotopy equivalence. We may therefore assume without loss of generality that $\overline{v} = F(v)$, for some morphism $v: Y \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $v \circ u$ be a composition of $u$ and $v$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $F(u)$ is homotopy inverse to $F(v)$, the morphism $F( v \circ u)$ is homotopic to $\operatorname{id}_{ F(C) } = F( \operatorname{id}_{C} )$. Since the map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X))$ is a homotopy equivalence, it follows that $v \circ u$ is homotopic to $\operatorname{id}_{C}$: that is, $v$ is a left homotopy inverse to $u$. A similar argument (with the roles of $u$ and $v$ reversed) shows that $v$ is also a right homotopy inverse to $u$. It follows that $u$ is an isomorphism. $\square$

Corollary 5.2.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then the induced map of cores $\operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is also fully faithful.

Proof. Fix objects $X,Y \in \operatorname{\mathcal{C}}^{\simeq }$. Our assumption that $F$ is fully faithful guarantees that the induced map $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes. By virtue of Proposition 5.2.2.6, $\theta $ restricts to a homotopy equivalence from the summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ spanned by the isomorphisms from $X$ to $Y$ to the summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ spanned by the isomorphisms from $F(X)$ to $F(Y)$. Unwinding the definitions, we conclude that $F^{\simeq }$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\simeq }}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}^{\simeq }}( F(X), F(Y) )$. $\square$

Definition 5.2.2.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The essential image of $F$ is the full subcategory of $\operatorname{\mathcal{D}}$ spanned by those objects $D \in \operatorname{\mathcal{D}}$ for which there exists an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $F(C) \simeq D$. We say that $F$ is essentially surjective if its essential image is the entire $\infty $-category $\operatorname{\mathcal{D}}$: that is, if the map of sets $\pi _0( \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{D}}^{\simeq } )$ is surjective.

Remark 5.2.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$. Then $\operatorname{\mathcal{D}}'$ is a replete full subcategory of $\operatorname{\mathcal{D}}$, and $F$ can be regarded as an essentially surjective functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}'$. Moreover, the essential image $\operatorname{\mathcal{D}}'$ is uniquely determined by these properties.

Remark 5.2.2.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Then $F$ is essentially surjective if and only if the induced functor of homotopy categories $f: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective (in the sense of classical category theory).

Remark 5.2.2.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Then $F$ is essentially surjective if and only if the induced map of Kan complexes $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is essentially surjective.

Example 5.2.2.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is essentially surjective if and only if the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is an essentially surjective functor of $\infty $-categories (in the sense of Definition 5.2.2.8).

Example 5.2.2.13. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is essentially surjective (in the sense of Definition 5.2.2.8) if and only if the induced map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is a surjection.

Remark 5.2.2.14 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $F$ is essentially surjective. Then $G$ is essentially surjective if and only if $G \circ F$ is essentially surjective. In particular, the collection of essentially surjective functors is closed under composition.

Theorem 5.2.2.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if it is fully faithful and essentially surjective.

We begin by considering the special case of Theorem 5.2.2.15 where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are Kan complexes.

Lemma 5.2.2.16. Let $f: X \rightarrow Y$ be a morphism of Kan complexes which is fully faithful and essentially surjective. Then $f$ is a homotopy equivalence.

Proof. Since $f$ is essentially surjective, the underlying map of connected components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective. We claim that it is also injective. To prove this, suppose that $x$ and $x'$ are vertices of $X$ such that $f(x)$ and $f(x')$ belong to the same connected component of $Y$. Then the morphism space $\operatorname{Hom}_{Y}( f(x), f(x') )$ is nonempty. Since $f$ is fully faithful, it induces a homotopy equivalence $\operatorname{Hom}_{X}(x,x') \rightarrow \operatorname{Hom}_{Y}(f(x), f(x') )$. It follows that $\operatorname{Hom}_{X}(x,x')$ is nonempty, so that $x$ and $x'$ belong to the same connected component of $X$. This completes the proof that $\pi _0(f)$ is a bijection.

By virtue of Whitehead's theorem (Theorem 3.2.6.1), it will suffice to show that for every vertex $x \in X$ having image $y =f(x) \in Y$ and every integer $n \geq 0$, the induced map $\theta : \pi _{n+1}( X, x) \rightarrow \pi _{n+1}( Y, y)$ is an isomorphism. Using Example 5.2.1.10, we can identify $\theta $ with the natural map $\pi _{n}( \operatorname{Hom}_{X}(x,x), \operatorname{id}_{x} ) \rightarrow \pi _{n}( \operatorname{Hom}_{Y}(y,y), \operatorname{id}_{y} )$, which is bijective by virtue of our assumption that $f$ induces a homotopy equivalence $\operatorname{Hom}_{X}(x,x) \rightarrow \operatorname{Hom}_{Y}(y,y)$. $\square$

Proof of Theorem 5.2.2.15. Assume first that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories. Then $F$ induces a homotopy equivalence of Kan complexes $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ (Remark 4.5.1.18). Passing to connected components, we conclude that the induced map $\pi _0( \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{D}}^{\simeq } )$ is bijective. In particular, $F$ is essentially surjective. We have a commutative diagram of Kan complexes

5.10
\begin{equation} \begin{gathered}\label{equation:restriction-to-endpoint-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{\theta _0} & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{D}})^{\simeq },} \end{gathered} \end{equation}

where the horizontal maps are homotopy equivalences (Theorem 4.5.4.1) and the vertical maps are Kan fibrations (Corollary 4.4.5.4). Applying Proposition 3.2.7.1, we conclude that for every vertex $(X,Y) \in \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq }$, the induced map of fibers

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & = & \{ (X,Y) \} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq } } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \\ & \rightarrow & \{ (X,Y) \} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{D}})^{\simeq } } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq } \\ & = & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \end{eqnarray*}

is a homotopy equivalence. It follows that $F$ is fully faithful.

Now suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which is fully faithful and essentially surjective. Using Corollary 5.2.2.7 and Remark 5.2.2.11, we see that the induced map $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is also fully faithful and essentially surjective, and is therefore a homotopy equivalence of Kan complexes (Lemma 5.2.2.16). It follows that the morphism $\theta _0$ in (5.10) is a homotopy equivalence of Kan complexes. Combining our assumption that $F$ is fully faithful with Proposition 3.2.7.1, we conclude that $\theta $ is also a homotopy equivalence. Applying Theorem 4.5.4.1, we conclude that $F$ is an equivalence of $\infty $-categories. $\square$

Corollary 5.2.2.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$. Then $F$ is fully faithful if and only if it induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$.

Corollary 5.2.2.18. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is fully faithful (when regarded as a functor of $\infty $-categories) if and only if it induces a homotopy equivalence from $X$ to a summand of $Y$.