10.2.3 Monomorphisms
Let $\operatorname{\mathcal{C}}$ be a category. Recall that a morphism $f: X_0 \rightarrow X$ of $\operatorname{\mathcal{C}}$ is a monomorphism if, for every object $C$ of $\operatorname{\mathcal{C}}$, the composition map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0) \xrightarrow { f \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, X) \]
is injective. This notion has an obvious counterpart in the setting of $\infty $-categories.
Definition 10.2.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. We say that $f$ is a monomorphism if, for every object $C \in \operatorname{\mathcal{C}}$, the composition map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \]
induces a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$.
Warning 10.2.3.2. Let $f: X_0 \rightarrow X$ be a map of Kan complexes. The assertion that $f$ is a monomorphism can be given two different interpretations:
- $(1)$
The map $f$ is a monomorphism in the ordinary category of $\operatorname{Set_{\Delta }}$ of simplicial sets.
- $(2)$
The map $f$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces.
Beware that these conditions are unrelated to one another. Condition $(2)$ is homotopy invariant: it is the requirement that $f$ restricts to a homotopy equivalence of $X_0$ with a summand of $X$ (Proposition 10.2.3.14). Condition $(1)$ is very far from being homotopy invariant: we can always arrange that it is satisfied by replacing $X$ by a homotopy equivalent Kan complex (see Exercise 3.1.7.11).
Notation 10.2.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ having source $X_0$ and target $X$. If $f$ is a monomorphism, we will sometimes visually emphasize this by denoting $f$ with a hooked arrow (that is, we will write $f: X_0 \hookrightarrow X$ in place of $f: X_0 \rightarrow X$). Beware that this convention can be ambiguous in some situations (for example if $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces; see Warning 10.2.3.2).
Example 10.2.3.4. Let $\operatorname{\mathcal{C}}$ be a category and let $f: X_0 \rightarrow X$ be a morphism in $\operatorname{\mathcal{C}}$. Then $f$ is a monomorphism in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.2.3.1) if and only if it is a monomorphism in the usual category-theoretic sense.
Example 10.2.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. Suppose that $X$ is subterminal. Then $f$ is a monomorphism if and only if $X_0$ is subterminal.
Example 10.2.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a final object ${\bf 1}$, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then there is a morphism $f: X \rightarrow {\bf 1}$, which is uniquely determined up to homotopy. It follows from Example 10.2.3.5 that $f$ is a monomorphism if and only if $X$ is subterminal.
Example 10.2.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every isomorphism in $\operatorname{\mathcal{C}}$ is a monomorphism.
Warning 10.2.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $i: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a left homotopy inverse $r: X \rightarrow X_0$. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then $i$ is automatically a monomorphism. In general, this is not necessarily true. For example, let $(X,x)$ be a pointed Kan complex, and regard the inclusion map $i: \{ x\} \rightarrow X$ as a morphism in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces. Then $i$ has a left homotopy inverse (given by the constant map $X \rightarrow \{ x\} $). However, $i$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{S}}$ only if $x$ belongs to a contractible connected component of $X$ (Proposition 10.2.3.14).
Variant 10.2.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that $f$ is an epimorphism if it is a monomorphism when viewed as a morphism of the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$: that is, if the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, C ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C ) \]
induces a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$, for each object $C \in \operatorname{\mathcal{C}}$. We will generally avoid this terminology, to avoid confusion with the notion of quotient morphism which we introduce in §10.2.4 (see Warning 10.2.4.10).
Proposition 10.2.3.14. Let $f: X_0 \rightarrow X$ be a map of Kan complexes. The following conditions are equivalent:
- $(1)$
The map $f$ is a monomorphism in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, in the sense of Definition 10.2.3.1.
- $(2)$
The map $f$ induces a homotopy equivalence of $X_0$ with a summand of $X$.
- $(3)$
For every vertex $x \in X$, the homotopy fiber $\{ x\} \times ^{\mathrm{h}}_{X} X_0$ is either empty or contractible.
Proof.
The implication $(1) \Rightarrow (2)$ is immediate from the definitions, and the equivalence $(2) \Leftrightarrow (3)$ follows from Remark 3.4.0.6. We will complete the proof by showing that $(2)$ implies $(1)$. Without loss of generality, we may assume that $X_0$ is a summand of $X$ and that $f$ is the inclusion map. For every Kan complex $C$, composition with $f$ induces an isomorphism of $\operatorname{Fun}(C,X_0)$ with the summand of $\operatorname{Fun}(C,X)$ spanned by those maps $g: C \rightarrow X$ which factor through $X_0$. Combining this observation with Remark 5.6.1.5, we deduce that the map $\operatorname{Hom}_{\operatorname{\mathcal{S}}}(C, X_0) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{S}}}(C,X)$ is a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{S}}}(C,X_0)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{S}}}(C,X)$.
$\square$
Proposition 10.2.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a monomorphism if and only if it is subterminal when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.
Proof.
By virtue of Proposition 10.2.3.14, the morphism $f$ is a monomorphism if and only if, for every object $C \in \operatorname{\mathcal{C}}$, every homotopy fiber of the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is either empty or contractible. For every vertex $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$, Corollary 4.6.9.18 identifies the corresponding homotopy fiber with the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( g, f )$. The desired result now follows by allowing $g$ to vary.
$\square$
Corollary 10.2.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a monomorphism if and only if it satisfies the following condition for each $n \geq 3$:
- $(\ast _ n)$
Let $\sigma : \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the composition
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subset \Lambda ^{n}_{n} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]
coincides with $f$. Then $\sigma $ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.
Proof.
By virtue of Remark 10.2.2.7, this is a reformulation of Proposition 10.2.3.15.
$\square$
Corollary 10.2.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\sigma $ denote the composite map
\[ \Delta ^1 \times \Delta ^1 \xrightarrow { (i,j) \mapsto ij } \Delta ^1 \xrightarrow {u} \operatorname{\mathcal{C}}, \]
which we depict as a diagram
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [r]^-{\operatorname{id}} \ar [d]^{\operatorname{id}} & X_0 \ar [d]^{f} \\ X_0 \ar [r]^-{f} & X. } \]
Then $f$ is a monomorphism if and only if $\sigma $ is a pullback square in $\operatorname{\mathcal{C}}$.
Proof.
Combine Proposition 10.2.3.15 with the characterization of subterminal objects supplied by Proposition 10.2.2.10 (see Proposition 7.6.3.14).
$\square$
Corollary 10.2.3.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty $-categories and let $f$ be a morphism in $\operatorname{\mathcal{C}}$. If $F(f)$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$, then $f$ is a monomorphism in $\operatorname{\mathcal{C}}$.
Proof.
Combine Corollary 10.2.3.17 with Corollary 7.1.5.17.
$\square$
Corollary 10.2.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \hookrightarrow Y$ be a monomorphism in $\operatorname{\mathcal{C}}$. If $f': X' \rightarrow Y'$ is a retract of $f$ (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$), then $f'$ is also a monomorphism.
Proof.
Combine Corollaries 10.2.3.17 and 8.5.1.11.
$\square$
Corollary 10.2.3.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves pullbacks. Then $F$ carries monomorphisms in $\operatorname{\mathcal{C}}$ to monomorphisms in $\operatorname{\mathcal{D}}$.
Example 10.2.3.23. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a left adjoint. Then $F$ carries subterminal objects of $\operatorname{\mathcal{C}}$ to subterminal objects of $\operatorname{\mathcal{D}}$, and carries monomorphisms in $\operatorname{\mathcal{C}}$ to monomorphisms in $\operatorname{\mathcal{D}}$. This follows from Corollaries 10.2.2.11, 10.2.3.21 and 7.1.3.21.
We now prove a variant of Proposition 10.2.3.14 for the $\infty $-category $\operatorname{\mathcal{QC}}$ of (small) $\infty $-categories.
Proposition 10.2.3.25. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The functor $F$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ which contains every isomorphism from $F(X)$ to $F(Y)$.
- $(3)$
The functor $F$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a replete subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$.
Proof.
We first show that $(1)$ implies $(2)$. By virtue of Corollary 5.3.7.7, we may assume without loss of generality that $F$ is an isofibration of $\infty $-categories. In this case, it follows from Exercise 7.6.4.12 that the diagonal inclusion $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ (formed in the ordinary category of simplicial sets) can be identified with the relative diagonal of $F$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. Combining this observation with Corollary 10.2.3.17, we deduce that $F$ is a monomorphism (in the $\infty $-category $\operatorname{\mathcal{QC}}$) if and only if $\delta $ is an equivalence of $\infty $-categories. In particular, if $F$ is a monomorphism, then $\delta $ is fully faithful: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}}( \delta (X), \delta (Y) ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
is a homotopy equivalence. Our assumption that $F$ is an isofibration guarantees that the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is Kan fibration (Proposition 4.6.1.21). Applying Corollary 3.2.8.6, we deduce that $F_{X,Y}$ restricts to a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. To complete the proof, it will suffice to show that this summand every isomorphism from $F(X)$ to $F(Y)$. In fact, we will prove something more precise: the induced map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a trivial Kan fibration from $\operatorname{\mathcal{C}}^{\simeq }$ to a summand of $\operatorname{\mathcal{D}}^{\simeq }$. This follows again from Corollary 3.2.8.6, since $F^{\simeq }$ is a Kan fibration (Proposition 4.4.3.7).
We now show that $(2)$ implies $(3)$. As above, we may assume that $F$ is an isofibration. Let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ denote the homotopy categories of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. We define a subcategory $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_{0} \subseteq \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as follows:
An object $\overline{X}$ of $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ belongs to the subcategory $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_{0}$ if and only if it is the image of an object $X$ of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
A morphism $\overline{u}: \overline{X} \rightarrow \overline{Y}$ of $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ belongs to the subcategory $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_0$ if and only if it is the image of a morphism $u$ of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
We first claim that the subcategory $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_0$ is well-defined: that is, if $\overline{u}: \overline{X} \rightarrow \overline{Y}$ and $\overline{v}: \overline{Y} \rightarrow \overline{Z}$ are composable morphisms of $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ which can be lifted to morphisms $u: X \rightarrow Y$ and $v: Y' \rightarrow Z$ of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then the composite morphism $\overline{v} \circ \overline{u}$ has the same property. Assumption $(2)$ guarantees that the identity morphism $\operatorname{id}_{ \overline{Y} }$ belongs to the image of the map
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, Y' ) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y') ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Y} ) ) \simeq \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{D}}}}( \overline{Y}, \overline{Y} ). \]
That is, there exists a morphism $e: Y \rightarrow Y'$ in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ satisfying $F(e) = \operatorname{id}_{ \overline{Y} }$. Replacing $v$ by the composition $v \circ e$, we can arrange that $Y = Y'$: that is, that $u$ and $v$ are composable morphisms in the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. It then follows that $\overline{v} \circ \overline{u} = F( v \circ u)$ is also morphism of $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_0$, as desired.
By virtue of Proposition 4.1.2.10, the subcategory $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}_0 \subseteq \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is the homotopy category of a (unique) subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$. Using condition $(2)$, we see that the subcategory $\operatorname{\mathcal{D}}_0$ is replete. By construction, the functor $F$ factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {F_0} \operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we can identify $\operatorname{Hom}_{\operatorname{\mathcal{D}}_0}( F(X), F(Y) )$ with the summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ given by the essential image of $F_{X,Y}$. Invoking assumption $(2)$, we see that the functor $F_0$ is fully faithful. By construction, $F_0$ is also surjective on objects, and is therefore an equivalence of $\infty $-categories (Theorem 4.6.2.19). This completes the proof of the implication $(2) \Rightarrow (3)$.
We now show that $(3)$ implies $(1)$. Assume that $F$ induces an equivalence from $\operatorname{\mathcal{C}}$ to a replete subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{C}}$; we wish to show that $F$ is a monomorphism. By virtue of Remark 10.2.3.12 (and Example 10.2.3.7), it will suffice to show that the inclusion map $\iota : \operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$ is a monomorphism in $\operatorname{\mathcal{QC}}$. Fix an $\infty $-category $\operatorname{\mathcal{B}}$, so that composition with the homotopy class $[\iota ]$ induces a map of Kan complexes $\theta : \operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}_0 ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. We wish to show that $\theta $ induces a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}_0 )$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. By virtue of Remark 5.6.4.5, it will suffice to prove the analogous assertion for the inclusion map $\operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}_0)^{\simeq } \hookrightarrow \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq }$, which follows immediately from Corollary 4.4.3.13.
$\square$
Corollary 10.2.3.26. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then $F$ is a monomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$.
Proof.
Let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the essential image of $F$. By virtue of Proposition 10.2.3.25, it will suffice to show that $F$ induces an equivalence from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_0$, which is a reformulation of the requirement that $F$ is fully faithful (Corollary 4.6.2.21).
$\square$
Warning 10.2.3.27. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a subcategory. Beware that, if we do not assume that $\operatorname{\mathcal{C}}_0$ is replete (or full), then the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ need not be a monomorphism in $\operatorname{\mathcal{QC}}$. For example, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}})$ is the nerve of a category $\operatorname{\mathcal{D}}$. Then the $0$-skeleton $\operatorname{\mathcal{C}}_0 = \operatorname{sk}_{0}( \operatorname{\mathcal{C}})$ is always subcategory of $\operatorname{\mathcal{C}}$ (namely, the subcategory spanned by the identity morphisms of $\operatorname{\mathcal{C}}$). However, the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is a monomorphism in $\operatorname{\mathcal{QC}}$ if and only if every isomorphism in $\operatorname{\mathcal{D}}$ is an identity morphism.
Definition 10.2.3.28. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. A subobject of $X$ is a subterminal object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$: that is, an object which is given by a monomorphism $f: X_0 \hookrightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Proposition 10.2.3.15). In this situation, we will sometimes abuse terminology by referring to $X_0$ as a subobject of $X$ and writing $X_0 \subseteq X$; in this case, we implicitly assume that a monomorphism $X_0 \hookrightarrow X$ has been specified.
Notation 10.2.3.29. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. We let $\operatorname{Sub}(X)$ denote the set $\operatorname{Sub}( \operatorname{\mathcal{C}}_{/X} )$ of isomorphism classes of subterminal objects of $\operatorname{\mathcal{C}}_{/X}$ (see Notation 10.2.2.12). If $f: X_0 \hookrightarrow X$ is a monomorphism, we write $[X_0] \in \operatorname{Sub}(X)$ for the isomorphism class of $f$. We will sometimes abuse notation by identifying the isomorphism class $[X_0]$ with the object $X_0$ itself: by virtue of Proposition 10.2.2.14, this identification is essentially harmless provided that $X_0$ is understood as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ (that is, provided that we remember the data of the monomorphism $f$). We will refer to $\operatorname{Sub}(X)$ as the set of subobjects of $X$, and we endow it with the partial ordering described in Notation 10.2.2.12: that is, if $f_0: X_0 \hookrightarrow X$ and $f_1: X_1 \hookrightarrow X$ are monomorphisms, we write $[X_0] \subseteq [X_1]$ if there exists a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [dr]_{f_0} \ar [rr]^{g} & & X_1 \ar [dl]^{f_1} \\ & Y & } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. In this case, $g$ is automatically a monomorphism (see Remark 10.2.3.12).
Example 10.2.3.30. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$. Using Proposition 10.2.3.14, we can identify $\operatorname{Sub}(X)$ with the partially ordered collection of all summands of $X$. Alternatively, we can identify $\operatorname{Sub}(X)$ as the collection of all subsets of the set $\pi _0(X)$ (see Exercise 1.1.6.16).
In particular, the collection of monomorphisms in $\operatorname{\mathcal{C}}$ is closed under pullbacks.
Construction 10.2.3.34 (Inverse Images). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products. Then every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ determines a pullback functor
\[ f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X} \quad \quad Y' \mapsto X \times _{Y} Y' \]
(see Proposition 7.6.3.16). The functor $f^{\ast }$ has a left adjoint, and therefore carries subterminal objects of $\operatorname{\mathcal{C}}_{/X}$ to subterminal objects of $\operatorname{\mathcal{C}}_{/Y}$. Passing to isomorphism classes, we obtain a map of partially ordered sets $f^{-1}: \operatorname{Sub}(Y) \rightarrow \operatorname{Sub}(X)$, given concretely by the formula $f^{-1} [Y_0] = [ Y_0 \times _{Y} X ]$. Since the functor $f^{\ast }$ preserves products, $f^{-1}$ is a homomorphism of lower semilattices: that is, it satisfies the identities
\[ f^{-1}( [Y_0] \cap [Y_1] ) = f^{-1}([Y_0]) \cap f^{-1}( [Y_1] ) \quad \quad f^{-1}([Y]) = [X]. \]