Definition 10.3.1.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set. A sieve on $\operatorname{\mathcal{C}}$ is a simplicial subset $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ for which the inclusion map $\operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is a right fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
10.3.1 Sieves
Let $\operatorname{\mathcal{C}}$ be a category. Recall that a sieve on $\operatorname{\mathcal{C}}$ is a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ satisfying the following condition:
If $f: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ and $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$, then $X$ also belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$.
This condition has a counterpart in the setting of $\infty $-categories.
Example 10.3.1.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the simplicial subsets $\emptyset , \operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}$ are sieves on $\operatorname{\mathcal{C}}$.
Remark 10.3.1.3 (Base Change). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $\operatorname{\mathcal{D}}^{0} \subseteq \operatorname{\mathcal{D}}$ be a sieve on $\operatorname{\mathcal{D}}$. Then the inverse image $\operatorname{\mathcal{C}}^{0} = F^{-1}( \operatorname{\mathcal{D}}^{0} )$ is a sieve on $\operatorname{\mathcal{C}}$.
Remark 10.3.1.4 (Transitivity). Let $\operatorname{\mathcal{C}}$ be a simplicial set containing simplicial subsets $\operatorname{\mathcal{C}}^{1} \subseteq \operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}^0$ is a sieve on $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{C}}^1$ is a sieve on $\operatorname{\mathcal{C}}^0$ if and only if it is a sieve on $\operatorname{\mathcal{C}}$.
Proposition 10.3.1.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then a simplicial subset $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is a sieve if and only if it satisfies the following condition:
Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. If the final vertex $\sigma (n)$ is contained in $\operatorname{\mathcal{C}}^{0}$, then $\sigma $ is contained in $\operatorname{\mathcal{C}}^{0}$.
Proof. For every integer $n \geq 0$, the inclusion map $\{ n\} \hookrightarrow \Delta ^ n$ is right anodyne (Example 4.3.7.11). If the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is a right fibration, then condition $(\ast )$ is a special case of Proposition 4.2.4.5. Conversely, suppose that condition $(\ast )$ is satisfied, and let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. For every integer $0 < i \leq n$, the horn $\Lambda ^{n}_{i}$ contains the final vertex $\{ n\} \subseteq \Delta ^ n$. Consequently, if the restriction $\sigma |_{ \Lambda ^{n}_{i} }$ factors (uniquely) through $\iota $, then condition $(\ast )$ guarantees that $\sigma $ factors (uniquely) through $\iota $. Allowing $n$ and $i$ to vary, we conclude that $\iota $ is a right fibration. $\square$
Corollary 10.3.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset. Then $\operatorname{\mathcal{C}}^{0}$ is a sieve on $\operatorname{\mathcal{C}}$ if and only if it is a full subcategory of $\operatorname{\mathcal{C}}$ which satisfies the following condition:
If $f: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ and $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$, then $X$ also belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$.
Proof. By definition, $\operatorname{\mathcal{C}}^{0}$ is a sieve on $\operatorname{\mathcal{C}}$ if and only if the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is a right fibration. In particular, this guarantees that $\iota $ is an inner fibration, so that $\operatorname{\mathcal{C}}^{0}$ is a subcategory of $\operatorname{\mathcal{C}}$. It also guarantees that a morphism $f: X \rightarrow Y$ is contained in $\operatorname{\mathcal{C}}^{0}$ if and only if the object $Y$ is contained in $\operatorname{\mathcal{C}}^{0}$, so that the subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is full and satisfies $(\ast )$. Conversely, if $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is a full subcategory satisfying condition $(\ast )$, then $\iota $ satisfies the criterion of Proposition 10.3.1.5 and is therefore a right fibration. $\square$
Example 10.3.1.7. Let $\operatorname{\mathcal{C}}$ be a category and let $S$ be a simplicial subset of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then $S$ is a sieve on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.3.1.1) if and only if it has the form $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^0 )$, where $\operatorname{\mathcal{C}}^0$ is sieve on $\operatorname{\mathcal{C}}$ (in the usual category-theoretic sense).
Corollary 10.3.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a sieve. Then $\operatorname{\mathcal{C}}^{0}$ is a replete full subcategory of $\operatorname{\mathcal{C}}$. In particular, $\operatorname{\mathcal{C}}^{0}$ is an $\infty $-category.
Remark 10.3.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a sieve, and let $f: K \rightarrow \operatorname{\mathcal{C}}^{0}$ be a diagram. If the simplicial set $K$ is nonempty, then the inclusion map $\operatorname{\mathcal{C}}^{0}_{/f} \hookrightarrow \operatorname{\mathcal{C}}_{/f}$ is an isomorphism. In particular, an extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}^{0}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{0}$ if and only if it is a limit diagram in $\operatorname{\mathcal{C}}$. See Corollary 9.3.1.21 for a more general statement.
Remark 10.3.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$. Then $U$ restricts to a functor $U^{0}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}^{0}$. Moreover, the following conditions are equivalent:
The object $X \in \operatorname{\mathcal{C}}$ is subterminal (see Definition 9.3.2.2).
The functor $U^{0}$ is a trivial Kan fibration.
The functor $U^{0}$ is an equivalence of $\infty $-categories.
The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Remark 9.3.2.14. Note that $U^0$ is a pullback of $U$, and therefore a right fibration (Proposition 4.3.6.1). In particular, $U^{0}$ is an isofibration (Example 4.4.1.11), so the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20.
Example 10.3.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a sieve, and let $X$ be an object of $\operatorname{\mathcal{C}}^{0}$. If $C_{\bullet }$ is any simplicial object of $\operatorname{\mathcal{C}}$ satisfying $C_{0} = X$, then $C_{\bullet }$ can also be regarded as a simplicial object of $\operatorname{\mathcal{C}}^{0}$. Applying Remark 10.3.1.9, we deduce that $C_{\bullet }$ is a Čechnerve of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}^{0}$ if and only if it is a Čechnerve of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Definition 10.2.4.1).
Proposition 10.3.1.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $U$ restricts to an isomorphism of $\operatorname{\mathcal{E}}$ with a sieve $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$.
The morphism $U$ is a right covering map (Definition 4.2.3.8) and, for every vertex $C \in \operatorname{\mathcal{C}}$, the fiber $U^{-1} \{ C\} $ has at most one element.
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 4.2.3.12. Conversely, suppose that condition $(2)$ is satisfied, and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$. Our assumption that $U$ is a right covering map guarantees the existence of a pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{U} \ar [d] & \operatorname{N}_{\bullet }( \int ^{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathscr {F}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) } \]
where $\mathscr {F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is the contravariant homotopy transport representation of $\operatorname{\mathcal{C}}$, given concretely by the formula $\mathscr {F}(C) = U^{-1} \{ C\} $ (see Corollary 5.2.7.4). Our assumption that each of the sets $\mathscr {F}(C)$ has at most one element guarantees that the right vertical map induces an isomorphism from $\int ^{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathscr {F}$ to the sieve $\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ spanned by those objects $C$ for which the fiber $U^{-1} \{ C\} $ is nonempty. Condition $(1)$ now follows from Example 10.3.1.7 and Remark 10.3.1.3. $\square$
Corollary 10.3.1.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. Then the construction $(\operatorname{\mathcal{D}}^{0} \subseteq \operatorname{\mathcal{D}}) \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}^{0} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \operatorname{\mathcal{C}}$ induces a bijection
\[ \{ \textnormal{Sieves $\operatorname{\mathcal{D}}^{0} \subseteq \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Sieves $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$} \} . \]
Remark 10.3.1.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the collection of sieves on $\operatorname{\mathcal{C}}$ is closed under the formation of intersections. In particular, for every vertex $X \in \operatorname{\mathcal{C}}$, there is a smallest sieve $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ containing $X$. We will refer to $\operatorname{\mathcal{C}}^{0}$ as the sieve generated by $X$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{\mathcal{C}}^{0}$ admits a more explicit description: it is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C$ for which there exists a morphism $f: C \rightarrow X$.
Remark 10.3.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Remark 10.3.1.14): that is, the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is nonempty. Then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.3.2.2) if and only if it is a final object of $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 4.6.7.1).
Remark 10.3.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Suppose that $\operatorname{\mathcal{C}}$ admits finite limits, so that $f$ admits a Čechnerve $\operatorname{\check{C}}(X/Y)_{\bullet }$ (Proposition 10.2.5.6). If $Y$ is a subterminal object of $\operatorname{\mathcal{C}}$, then the underlying simplicial object of $\operatorname{\check{C}}(X/Y)_{\bullet }$ is also a Čechnerve of the object $X$. This follows by combining Remark 10.3.1.10 with Example 10.3.1.11.
Remark 10.3.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a $2$-simplex where $g$ is a monomorphism. Suppose that $\operatorname{\mathcal{C}}$ admits pullbacks, so that the morphisms $f$ and $h$ admit Čechnerves $\operatorname{\check{C}}_{\bullet }(X/Y)$ and $\operatorname{\check{C}}_{\bullet }(X/Z)$ (Proposition 10.2.5.6). Then the underlying simplicial objects of $\operatorname{\check{C}}_{\bullet }(X/Y)$ and $\operatorname{\check{C}}_{\bullet }(X/Z)$ are canonically isomorphic. To see this, let us regard (10.19) as morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$. Since the forgetful functor $\operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ preserves pullbacks (Corollary 7.1.6.20), we can identify $\operatorname{\check{C}}_{\bullet }(X/Y)$ with the image of $\operatorname{\check{C}}_{\bullet }( \widetilde{X} / \widetilde{Y} )$. The desired result now follows by applying Example 10.3.1.16 to the object $\widetilde{Y} \in \operatorname{\mathcal{C}}_{/Z}$ (which is subterminal by virtue of Remark 9.3.4.16).
10.19
\begin{equation} \begin{gathered}\label{equation:monomorphism-of-Cech-nerves} \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z, } \end{gathered} \end{equation}
It will often be convenient to work with a variant Definition 10.3.1.1.
Definition 10.3.1.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. A sieve on $Y$ is a sieve on the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$; that is, a full subcategory $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ satisfying the following condition:
For every $2$-simplex
in the $\infty $-category $\operatorname{\mathcal{C}}$, if $f$ is contained in the subcategory $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$, then $f'$ is also contained in $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$.
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [rr] \ar [dr]_{f'} & & X \ar [dl]^{f} \\ & Y & } \]
Example 10.3.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. By virtue of Remark 10.3.1.14, there is a smallest sieve $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ on the object $Y$ which contains the morphism $f$. We will refer to $\operatorname{\mathcal{C}}^{0}_{/Y}$ as the sieve generated by $f$. Concretely, a morphism $e: C \rightarrow Y$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if there exists a commutative diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Stated more informally, a morphism $e$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if it factors through $f$.
\[ \xymatrix@R =50pt@C=50pt{ C \ar [dr]_{e} \ar [rr] & & X \ar [dl]^{f} \\ & Y & } \]
Remark 10.3.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on an object $Y$. Then $\operatorname{\mathcal{C}}^{0}_{/Y}$ coincides with $\operatorname{\mathcal{C}}_{/Y}$ if and only if it contains the identity morphism ${\operatorname{id}}_{Y}: Y \rightarrow Y$. In particular, if $\operatorname{\mathcal{C}}^{0}_{/Y}$ is the sieve generated by a morphism $f: X \rightarrow Y$ (Example 10.3.1.19), then $\operatorname{\mathcal{C}}^{0}_{/Y} = \operatorname{\mathcal{C}}_{/Y}$ if and only if the morphism $f$ admits a right homotopy inverse $s: Y \rightarrow X$.
Remark 10.3.1.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on $Y$. The condition that a morphism $f: X \rightarrow Y$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$ depends only on the isomorphism class of $f$ as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. In particular, if $X$ is fixed, then the condition that $f$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$ depends only on the homotopy class $[f] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y)$.
We have the following variant of Corollary 10.3.1.13:
Proposition 10.3.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let $Y$ be an object of $\operatorname{\mathcal{C}}$ (which we also regard as an object of $\operatorname{\mathcal{D}}$). Then the construction $(\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}) \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}^{0}_{/Y} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}_{/Y}) } \operatorname{\mathcal{C}}_{/Y}$ induces a bijection
\[ \{ \textnormal{Sieves $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$} \} \xrightarrow {\sim } \{ \textnormal{Sieves $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$} \} . \]
Warning 10.3.1.23. Proposition 10.3.1.22 is not a special case of Corollary 10.3.1.13, because the slice category $\operatorname{\mathcal{D}}_{/Y}$ is usually not equivalent to the homotopy category of $\operatorname{\mathcal{C}}_{/Y}$.
Proof of Proposition 10.3.1.22. Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. We wish to show that there is a unique sieve $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$ with the following property: a morphism $f: X \rightarrow Y$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if the homotopy class $[f]$ belongs to the sieve $\operatorname{\mathcal{D}}^{0}_{/Y}$. The uniqueness assertion is immediate. To prove existence, we define $\operatorname{\mathcal{D}}^{0}_{/Y}$ to be the full subcategory of $\operatorname{\mathcal{D}}_{/Y}$ spanned by those homotopy classes $[f]: X \rightarrow Y$ such that $f$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$; by virtue of Remark 10.3.1.21, this condition depends only on the homotopy class $[f]$ and not on the choice of representative $f$. To complete the proof, it will suffice to show that the subcategory $\operatorname{\mathcal{D}}^{0}_{/Y} \subseteq \operatorname{\mathcal{D}}_{/Y}$ is a sieve. Suppose we are given a commutative diagram
10.20
\begin{equation} \begin{gathered}\label{equation:sieve-in-homotopy} \xymatrix@R =50pt@C=50pt{ X' \ar [dr]_{ [f'] } \ar [rr] & & X \ar [dl]^{ [f] } \\ & Y & } \end{gathered} \end{equation}
in the homotopy category $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We wish to show that if $[f]$ belongs to $\operatorname{\mathcal{D}}^{0}_{/Y}$, then $[f']$ also belongs to $\operatorname{\mathcal{D}}^{0}_{/Y}$. This follows from our assumption that $\operatorname{\mathcal{C}}^{0}_{/Y}$ is a sieve on $Y$, since (10.20) can be lifted to a $2$-simplex in the $\infty $-category $\operatorname{\mathcal{C}}$. $\square$
Notation 10.3.1.24 (Pullback Sieves). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on the object $Y$. We let $f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ denote the full subcategory of $\operatorname{\mathcal{C}}_{/X}$ spanned by those objects $e: C \rightarrow X$ for which the composition $(f \circ e): C \rightarrow Y$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$. By virtue of Remark 10.3.1.21, this condition is independent of the choice of composition $f \circ e$. The subcategory $f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ is a sieve on the object $X$, which we will refer to as the pullback of $\operatorname{\mathcal{C}}^{0}_{/Y}$ along the morphism $f$.
Example 10.3.1.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pullback diagram and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve on $Y$ generated by the morphism $f$. Then the pullback $u^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ is the sieve on $Y'$ generated by the morphism $f'$. In other words, a morphism $[v]: C \rightarrow X'$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ factors through $[f']$ if and only if the composite morphism $[u] \circ [v]$ factors through $[f]$ (see Warning 7.6.2.3).
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r]^-{u} & Y, } \]
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is dense if the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Definition 8.4.1.5). We now consider a slight variant of this condition.
Definition 10.3.1.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We say that a sieve $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ on $X$ is dense if the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/X}$.
Warning 10.3.1.27. The terminology of Definition 10.3.1.26 has the potential to create confusion. Since the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ creates colimits (Proposition 7.1.4.20), a sieve $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ which is dense in the sense of Definition 10.3.1.26 is also dense when regarded as a full subcategory of $\operatorname{\mathcal{C}}_{/X}$ (in the sense of Definition 8.4.1.5). Beware that the converse is false in general (Example 10.3.1.28). However, it is true if $\operatorname{\mathcal{C}}$ admits finite products (Proposition 10.3.1.29).
Example 10.3.1.28. Let $\operatorname{\mathcal{C}}$ be the $1$-dimensional simplicial set associated to the directed graph depicted in the diagram and let $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ be the sieve spanned by the objects $A$ and $B$. Then $\operatorname{\mathcal{C}}^{0}_{/X}$ is dense when regarded as a full category of $\operatorname{\mathcal{C}}_{/X}$ (in the sense of Definition 8.4.1.5), but not when regarded as a sieve on $X$ (in the sense of Definition 10.3.1.26).
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X \\ Y & B, \ar [l] \ar [u] } \]
Proposition 10.3.1.29. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pairwise products and let $X \in \operatorname{\mathcal{C}}$. Then a sieve $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ is dense (in the sense of Definition 10.3.1.26) if and only if it is dense when regarded as a subcategory of $\operatorname{\mathcal{C}}_{/X}$ (in the sense of Definition 8.4.1.5).
Proof. Assume that $\operatorname{\mathcal{C}}^{0}_{/X}$ is a dense subcategory of $\operatorname{\mathcal{C}}_{/X}$; we will show that it is dense when regarded as a sieve (for the reverse implication, see Warning 10.3.1.27). By assumption, the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/X}$ is left Kan extended from $\operatorname{\mathcal{C}}^0_{/X}$. We wish to show that the forgetful functor $U: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is also left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/X}$. To prove this, it suffices to show that the functor $U$ preserves colimits. This is a special case of Corollary 7.1.4.22, since the functor $U$ admits a right adjoint (given on objects by the construction $Y \mapsto X \times Y$; see Proposition 7.6.1.14). $\square$
Example 10.3.1.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ is a dense sieve on $X$ (see Example 7.3.3.8).
Remark 10.3.1.31. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}^{1}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ be sieves on an object $X$. If $\operatorname{\mathcal{C}}^{0}_{/X}$ is dense, then $\operatorname{\mathcal{C}}^{1}_{/X}$ is also dense. See Proposition 7.3.8.6.
Remark 10.3.1.32. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on $X$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, which we regard as an object of $\operatorname{\mathcal{C}}_{/Y}$, and let $\operatorname{\mathcal{C}}^{0}_{/X} = f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y})$ be the pullback sieve (Notation 10.3.1.24). Then the forgetful functor $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/Y}$ at $f$ if and only if the following condition is satisfied:
The composite map
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.
\[ ( \operatorname{\mathcal{C}}_{/X}^{0} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }_{/X} \rightarrow \operatorname{\mathcal{C}} \]
In particular, the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ is dense if and only if it satisfies condition $(\ast _{f})$ for every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.
Proposition 10.3.1.33. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. For every dense sieve $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$, the pullback sieve $f^{\ast } \operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/X}$ is also dense.
Proof. This is an immediate consequence of the criterion of Remark 10.3.1.32. $\square$
Proposition 10.3.1.34 (Transitivity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y}, \operatorname{\mathcal{C}}^{1}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be sieves on $Y$. Assume that:
The sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ is dense.
For each morphism $f: X \rightarrow Y$ which belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$, the pullback sieve $f^{\ast }( \operatorname{\mathcal{C}}^{1}_{/Y} ) \subseteq \operatorname{\mathcal{C}}_{/X}$ is dense.
Then $\operatorname{\mathcal{C}}^{1}_{/Y}$ is also a dense sieve.
Proof. Let $U: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map; we wish to show that $U$ is left Kan extended from $\operatorname{\mathcal{C}}^{1}_{/Y}$. Assumption $(1)$ guarantees that $U$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/Y}$. By virtue of Corollary 7.3.8.8, it will suffice to show that $U|_{ \operatorname{\mathcal{C}}^{0}_{/Y} }$ is left Kan extended from the intersection $\operatorname{\mathcal{C}}^{01}_{/Y} = \operatorname{\mathcal{C}}^{0}_{/Y} \cap \operatorname{\mathcal{C}}^{1}_{/Y}$. Fix a morphism $f: X \rightarrow Y$ which belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$; we wish to show that $U$ is left Kan extended from $\operatorname{\mathcal{C}}^{01}_{/Y}$ at $f$. This follows from our assumption that $f^{\ast } \operatorname{\mathcal{C}}^{01}_{/Y} = f^{\ast } \operatorname{\mathcal{C}}^{1}_{/Y}$ is a dense sieve on $X$. $\square$