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4.6.4 Oriented Fiber Products

Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories. To every pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, one can associate the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, whose objects are triples $(C, D, \eta )$ where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$ (Notation 2.1.4.19). This construction has a counterpart in the setting of $\infty $-categories.

Definition 4.6.4.1 (The Oriented Fiber Product). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the simplicial set given by the iterated fiber product

\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}. \]

We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$.

As our notation suggests, we will be primarily interested in the special case of Definition 4.6.4.1 where the simplicial sets $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories.

Proposition 4.6.4.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category, and suppose we are given morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Then the projection map $\theta : \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration of simplicial sets.

Proof. By construction, we have a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d]^{\theta } & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}) \ar [d]^{\theta _0} \\ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{E}}). } \]

Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, the restriction map $\theta _0$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3). Invoking Remark 4.5.7.7, we conclude that $\theta $ is an isofibration of simplicial sets. $\square$

Corollary 4.6.4.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category.

Proof. By virtue of Proposition 4.6.4.2, the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an isofibration. Since $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category (Remark 4.5.7.3). $\square$

Example 4.6.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between ordinary categories, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ denote the oriented fiber product of Notation 2.1.4.19. Since the nerve construction is compatible with the formation of inverse limits and functor categories, we have a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}) \simeq ( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ). \]

Consequently, Definition 4.6.4.1 can be viewed as a generalization of the classical oriented fiber product.

Example 4.6.4.5. Let $X$, $Y$, and $Z$ be topological spaces, and suppose we are given a pair of continuous functions $f: X \rightarrow Z$ and $g: Y \rightarrow Z$. We let $X \times ^{h}_{Z} Y$ denote the set of all triples $(x,y,\eta )$ where $x$ is a point of $X$, $y$ is a point of $Y$, and $\eta : [0,1] \rightarrow Z$ is a continuous function satisfying $\eta (0) = f(x)$ and $\eta (1) = g(y)$. We will refer to $X \times ^{h}_{Z} Y$ as the homotopy fiber product of $X$ and $Y$ over $Z$. The homotopy fiber product $X \times ^{h}_{Z} Y$ carries a natural topology, which is defined by viewing it as a subspace of the product $X \times Y \times \operatorname{Hom}_{\operatorname{Top}}( [0,1], Z)$ (where we endow the path space $\operatorname{Hom}_{\operatorname{Top}}([0,1],Z)$ with the compact-open topology). We then have a canonical isomorphism of simplicial sets

\[ \operatorname{Sing}_{\bullet }( X \times _{Z}^{h} Y) \simeq \operatorname{Sing}_{\bullet }(X) \operatorname{\vec{\times }}_{ \operatorname{Sing}_{\bullet }(Z) } \operatorname{Sing}_{\bullet }(Y) \]

where the right hand side is the oriented fiber product of Definition 4.6.4.1.

Example 4.6.4.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets. Then the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\Delta ^0} \operatorname{\mathcal{D}}$ can be identified with the cartesian product $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$.

Remark 4.6.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ and $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$ be the opposite morphisms. Then we have a canonical isomorphism of simplicial sets

\[ (\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq (\operatorname{\mathcal{D}}^{\operatorname{op}} \operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} ). \]

Remark 4.6.4.8. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, which we identify with a vertex of the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. For any simplicial set $J$, we have canonical isomorphisms

\[ \operatorname{Fun}(J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} ) \simeq \operatorname{Fun}_{ K/ }( J \diamond K, \operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}(J, \{ F\} \operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}_{ K/ }( K \diamond J, \operatorname{\mathcal{C}}), \]

where $J \diamond K$ and $K \diamond J$ denote the simplicial sets defined in Notation 4.5.5.3. Restricting to vertices, we obtain bijections

\[ \{ \textnormal{Morphisms $J \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $} \} \simeq \{ \textnormal{Morphisms $\overline{F}: J \diamond K \rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}|_{K} = F$} \} \]
\[ \{ \textnormal{Morphisms $J \rightarrow \{ F\} \operatorname{\vec{\times }}_{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \operatorname{\mathcal{C}}$} \} \simeq \{ \textnormal{Morphisms $\overline{F}': K \diamond J \rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}'|_{K} = F$} \} . \]

Example 4.6.4.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we identify with morphisms of simplicial sets $X,Y: \Delta ^{0} \rightarrow \operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1 is the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.

The following result is a relative version of Proposition 4.6.1.8:

Proposition 4.6.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. We will prove the second assertion; the first follows by a similar argument. Let $A \hookrightarrow B$ be a right anodyne morphism of simplicial sets; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]

admits a solution. Unwinding the definitions, we are reduced to showing that a map of simplicial sets

\[ \sigma _0: B \coprod _{A} (A \diamond \{ X\} ) \rightarrow \operatorname{\mathcal{C}} \]

can be extended to a map $\sigma : B \diamond \{ X\} \rightarrow \operatorname{\mathcal{C}}$ (see Notation 4.5.5.3). By virtue of Lemma 4.5.6.2, it will suffice to show that the inclusion map

\[ \iota : B \coprod _{A} ( A \diamond \{ X\} ) \hookrightarrow B \diamond \{ X\} \]

is a categorical equivalence of simplicial sets, which follows from Corollary 4.5.5.14. $\square$

Corollary 4.6.4.11. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category containing an object $X$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then the projection map $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration.

Proof. Unwinding the definition, we have pullback diagrams

\[ \xymatrix@R =50pt@C=50pt{ \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\ar [r] \ar [d] & \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \{ X\} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}& \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}. } \]

The desired result now follows by combining Proposition 4.6.4.10 with Remark 4.2.1.8. $\square$

If $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between ordinary categories, then Remark 4.3.1.11 supplies canonical isomorphisms

\[ \operatorname{\mathcal{C}}_{/F} \simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{C}}_{F/} \simeq \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]

Our next goal is to establish a similar result in the setting of $\infty $-categories. Here the situation is a bit more subtle: if $F: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram in an $\infty $-category $\operatorname{\mathcal{C}}$, then the simplicial sets $\operatorname{\mathcal{C}}_{/F}$ and $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ are generally not isomorphic. However, we will show that they are equivalent as $\infty $-categories.

Construction 4.6.4.12 (The Slice Diagonal Morphism). Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $c: \operatorname{\mathcal{C}}_{/F} \diamond K \rightarrow \operatorname{\mathcal{C}}_{/F} \star K$ be the comparison morphism of Notation 4.5.5.3. By virtue of Remark 4.6.4.8, the composite map

\[ \operatorname{\mathcal{C}}_{/F} \diamond K \xrightarrow {c} \operatorname{\mathcal{C}}_{/F} \star K \rightarrow \operatorname{\mathcal{C}} \]

determines a morphism of simplicial sets $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we will refer to as the slice diagonal morphism. Similarly, the composition

\[ K \diamond \operatorname{\mathcal{C}}_{F/} \rightarrow K \star \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}} \]

determines a morphism of simplicial sets $\delta _{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$, which we will refer as the coslice diagonal morphism.

Remark 4.6.4.13. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. For every simplicial set $J$, composition with the slice diagonal $\delta _{/F}$ of Construction 4.6.4.12 determines a map of sets

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( J, \operatorname{\mathcal{C}}_{/F} ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} ). \]

Under the bijection of Remark 4.6.4.8, this identifies with the map

\[ \operatorname{Hom}_{(\operatorname{Set_{\Delta }})_{K/} }( J \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{(\operatorname{Set_{\Delta }})_{K/} }( J \diamond K, \operatorname{\mathcal{C}}) \]

given by precomposition with the comparison map $c_{J,K}: J \diamond K \twoheadrightarrow J \star K$ of Notation 4.5.5.3.

Remark 4.6.4.14. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the slice and coslice diagonal morphisms

\[ \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]

are monomorphisms of simplicial sets. This follows from Remark 4.6.4.13, together with the observation that for every simplicial set $J$, the comparison maps

\[ c_{J,K}: J \diamond K \twoheadrightarrow J \star K \quad \quad c_{K,J}: K \diamond J \twoheadrightarrow K \star J \]

are epimorphisms (see Exercise 4.5.5.5)

Exercise 4.6.4.15. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $f$ can be identified with a vertex of the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, which (to avoid confusion) we will temporarily denote by $F$. Applying Construction 4.6.4.12 to the inclusion map $\{ F \} \hookrightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$, we obtain a monomorphism of simplicial sets $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} $, which induces a monomorphism

\[ u: \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} . \]

Show that the slice diagonal morphism $\delta _{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ of Construction 4.6.4.12 factors (uniquely) through $u$. In particular, $\delta _{/f}$ determines a morphism of simplicial sets $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$. Similarly, the coslice diagonal morphism $\delta _{f/}$ induces a morphism of simplicial sets $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{F/} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$.

We can now formulate the main result of this section.

Theorem 4.6.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the slice and coslice diagonal maps

\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \delta _{F/}: \operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]

are equivalences of $\infty $-categories.

Corollary 4.6.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C \in \operatorname{\mathcal{C}}$ be an object. Then the slice and coslice diagonal maps

\[ \delta _{/C}: \operatorname{\mathcal{C}}_{/C} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} \quad \quad \delta _{C/}: \operatorname{\mathcal{C}}_{C/} \hookrightarrow \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}} \]

are equivalences of $\infty $-categories.

Corollary 4.6.4.18. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. Then the induced functors

\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]

are equivalences of $\infty $-categories.

Proof. We will show that $G'$ is an equivalence of $\infty $-categories; the analogous statement for $G''$ follows by a similar argument. Note that we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/F} \ar [r]^-{G'} \ar [d] & \operatorname{\mathcal{D}}_{ / (G \circ F) } \ar [d] \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \ar [r]^-{\overline{G}'} & \operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) } \{ G \circ F \} , } \]

where the vertical maps are equivalences of $\infty $-categories by virtue of Theorem 4.6.4.16. It will therefore suffice to show that $\overline{G}'$ is an equivalence of $\infty $-categories. This follows by applying Corollary 4.5.4.7 to the diagram

\[ \xymatrix { \operatorname{Fun}(\Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [r]^-{G \circ } \ar [d] & \operatorname{Fun}(\Delta ^0 \diamond K, \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \ar [r]^-{G \circ } & \operatorname{Fun}(K,\operatorname{\mathcal{D}}); } \]

here the vertical maps are isofibrations by virtue of Corollary 4.4.5.3, and the horizontal maps are equivalences by virtue of Remark 4.5.1.16. $\square$

Corollary 4.6.4.19. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be fully faithful functor of $\infty $-categories and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the induced functors

\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]

are also fully faithful.

Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $G$ (Definition 4.6.2.9), so that $G$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.19). By virtue of Corollary 4.6.4.18, the functors $G'$ and $G''$ restrict to equivalences

\[ \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}'_{/(G \circ F)} \quad \quad \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}}'_{(G \circ F)/} \]

We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $G: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is the inclusion of a full subcategory. In this case, the functors $G'$ and $G''$ are also the inclusions of full subcategories, hence fully faithful (Example 4.6.2.2). $\square$

Remark 4.6.4.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The slice diagonal morphism $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ carries each $n$-simplex of $\operatorname{\mathcal{C}}_{/F}$ to an $n$-simplex $\sigma $ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we can identify with a map $\Delta ^0 \diamond K \rightarrow \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})$. It is not difficult to see that this map factors (uniquely) through the comparison map $c: \Delta ^0 \diamond K \twoheadrightarrow K^{\triangleleft }$ of Notation 4.5.5.3, and can therefore also be viewed as an $n$-simplex of the simplicial set $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $. Consequently, $\delta _{/F}$ factors as a composition

\[ \operatorname{\mathcal{C}}_{/F} \xrightarrow { \delta '_{/F} } \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \xrightarrow {\iota } \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} , \]

where $\iota $ is a monomorphism of simplicial sets given by precomposition with $c$. Since $c$ is a categorical equivalence of simplicial sets (Theorem 4.5.5.8), the functor $\iota $ is an equivalence of $\infty $-categories: this follows by applying Corollary 4.5.4.7 to the diagram

\[ \xymatrix { \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) \ar [rr]^{ \circ c} \ar [dr] & & \operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), & } \]

since the vertical maps are isofibrations (Corollary 4.4.5.3). It follows from Theorem 4.6.4.16 that the functor

\[ \delta '_{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{Fun}(K^{\triangleleft },\operatorname{\mathcal{C}}) \times _{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \]

is also an equivalence of $\infty $-categories. Similarly, the coslice diagonal morphism $\delta _{F/}$ factors through an equivalence of $\infty $-categories

\[ \delta '_{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}). \]

We now turn to the proof of Theorem 4.6.4.16. As we will see, it is essentially a reformulation of Theorem 4.5.5.8.

Lemma 4.6.4.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram indexed by a simplicial set $K$. Suppose we are given a pair of diagrams $e_0, e_1: J \rightarrow \operatorname{\mathcal{C}}_{/F}$ indexed by a simplicial set $J$, which we identify with diagrams $F_0, F_1: J \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $F_0|_{K} = F = F_{1}|_{K}$. The following conditions are equivalent:

$(1)$

The diagrams $e_0$ and $e_1$ are isomorphic when regarded as objects of the diagram $\infty $-category $\operatorname{Fun}(J, \operatorname{\mathcal{C}}_{/F} )$.

$(2)$

The diagrams $F_0$ and $F_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{ K / }(J \star K, \operatorname{\mathcal{C}})$.

Proof. Choose a categorical mapping cylinder

\[ J \coprod J \xrightarrow {(s_0, s_1)} \overline{J} \xrightarrow {\pi } J \]

for the simplicial set $J$ (Definition 4.6.3.3). Using Corollary 4.5.5.9, we deduce that the resulting diagram

\[ (J \star K) \coprod _{K} (J \star K) \xrightarrow {(s'_0, s'_1)} \overline{J} \star K \xrightarrow {\pi '} J \star K \]

is a categorical mapping cylinder for the join $J \star K$ relative to $K$. Using the criterion of Corollary 4.6.3.11, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

There exists a diagram $\overline{e}: \overline{J} \rightarrow \operatorname{\mathcal{C}}_{/F}$ satisfying $e_0 = \overline{e} \circ s_0$ and $e_1 = \overline{e} \circ s_1$.

$(2')$

There exists a diagram $\overline{F}': \overline{J} \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $F_0 = \overline{F} \circ s'_0$ and $F_1 = \overline{F} \circ s'_1$.

The equivalence of $(1')$ and $(2')$ follows immediately from the universal property of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/F}$. $\square$

Variant 4.6.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and suppose we are given a pair of diagrams

\[ e_0, e_1: J \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} , \]

which we identify with morphisms of simplicial sets $F_0, F_1: J \diamond K \rightarrow \operatorname{\mathcal{C}}$ extending $F$ (Remark 4.6.4.8). The following conditions are equivalent:

$(1)$

The diagrams $e_0$ and $e_1$ are isomorphic when regarded as objects of the diagram $\infty $-category

\[ \operatorname{Fun}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} ). \]
$(2)$

The diagrams $F_0$ and $F_1$ are isomorphic when regarded as objects of the $\infty $-category

\[ \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}}). \]

Proof. We proceed as in Lemma 4.6.4.21. Choose a categorical mapping cylinder

\[ J \coprod J \xrightarrow {(s_0, s_1)} \overline{J} \xrightarrow {\pi } J \]

for the simplicial set $J$ (Definition 4.6.3.3). Using Remark 4.5.5.7, we see that the induced diagram

\[ (J \diamond K) \coprod _{K} (J \diamond K) \xrightarrow {(s'_0, s'_1)} \overline{J} \diamond K \xrightarrow {\pi '} J \diamond K \]

is a categorical mapping cylinder for the simplicial set $J \diamond K$ relative to $K$. Using the criterion of Corollary 4.6.3.11, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

There exists a diagram $\overline{e}: \overline{J} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $ satisfying $\overline{e} \circ s_0 = e_0$ and $\overline{e} \circ s_1 = e_1$.

$(2')$

There exists a diagram $\overline{F}: \overline{J} \diamond K \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F} \circ s'_0 = F_0$ and $\overline{F} \circ s'_1 = F_1$.

The equivalence of $(1')$ and $(2')$ follows from Remark 4.6.4.8. $\square$

Proof of Theorem 4.6.4.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we regard as an object of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. We will show that the slice diagonal morphism

\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \]

is an equivalence of $\infty $-categories; the corresponding assertion for the coslice diagonal morphism follows by a similar argument. Fix a simplicial set $J$; we wish to show that the induced map of sets

\[ \theta : \pi _0( \operatorname{Fun}(J, \operatorname{\mathcal{C}}_{/F} )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} )^{\simeq } ) \]

is a bijection. Using Lemma 4.6.4.21, Variant 4.6.4.22, and Remark 4.6.4.13, we can identify $\theta $ with the map of sets

\[ \pi _0( \operatorname{Fun}_{K/}( J \star K, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}})^{\simeq } ) \]

induced by precomposition with the comparison map $c_{J,K}: J \diamond K \twoheadrightarrow J \star K$ of Notation 4.5.5.3. It will therefore suffice to show that composition with $c_{J,K}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{K/}( J \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}})$. This follows by applying Corollary 4.5.4.7 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( J \star K, \operatorname{\mathcal{C}}) \ar [r]^-{\circ c_{J,K}} \ar [d] & \operatorname{Fun}( J \diamond K, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( K, \operatorname{\mathcal{C}}) \ar@ {=}[r] & \operatorname{Fun}( K, \operatorname{\mathcal{C}}); } \]

here the vertical maps are isofibrations (Corollary 4.4.5.3) and the upper horizontal map is an equivalence of $\infty $-categories because the morphism $c_{J,K}$ is a categorical equivalence (Theorem 4.5.5.8). $\square$

Variant 4.6.4.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we identify with an object $F$ of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then the functors

\[ \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \quad \quad \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]

of Exercise 4.6.4.15 are equivalences of $\infty $-categories.

Proof. We will show that the slice diagonal $\delta _{/f}$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$; the analogous assertion for coslice $\infty $-categories follows by a similar argument. By virtue of Theorem 4.6.4.16, it will suffice to show that the inclusion map

\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \]

is an equivalence of $\infty $-categories. By construction, this map fits into a commutative diagram of $\infty $-categories

\[ \xymatrix { \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \ar [r] \ar [d]^{\iota } & \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \ar [d]^{U} \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \ar [r] \ar [d] & \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), } \]

where the upper square and lower square are both pullback diagrams. Note that the morphisms $V$ and $V \circ U$ are both right fibrations (Propositions 4.6.4.10 and 4.3.6.1), and therefore isofibrations (Example 4.4.1.10). By virtue of Corollary 4.5.4.5, it will suffice to show that the morphism $U$ is an equivalence of $\infty $-categories, which is a special case of Theorem 4.6.4.16. $\square$