These tags became irrelevant because they refer to an older organization of various materials.
*** work this in ***
Corollary 9.5.0.1. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{q} & \operatorname{\mathcal{C}}' \ar [d]^-{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}'. } \]
Assume that:
- $(1)$
The functors $q$ and $q'$ are isofibrations.
- $(2)$
The isofibration $q$ is locally cartesian and the functor $F$ carries locally $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to locally $q'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.
- $(3)$
The functor $\overline{F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is an equivalence of $\infty $-categories.
Then $F$ is an equivalence of $\infty $-categories if and only if, for every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D) \in \operatorname{\mathcal{D}}'$, the induced map of fibers $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is an equivalence of $\infty $-categories. Moreover, if this condition is satisfied, then $q'$ is also a locally cartesian fibration.
Proof.
If $F$ is an equivalence of $\infty $-categories, then Corollary 4.5.2.27 guarantees that each $F_{D}$ is an equivalence of $\infty $-categories. The converse follows by combining Proposition 5.1.6.7.
$\square$
*** snip
Corollary 9.5.0.2. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets, let $K$ be a simplicial set, and let $q': S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow S$ be the projection map onto the first factor. Then:
- $(1)$
The morphism $q'$ is a locally cartesian fibration of simplicial sets.
- $(2)$
Let $e$ be an edge of the simplicial set $S \times _{ \operatorname{Fun}(B,S)} \operatorname{Fun}(B,X)$. Then $e$ is locally $q'$-cartesian if and only if, for every vertex $b \in B$, the image of $e$ under the evaluation functor $S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ is locally $q$-cartesian.
Proof.
By virtue of Remark 5.1.5.6, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration. The desired result now follows by combining Theorem 5.2.1.1 with Remark 5.1.4.6.
$\square$
Lemma 9.5.0.4. Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a fully faithful functor of $\infty $-categories. Suppose that every object $Y \in \widehat{\operatorname{\mathcal{C}}}$ is a retract of $H(X)$, for some object $X \in \operatorname{\mathcal{C}}$. Then, for any $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $H$ determines a fully faithful functor $\theta : \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Moreover, the essential image of $\theta $ consists of those functors $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfy the following condition:
- $(\ast )$
For every functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$, if $H \circ F$ is a split idempotent in $\widehat{\operatorname{\mathcal{C}}}$, then $G \circ F$ is a split idempotent in $\operatorname{\mathcal{D}}$.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is a full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ and that $H$ is the inclusion functor. Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admit an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$. It follows from Proposition 8.5.1.7 that, in this case, the functor $\widehat{G}$ is automatically left (and right) Kan extended from $\operatorname{\mathcal{C}}$. Applying Corollary 7.3.6.15, we deduce that the restriction functor $\theta : \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is a trivial Kan extension. Note that any functor $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ carries split idempotents in $\widehat{\operatorname{\mathcal{C}}}$ to split idempotents in $\operatorname{\mathcal{D}}$, so that $G = \widehat{G}|_{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast )$. To complete the proof, it will suffice to prove the reverse implication. Fix a functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfies condition $(\ast )$; we wish to show that $G$ admits an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$.
Choose an uncountable regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Using Proposition 8.5.4.7, we can choose a fully faithful functor $H': \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where the $\infty $-category $\widehat{\operatorname{\mathcal{D}}}$ admits $\kappa $-small colimits. Replacing $\operatorname{\mathcal{D}}$ by the essential image of $H'$, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{D}}}$. Invoking Proposition 7.6.7.13, we deduce that the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ admits a left Kan extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$. We will complete the proof by showing that $\widehat{G}$ factors through $\operatorname{\mathcal{D}}$.
Fix an object $Y \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that $\widehat{G}(Y)$ belongs to $\operatorname{\mathcal{D}}$. By assumption, there exists a retraction diagram
9.5
\begin{equation} \begin{gathered}\label{equation:universal-mapping-property-of-idempotent-completion} \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{i} & \\ X \ar [ur]^{r} \ar [rr]^{ \operatorname{id}_{X} } & & X } \end{gathered} \end{equation}
in $\widehat{\operatorname{\mathcal{C}}}$, where the object $X$ belongs to $\operatorname{\mathcal{C}}$. Using Corollary 8.5.1.24, we can extend (9.5) to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \overline{\operatorname{\mathcal{C}}}$. Then $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) }$ is an idempotent in $\operatorname{\mathcal{C}}$ which splits in $\widehat{\operatorname{\mathcal{C}}}$. Invoking assumption $(\ast )$, we deduce that $G \circ F$ is a split idempotent in $\operatorname{\mathcal{D}}$. That is, there exists a functor $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{D}}$ satisfying $\overline{F}' |_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } = G \circ F$. Applying Corollary 8.5.3.10, we deduce that $\widehat{G} \circ \overline{F}$ is isomorphic to $\overline{F}'$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \widehat{\operatorname{\mathcal{D}}} )$. Evaluating on the final object of $\operatorname{Ret}$, we deduce that $\widehat{G}(Y)$ is isomorphic to an object of $\operatorname{\mathcal{D}}$ and therefore belongs to $\operatorname{\mathcal{D}}$ (since the full subcategory $\operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ was assumed to be replete).
$\square$
Proposition 9.5.0.10. The contents of this tag are now at Proposition 5.4.9.14.
Proposition 9.5.0.11. The contents of this tag are now (mostly) at Proposition 9.5.0.11.
Example 9.5.0.13. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (Construction 5.6.1.1). Then:
The Kan complex $X$ is an initial object of the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if it is empty.
The Kan complex $X$ is a final object of the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if it is contractible.
Corollary 9.5.0.14. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the pullback functor
\[ U^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} \]
has a right adjoint, given on objects by the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$.
Corollary 9.5.0.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors having restrictions $F_0 = F|_{\operatorname{\mathcal{C}}^{0}}$ and $G_0 = G|_{\operatorname{\mathcal{C}}^{0}}$, and suppose that $U \circ F = U \circ G$. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0 ) \]
is a homotopy equivalence of Kan complexes.
Corollary 9.5.0.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(\mathscr {H}, \alpha )$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:
- $(1)$
The morphism $f$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$
- $(2)$
The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$.
- $(3)$
The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$.
Corollary 9.5.0.18. The contents of this tag are now contained in Proposition 7.6.7.9.
Example 9.5.0.20. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1), and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$ (which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$). Specializing Remark 8.1.8.5 to the case where $\operatorname{\mathcal{A}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{$n$-simplices of $\operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$} \} \ar [d]^{\sim } \\ \{ \textnormal{$n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$} \} . } \]
In other words, we can identify $n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$ with commutative diagrams
\[ \xymatrix@C =20pt{ f_{0,0} \ar@ {=>}[dr] & & f_{1,1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n-1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n,n} \ar@ {=>}[dl] \\ & \cdots \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dr] \ar@ {=>}[dl] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] & \\ & & f_{0,n-2} \ar@ {=>}[dr] & & f_{1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{2,n} \ar@ {=>}[dl] & & \\ & & & f_{0,n-1} \ar@ {=>}[dr] & & f_{1,n} \ar@ {=>}[dl] & & & \\ & & & & f_{0,n} & & & & } \]
in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
Proposition 9.5.0.22. See Proposition 5.7.6.19.
Question 9.5.0.24. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Can the $\infty $-category $\operatorname{\mathcal{E}}$ be reconstructed (up to equivalence) from the $\infty $-category $\operatorname{\mathcal{C}}$ and the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$?
Theorem 9.5.0.25. See Theorem 5.3.5.7.
Proposition 9.5.0.26. For a stronger result, see Theorem 5.3.5.7.
Exercise 9.5.0.27. This exercise is now carried out in the text (Theorem 9.5.0.25).
Exercise 9.5.0.28. Suppose we are given a finite sequence of $\infty $-categories $\{ \operatorname{\mathcal{E}}(m) \} _{0 \leq im \leq n}$ and functors
\[ \operatorname{\mathcal{E}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{E}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {F(n)} \operatorname{\mathcal{E}}(n). \]
Let $\operatorname{\mathcal{E}}$ denote the iterated relative join
\[ ((( \operatorname{\mathcal{E}}(0) \star _{\operatorname{\mathcal{E}}(1)} \operatorname{\mathcal{E}}(1) ) \star _{\operatorname{\mathcal{E}}(2)} \operatorname{\mathcal{E}}(2)) \star \cdots ) \star _{\operatorname{\mathcal{E}}(n)} \operatorname{\mathcal{E}}(n). \]
Show that the associated projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is a cocartesian fibration whose homotopy transport representation $\operatorname{hTr}_{U}: [n] \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is the diagram
\[ \operatorname{\mathcal{E}}(0) \xrightarrow { [F(1)] } \operatorname{\mathcal{E}}(1) \xrightarrow { [F(2)] } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {[F(n)]} \operatorname{\mathcal{E}}(n). \]
For a more general statement, see Proposition 9.6.0.6 and Remark 5.3.3.21.
Variant 9.5.0.30. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\sigma $ be an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_ n \]
in the category $\operatorname{\mathcal{C}}$. Using Remark 5.3.3.7 and Example 5.3.3.12, we obtain an isomorphism of simplicial sets
\[ \Delta ^ n \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq ((( \mathscr {F}(C_0) \star _{ \mathscr {F}(C_1)} \mathscr {F}(C_1)) \star _{ \mathscr {F}(C_2)} \mathscr {F}(C_2)) \star \cdots ) \star _{\mathscr {F}(C_ n)} \mathscr {F}(C_ n). \]
Example 9.5.0.31. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ be a morphism of simplicial sets, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be the simplicial set given by Definition 5.7.2.1, so that the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map (see Example 5.7.2.8). Then the covariant transport representation $\operatorname{Tr}_{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} / \operatorname{\mathcal{C}}}$ is canonically isomorphic to the functor $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ induced by $\mathscr {F}$.
From this perspective, the negative answers to Questions 9.5.0.24 and 5.2.0.7 are due to the fact that a functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ cannot generally be lifted to a functor of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, and that such a lifting need not be unique when it exists.
Proposition 9.5.0.33. The contents of this tag are now (mostly) at Proposition 5.3.1.21.
Corollary 9.5.0.34. See Proposition 5.3.1.21.
Example 9.5.0.35. The contents of this tag are now at Corollary 5.7.7.6.
Corollary 9.5.0.36. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and suppose that the object $X$ belongs to a full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ (see Definition 6.2.2.1).
- $(2)$
The morphism $f$ is final when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$.
In particular, an object $X \in \operatorname{\mathcal{C}}'$ is a $\operatorname{\mathcal{C}}'$-reflection of $Y \in \operatorname{\mathcal{C}}$ if and only if it represents the right fibration $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$.
Proof.
Let us regard the object $Y \in \operatorname{\mathcal{C}}$ as fixed, and let $\theta : \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}'$ be the right fibration given by projection onto the first factor. Using Example 5.2.8.13, we can identify the enriched homotopy transport representation of $\theta $ with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $(X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}'}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. The desired result now follows from the criterion of Proposition 5.7.6.21.
$\square$
Variant 9.5.0.38. See Definition 9.4.0.8.
Exercise 9.5.0.41 (Invariance of the Homotopy Fiber Product). The contents of this tag are now at Corollary 3.4.2.15.
Proposition 9.5.0.42. The contents of this tag are now located at Corollary 3.4.2.13.
Warning 9.5.0.43. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ C \ar [d] & A \ar [d] \ar [l] \ar [r] & B \ar [d] \\ C' & A' \ar [l] \ar [r] & B' } \]
in which the vertical maps are weak homotopy equivalences. Then the induced map $C \coprod _{A} B \rightarrow C' \coprod _{A'} B'$ need not be a weak homotopy equivalence. For example, the pushout of the upper half of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [d] & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] \\ \Delta ^0 & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] & \Delta ^0 } \]
is not weakly contractible (it has nontrivial homology in degree $1$), but the pushout of the lower half is isomorphic to $\Delta ^0$.
Exercise 9.5.0.44 (Symmetry). Let $T \rightarrow S \leftarrow X$ be a diagram of simplicial sets. Show that the homotopy fiber products $T \times _{S}^{h} X$ and $X \times _{S}^{h} T$ have the same weak homotopy type (see Proposition 3.4.1.9 for a related statement).
Corollary 9.5.0.45. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix { \operatorname{\mathcal{C}}' \ar [r]^-{F'} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [r]^-{G} & \operatorname{\mathcal{E}}} \]
where both squares are pullbacks. Assume that $G$ and $G \circ F$ are isofibrations. If $F$ is an equivalence of $\infty $-categories, then $F'$ is an equivalence of $\infty $-categories.
Proof.
We will verify that $F'$ satisfies the criterion of Theorem 4.5.7.1. Let $X$ be a simplicial set, and consider the commutative diagram of Kan complexes
\[ \xymatrix { \operatorname{Fun}(X,\operatorname{\mathcal{C}}')^{\simeq } \ar [r]^-{F'_ X} \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{D}}')^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{E}}')^{\simeq } \ar [d] \\ \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{F_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{D}})^{\simeq } \ar [r]^-{G_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{E}})^{\simeq }. } \]
We wish to show that the morphism $F'_{X}$ is a homotopy equivalence. Using Corollaries 4.4.5.6 and 4.4.3.18, we see that the right square and outer rectangle are homotopy pullback diagrams. It follows that the left square is also a homotopy pullback diagram (Proposition 3.4.1.11). Since $F$ is an equivalence of $\infty $-categories, the morphism $F_{X}$ is a homotopy equivalence of Kan complexes (Theorem 4.5.7.1). Applying Corollary 3.4.1.5, we deduce that $F'_{X}$ is also a homotopy equivalence of Kan complexes.
$\square$
Corollary 9.5.0.46. The contents of this tag are now contained in Corollary 7.1.4.21.
Definition 9.5.0.47. Let $\operatorname{\mathcal{C}}$ be a category. An initial object of $\operatorname{\mathcal{C}}$ is an object $Y \in \operatorname{\mathcal{C}}$ with the property that, for every object $Z \in \operatorname{\mathcal{C}}$, there is a unique morphism $Y \rightarrow Z$: that is, the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ has exactly one element. A final object of $\operatorname{\mathcal{C}}$ is an object $Y \in \operatorname{\mathcal{C}}$ with the property that, for every object $X \in \operatorname{\mathcal{C}}$, the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ has exactly one element.
Exercise 9.5.0.50. Let $U: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $\operatorname{\mathcal{C}}_{/U}$ be the slice category of Construction 4.3.1.8. By virtue of Remark 4.3.1.11, the objects of $\operatorname{\mathcal{C}}_{/U}$ can be identified with pairs $(Y, \alpha )$, where $Y$ is an object of $\operatorname{\mathcal{C}}$ and $\alpha : \underline{Y} \rightarrow U$ is a natural transformation of functors. Show that $\alpha $ exhibits $Y$ as a limit of $U$ (in the sense of Definition 7.1.0.1) if and only if the pair $(Y, \alpha )$ is a final object of the category $\operatorname{\mathcal{C}}_{/U}$. Similarly, show that a natural transformation $\beta : U \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $U$ if and only if the pair $(Y,\beta )$ determines an initial object of the coslice category $\operatorname{\mathcal{C}}_{U/}$.
Example 9.5.0.51. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. The following conditions are equivalent:
The objects $X$ and $Y$ are isomorphic.
The object $Y$ is a limit of the diagram $\{ X \} \hookrightarrow \operatorname{\mathcal{C}}$.
The object $Y$ is a colimit of the diagram $\{ X\} \hookrightarrow \operatorname{\mathcal{C}}$.
See Example 7.1.1.5 for a more precise statement.
Lemma 9.5.0.52. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $\operatorname{\mathcal{E}}$ is an $\infty $-category, then the natural map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an inner fibration of simplicial sets.
Warning 9.5.0.53. The statement of Theorem 5.7.0.2 assumes that $\operatorname{\mathcal{Q}}$ is a full subcategory of $\operatorname{\mathcal{QC}}$. However, it will sometimes be convenient to apply Theorem 5.7.0.2 when $\operatorname{\mathcal{Q}}$ is an enlargement of $\operatorname{\mathcal{QC}}$, whose objects include $\infty $-categories which are not necessarily small.
Definition 9.5.0.54. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of small $\infty $-categories (Construction 5.6.4.1), and let $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$ be a full subcategory. We will say that a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is $\operatorname{\mathcal{Q}}$-small if, for every object $C \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to an $\infty $-category which belongs to $\operatorname{\mathcal{Q}}$.
Example 9.5.0.55. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$, and let $\widetilde{\operatorname{\mathcal{Q}}}$ denote the fiber product $\operatorname{\mathcal{Q}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$: that is, the $\infty $-category of elements of the inclusion map $\operatorname{\mathcal{Q}}\hookrightarrow \operatorname{\mathcal{QC}}$. Then the projection map $V: \widetilde{\operatorname{\mathcal{Q}}} \rightarrow \operatorname{\mathcal{Q}}$ is a $\operatorname{\mathcal{Q}}$-small cocartesian fibration of $\infty $-categories. This follows from Example 5.7.2.18: for every object $Q \in \operatorname{\mathcal{Q}}$, the fiber $\{ Q\} \times _{ \operatorname{\mathcal{Q}}} \widetilde{\operatorname{\mathcal{Q}}}$ is an $\infty $-category equivalent to $Q$.
Exercise 9.5.0.56. Deduce Theorem 5.7.0.2 from Corollary 9.6.0.46.
Proposition 9.5.0.62. The contents of this tag are now at Proposition 6.3.3.9.
Variant 9.5.0.65. The contents of this tag are now at Corollary 5.7.5.20.
Corollary 9.5.0.66. The contents of this tag are now at Remark None.
Example 9.5.0.67. The contents of this tag are now at Remark 4.3.1.6.
Variant 9.5.0.68. The contents of this tag are now contained in Definition 4.2.3.1.
Construction 9.5.0.69. The contents of this tag are now at Definition 5.7.1.4 (for functors) and Definition 9.10.4.1 (for lax functors).
Corollary 9.5.0.70. The contents of this tag are now contained in Theorem 5.1.6.1.
Example 9.5.0.71. The contents of this tag are now (mostly) at Remark 5.7.1.12.
Variant 9.5.0.72. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor. The following conditions are equivalent:
The functor $U$ is an opfibration in groupoids (Definition 4.2.2.1).
The functor $U$ is a cocartesian fibration and every morphism of $\operatorname{\mathcal{E}}$ is $U$-cocartesian.
The functor $U$ is a cocartesian fibration and, for every object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a groupoid.
Example 9.5.0.73. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{E}}$. Assume that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:
The morphism $f$ is $U$-cartesian.
The morphism $f$ is $U$-cocartesian.
The morphism $f$ is an isomorphism in $\operatorname{\mathcal{E}}$.
In particular, every isomorphism in $\operatorname{\mathcal{E}}$ is both $U$-cartesian and $U$-cocartesian.
In particular, the collections of $U$-cartesian and $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ are closed under composition.
Example 9.5.0.77. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories.
Warning 9.5.0.82. The converse of Remark 9.5.0.81 is generally false: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor having the property that each fiber $\operatorname{\mathcal{C}}_{D}$ is a groupoid, then $F$ need not be a fibration in groupoids. For example, this condition is also satisfied whenever $F$ is an opfibration in groupoids, but an opfibration in groupoids need not be a fibration in groupoids. Roughly speaking, one can think of a fibration in groupoids $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ as encoding a family of groupoids $\{ \operatorname{\mathcal{C}}_{D} \} $ having a contravariant dependence on the object $D \in \operatorname{\mathcal{D}}$, and an opfibration in groupoids $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ as encoding a family of groupoids $\{ \operatorname{\mathcal{C}}_{D} \} $ having a covariant dependence on the object $D \in \operatorname{\mathcal{D}}$ (for a more precise formulation of this idea, we refer the reader to §
).