Remark 9.5.0.1. In the statement of Theorem 5.6.6.14, we can replace the smallness assumption on $\operatorname{\mathcal{C}}$ by the weaker assumption that for every object $Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y)$ is essentially small. Note that this latter condition cannot be omitted: if $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$, then $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is homotopy equivalent to the small Kan complex $\mathscr {F}(Y)$.
9.5 Tags Lost in Reorganization
These tags became irrelevant because they refer to an older organization of various materials.
Example 9.5.0.2. 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 8.1.5.5 to the case where $\operatorname{\mathcal{A}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain a bijection In other words, we can identify $n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$ with commutative diagrams in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
Remark 9.5.0.3. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Using Example 9.5.0.2, we can identify $n$-simplices $\sigma $ of the simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams . Allowing $[n] \in \operatorname{{\bf \Delta }}$ to vary, we obtain canonical isomorphisms of simplicial sets
Proposition 9.5.0.4. See Proposition 5.6.6.20.
Remark 9.5.0.5. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Stated more informally, Proposition None asserts that $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{C}}) )$ can be identified with commutative diagrams in the category $\operatorname{\mathcal{C}}$.
Question 9.5.0.6. 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 9.5.0.7. See Theorem 5.3.5.7.
Proposition 9.5.0.8. For a stronger result, see Theorem 5.3.5.7.
Exercise 9.5.0.9. This exercise is now carried out in the text (Theorem 9.5.0.7).
Exercise 9.5.0.10. Suppose we are given a finite sequence of $\infty $-categories $\{ \operatorname{\mathcal{E}}(m) \} _{0 \leq im \leq n}$ and functors Let $\operatorname{\mathcal{E}}$ denote the iterated relative join 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 For a more general statement, see Proposition 5.3.3.15 and Remark 5.3.3.19.
Remark 9.5.0.11. In the special case $\operatorname{\mathcal{C}}= [1]$, Proposition 5.3.3.15 reduces to Proposition 5.2.3.15. In the case $\operatorname{\mathcal{C}}= [n]$, it reduces to *** (see Variant 9.5.0.12).
Variant 9.5.0.12. 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 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
Example 9.5.0.13. 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 5.6.2.1, so that the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map (see Example 5.6.2.8). 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 $\mathscr {F}$.
Remark 9.5.0.14. We will see later that if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories, then the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ can be refined to a functor of $\infty $-categories $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which we will refer to as the covariant transport representation of $U$ (see Definition 5.6.5.1). Moreover, the refined analogues of Questions 9.5.0.6 and Question 5.2.0.7 both have positive answers:
A cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ can be recovered (up to equivalence) from the transport representation $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.
Every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ can be obtained (up to isomorphism) as the transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Moreover, there is an explicit realization of $\operatorname{\mathcal{E}}$ as the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Definition 5.6.2.1).
From this perspective, the negative answers to Questions 9.5.0.6 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 9.5.0.15. The contents of this tag are now (mostly) at Proposition 5.3.1.21.
Corollary 9.5.0.16. See Proposition 5.3.1.21.
Example 9.5.0.17. The contents of this tag are now at Corollary 5.6.7.6.
Corollary 9.5.0.18. 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:
The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ (see Definition 6.2.2.1).
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 5.6.6.22. $\square$
Remark 9.5.0.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Construction 4.6.8.13). Every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ induces an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$. Moreover, if $X$ is an object of $\operatorname{\mathcal{C}}$, then every vertex $x \in \mathscr {F}(X)$ determines a natural transformation of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, \bullet ) \rightarrow \mathrm{h} \mathit{\mathscr {F}}(\bullet )$, which is an isomorphism if and only if $x$ exhibits $\mathscr {F}$ as corepresented by $X$. Consequently, $\mathscr {F}$ is corepresentable by $X$ if and only if $\mathrm{h} \mathit{\mathscr {F}}$ is isomorphic to $\underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, \bullet )$ as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor.
Variant 9.5.0.20. See Definition 9.4.0.3.
Remark 9.5.0.21. Suppose we are given a commutative diagram of simplicial sets Using Exercise 3.1.7.11, we can factor $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a trivial Kan fibration (and therefore also a categorical equivalence, by virtue of Proposition 4.5.3.11). Combining Propositions 4.5.4.11 and 4.5.4.9, we conclude that diagram (9.4) is a categorical pushout square if and only if the induced map $u: C \coprod _{A} B' \rightarrow D$ is a categorical equivalence In particular, the condition that $u$ is a categorical equivalence does not depend on the choice of factorization $f = w \circ f'$.
Remark 9.5.0.22. Suppose we are given a commutative diagram of simplicial sets Using Exercise 3.1.7.11, we can factor $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a weak homotopy equivalence (in fact, we can even arrange that $w$ is a trivial Kan fibration). Combining Propositions 3.4.2.11 and 3.4.2.9, we conclude that diagram (9.5) is a homotopy pushout square if and only if the induced map $u: C \coprod _{A} B' \rightarrow D$ is a weak homotopy equivalence. In particular, the condition that $u$ is a weak homotopy equivalence does not depend on the choice of factorization $f = w \circ f'$.
Exercise 9.5.0.23 (Invariance of the Homotopy Fiber Product). The contents of this tag are now at Corollary 3.4.2.15.
Proposition 9.5.0.24. The contents of this tag are now located at Corollary 3.4.2.13.
Warning 9.5.0.25. Suppose we are given a commutative diagram of simplicial sets 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 is not weakly contractible (it has nontrivial homology in degree $1$), but the pushout of the lower half is isomorphic to $\Delta ^0$.
Exercise 9.5.0.26 (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 9.5.0.27. Suppose we are given a commutative diagram of $\infty $-categories 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
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 9.5.0.28. The contents of this tag are now contained in Corollary 7.1.4.21.
Definition 9.5.0.29. 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.
Remark 9.5.0.30. Let $\operatorname{\mathcal{C}}$ be a category. Then an object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is a colimit of the unique diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$; here $\emptyset $ denotes the category with no objects. Similarly, an object $Y \in \operatorname{\mathcal{C}}$ is final if and only if it is a limit of the unique diagram $\emptyset \rightarrow \operatorname{\mathcal{C}}$.
Remark 9.5.0.31. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let be the induced maps. Then an extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $q$ is a $U$-limit diagram if and only if it is $U_{/f}$-final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$. Similarly, an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if is $U_{f/}$-initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$.
Exercise 9.5.0.32. 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 9.5.0.33. 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 9.5.0.34. 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 9.5.0.35. The statement of Theorem 5.6.0.3 assumes that $\operatorname{\mathcal{Q}}$ is a full subcategory of $\operatorname{\mathcal{QC}}$. However, it will sometimes be convenient to apply Theorem 5.6.0.3 when $\operatorname{\mathcal{Q}}$ is an enlargement of $\operatorname{\mathcal{QC}}$, whose objects include $\infty $-categories which are not necessarily small.
Definition 9.5.0.36. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of small $\infty $-categories (Construction 5.5.4.1), 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 9.5.0.37. 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 5.6.2.18: 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 9.5.0.38. Deduce Theorem 5.6.0.3 from Corollary 9.6.0.35.
Remark 9.5.0.39. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then $\mathscr {F}$ is a covariant transport representation of $U$ if and only if there exists a morphism of simplicial sets $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (so that, in particular, the composite map $\operatorname{\mathcal{E}}\xrightarrow {G} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is equal to $U$). In this case, we will say that the morphism $G$ exhibits $\mathscr {F}$ as a covariant transport representation of $U$.
Remark 9.5.0.40. In the statement of Theorem 5.6.0.3, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category, or that it is small.
Remark 9.5.0.41 (Base Change). Suppose we are given a pullback diagram of simplicial sets where $U$ and $U'$ are inner fibrations. Let $f: C \rightarrow D$ be a morphism of $\operatorname{\mathcal{C}}$ having image $f': C' \rightarrow D'$ in $\operatorname{\mathcal{C}}'$. Then a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if and only if it is given by covariant transport along $f'$, when regarded as a functor from $\operatorname{\mathcal{E}}'_{C'}$ to $\operatorname{\mathcal{E}}'_{D'}$.
Remark 9.5.0.42 (Two-out-of-Six). Let $f: W_{} \rightarrow X_{}$, $g: X_{} \rightarrow Y_{}$, and $h: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If $g \circ f$ and $h \circ g$ are homotopy equivalences, then $f$, $g$, and $h$ are all homotopy equivalences.
Remark 9.5.0.43 (Two-out-of-Six). Let $f: W_{} \rightarrow X_{}$, $g: X_{} \rightarrow Y_{}$, and $h: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If $g \circ f$ and $h \circ g$ are weak homotopy equivalences, then $f$, $g$, and $h$ are all weak homotopy equivalences.
Proposition 9.5.0.44. The contents of this tag are now at Proposition 6.3.3.9.
Remark 9.5.0.45. The conclusion of Proposition 5.2.2.8 continues to hold under the more general assumption that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration (see Definition and Proposition
).
Remark 9.5.0.46. For historical reasons, it is traditional to place more emphasis on the duals of the notions introduced above. We say that a functor of categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration (fibration in sets, fibration in groupoids) if the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian fibration (opfibration in sets, opfibration in groupoids). If $U$ is a cartesian fibration, then there exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ and an isomorphism of $\operatorname{\mathcal{E}}$ with the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, characterized by the formula $(\int ^{\operatorname{\mathcal{C}}} \mathscr {F})^{\operatorname{op}} = \int _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}^{\operatorname{op}}$ (see Remark 5.6.1.6).
Variant 9.5.0.47. The contents of this tag are now at Corollary 5.6.5.20.
Corollary 9.5.0.48. The contents of this tag are now at Proposition 9.6.0.5.
Corollary 9.5.0.49. The contents of this tag are now at Remark None.
Example 9.5.0.50. The contents of this tag are now at Remark 4.3.1.6.
Variant 9.5.0.51. The contents of this tag are now contained in Definition 4.2.3.1.
Construction 9.5.0.52. The contents of this tag are now at Definition 5.6.1.4 (for functors) and Definition 9.10.5.1 (for lax functors).
Corollary 9.5.0.53. The contents of this tag are now contained in Theorem 5.1.5.1.
Example 9.5.0.54. The contents of this tag are now (mostly) at Remark 5.6.1.12.
Variant 9.5.0.55. 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 9.5.0.56. 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.
Remark 9.5.0.57. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the induced functor of opposite categories. Let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$, which we identify with a morphism $f^{\operatorname{op}}: Y \rightarrow X$ in the opposite category $\operatorname{\mathcal{E}}^{\operatorname{op}}$. Then $f$ is $U$-cartesian if and only if $f^{\operatorname{op}}$ is $U^{\operatorname{op}}$-cocartesian.
Remark 9.5.0.58. Suppose we are given a pullback diagram of categories and let $f: X \rightarrow Y$ be a morphism of the category $\operatorname{\mathcal{E}}'$. If $V(f)$ is a $U$-cartesian morphism of $\operatorname{\mathcal{E}}$, then $f$ is a $U'$-cartesian morphism of $\operatorname{\mathcal{E}}'$. Similarly, if $V(f)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$, then $f$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}'$.
Remark 9.5.0.59. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and suppose we are given a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{E}}$. Then:
If $g$ is $U$-cartesian, then $f$ is $U$-cartesian if and only if $g \circ f$ is $U$-cartesian.
If $f$ is $U$-cocartesian, then $g$ is $U$-cocartesian if and only if $g \circ f$ is $U$-cocartesian.
In particular, the collections of $U$-cartesian and $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ are closed under composition.
Example 9.5.0.60. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories.
Remark 9.5.0.61. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Then $U$ is a cartesian fibration if and only if the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian fibration.
Remark 9.5.0.62. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of categories. Then $U$ is an isofibration (Definition 4.4.1.1). If $Y$ is an object of $\operatorname{\mathcal{C}}$ and $\overline{f}: \overline{X} \rightarrow U(Y)$ is an isomorphism in the category $\operatorname{\mathcal{C}}$, then our assumption that $U$ is a cartesian fibration guarantees that we can choose a $U$-cartesian morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$, and the morphism $f$ is automatically an isomorphism (Example 9.5.0.56).
Remark 9.5.0.63. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories, and let $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$. Then $\operatorname{N}_{\bullet }(F)$ is a right fibration if and only if $F$ is a fibration in groupoids (see Definition 4.2.2.1 and Proposition 4.2.2.9). Similarly, $\operatorname{N}_{\bullet }(F)$ is a left fibration if and only if $F$ is an opfibration in groupoids.
Remark 9.5.0.64. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. For each object $D \in \operatorname{\mathcal{D}}$, let $\operatorname{\mathcal{C}}_{D} = \{ D \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ denote the corresponding fiber of $F$ (more concretely, $\operatorname{\mathcal{C}}_{D}$ is the subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C \in \operatorname{\mathcal{C}}$ satisfying $F(C) = D$, and those morphisms $u: C \rightarrow C'$ satisfying $F(u) = \operatorname{id}_{D}$). It follows from Remark 4.2.2.8 that if $F$ is a fibration in groupoids, then the projection map $\operatorname{\mathcal{C}}_{D} \rightarrow \{ D\} $ is also a fibration in groupoids, so that the category $\operatorname{\mathcal{C}}_{D}$ is a groupoid (Example 4.2.2.7). This observation motivates the terminology of Definition 4.2.2.1: if $F$ is a fibration in groupoids, then one can think of the category $\operatorname{\mathcal{C}}$ as the total space of a “family” of groupoids $\{ \operatorname{\mathcal{C}}_{D} \} _{D \in \operatorname{\mathcal{D}}}$ which is parametrized by the category $\operatorname{\mathcal{D}}$.
Warning 9.5.0.65. The converse of Remark 9.5.0.64 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 §).
Remark 9.5.0.66. Let $S$ be a simplicial set, and let $(\operatorname{Set_{\Delta }})_{/S}$ denote the slice category of simplicial sets $X$ equipped with a morphism $q_{X}: X \rightarrow S$. Then we can regard $(\operatorname{Set_{\Delta }})_{/S}$ as a simplicially enriched category, with mapping simplicial sets given by
Remark 9.5.0.67. For relative versions of Corollaries 4.3.6.11 and 4.3.6.13, we refer the reader to §9.7.0.10.
Remark 9.5.0.68. In the special case $S = \Delta ^{0}$, Proposition 4.3.6.8 reduces to Corollary 4.3.6.11, Corollary 4.3.6.9 reduces to Proposition 4.3.6.1, and Proposition 4.3.6.12 reduces to Corollary 4.3.6.13.
Remark 9.5.0.69 (Two-out-of-Six). Let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$, $G: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$, and $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors between $\infty $-categories. If $G \circ F$ and $H \circ G$ are equivalences of $\infty $-categories, then $F$, $G$, and $H$ are equivalences of $\infty $-categories.
Remark 9.5.0.70 (Two-out-of-Six). Let $f: W \rightarrow X$, $g: X \rightarrow Y$, and $h: Y \rightarrow Z$ be morphisms of simplicial sets. If $g \circ f$ and $h \circ g$ are categorical equivalences, then $f$, $g$, and $h$ are categorical equivalences.