Kerodon

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 $\mathscr {F}(Y)$.

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

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 $\operatorname{\mathcal{C}}$.

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).

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.

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.

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 $f = w \circ f'$.

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 $\operatorname{\mathcal{C}}_{f/}$.

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.

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).

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.

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 $\operatorname{\mathcal{D}}$.

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.