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9.3 Tags without Context

The tags in this section were removed because they reference other tags which were removed.

Proposition 9.3.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories which is corepresented by an object $X \in \operatorname{\mathcal{C}}$. Let $Y$ be another object of $\operatorname{\mathcal{C}}$. Then $Y$ corepresents the left fibration $U$ if and only if it is isomorphic to $X$.

Proof. Since $U$ is corepresented by $X$, there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{X} ) = X$. If $U$ is also corepresented by $Y$, then we can choose another initial object $\widetilde{Y} \in \operatorname{\mathcal{E}}$ satisfying $U( \widetilde{Y} ) = Y$. Applying Corollary 4.6.6.16, we deduce that there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Then $e = U( \widetilde{e} )$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$.

For the converse, suppose that there exists an isomorphism $e: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $U$ is a left fibration, we can lift $e$ to a morphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$. Applying Proposition 4.4.2.11, we see that $\widetilde{e}$ is also an isomorphism, so that $\widetilde{Y}$ is also an initial object of $\widetilde{\operatorname{\mathcal{C}}}$ (Corollary 4.6.6.16). It follows that the left fibration $U$ is corepresentable by the object $Y = U( \widetilde{Y} )$. $\square$

Corollary 9.3.0.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. Then $U$ is corepresented by an object $X \in \operatorname{\mathcal{C}}$ if and only if it is equivalent to the left fibration $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$, in the sense of Definition 5.1.6.1.

Corollary 9.3.0.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the construction $X \mapsto \operatorname{\mathcal{C}}_{X/}$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Objects of $\operatorname{\mathcal{C}}$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Corepresentable left fibrations $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence}.} \]

Example 9.3.0.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. Then the left fibrations

\[ \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}\quad \quad \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}} \]

are corepresentable by $X$, and the right fibrations

\[ \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}} \]

are representable by $X$. See Proposition 4.6.6.23.

Remark 9.3.0.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories. Then $U$ is representable by an object $X \in \operatorname{\mathcal{C}}$ if and only if the left fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is corepresentable by $X$.

Example 9.3.0.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the left fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is corepresentable if and only if $\operatorname{\mathcal{C}}$ has an initial object, and the right fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is representable if and only if $\operatorname{\mathcal{C}}$ has a final object.

Corollary 9.3.0.7. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. Then $q$ is a locally cartesian fibration if and only if, for each edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, there exists a functor $\operatorname{\mathcal{C}}_{Y} \rightarrow \operatorname{\mathcal{C}}_{X}$ given by contravariant transport along $e$.

Example 9.3.0.8. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of categories (Definition 5.0.0.3), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a cartesian fibration of $\infty $-categories (Example 5.1.4.2). Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is given by the composition

\[ \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { \chi _{q} } \operatorname{Pith}(\mathbf{Cat}) \rightarrow \mathrm{h} \mathit{\operatorname{Cat}} \xrightarrow { \operatorname{N}_{\bullet } } \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}. \]

Here $\chi _{q}$ denotes the transport representation of Construction 9.10.2.4 (with respect to any cleavage of the fibration $q$), the second functor is the truncation map of Remark 9.6.0.47, and $\operatorname{N}_{\bullet }$ is the fully faithful functor of Remark 4.5.1.3. Stated more informally, the homotopy transport representation $\operatorname{hTr}_{ \operatorname{N}_{\bullet }(q)}$ of Construction 5.2.5.7 can be obtained from the transport representation $\chi _{ \operatorname{N}_{\bullet }(q)}$ of Construction 9.10.2.4 by passing from the $2$-category $\mathbf{Cat}$ to its homotopy category.

Example 9.3.0.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories which is a fibration in sets (Definition 4.2.3.1), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a right fibration, and in particular a cartesian fibration. Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}$ of Construction 5.2.5.7 is given by the composition

\[ \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { \chi _{q} } \operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty } }, \]

where $\chi _{q}$ is the transport representation of Construction 9.10.6.7 and $\operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is the fully faithful embedding which associates to each set $X$ the associated discrete simplicial set, regarded as an $\infty $-category.

Definition 9.3.0.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f: C \rightarrow D$ be a morphism in the category $\operatorname{\mathcal{C}}$. A cleavage of $U$ over $f$ is a function

\[ \{ \textnormal{Objects of $\operatorname{\mathcal{E}}_{D}$} \} \rightarrow \{ \textnormal{Morphisms of $\operatorname{\mathcal{E}}$} \} \quad \quad Y \mapsto \widetilde{f}_{Y} \]

which associates to each object $Y \in \operatorname{\mathcal{E}}_{D}$ a locally $U$-cartesian morphism $\widetilde{f}_{Y}: X \rightarrow Y$ satisfying $U( \widetilde{f}_ Y) = f$.

A cleavage of $U$ consists of a choice, for each morphism $f$ of $\operatorname{\mathcal{C}}$, of a cleavage of $U$ over $f$. We will denote a cleavage of $U$ by $(f,Y) \mapsto \widetilde{f}_{Y}$.

Remark 9.3.0.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Then a cleavage of $U$ exists if and only if $U$ is a locally cartesian fibration (Definition 9.10.3.8).

Example 9.3.0.12. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a unitary lax functor, and let $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the forgetful functor of Notation 5.6.1.11. If $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $Y$ is an object of the category $\mathscr {F}(D)$, let $\widetilde{f}_{Y}$ denote the pair $(f, \operatorname{id}_{ \mathscr {F}(f)(Y) } )$, which we regard as a morphism from $( C, \mathscr {F}(f)(Y) )$ to $(D, Y)$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$. Then the construction $(f,Y) \mapsto \widetilde{f}_{Y}$ is a cleavage of $U$ (see Example 9.10.3.7), which we will refer to as the tautological cleavage.

Warning 9.3.0.13. The conclusion of Proposition 5.2.5.1 is generally not satisfied if $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is only assumed to be a locally cartesian fibration of simplicial sets. We will return to this point in ยง (see Proposition ).

Remark 9.3.0.14. In the situation of Definition 5.2.2.14, suppose that $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a right fibration. Then condition $(3)$ is superfluous (every edge of $\operatorname{\mathcal{C}}$ is $q$-cartesian by virtue of Example 5.1.1.3).

Remark 9.3.0.15. This tag refers to a construction whose contents is now contained in Example example:monoidal-category-of-cocycle (and uses the language of monoidal categories, rather than $2$-categories).

Let $G$ be a group, let $\Gamma $ be an abelian group equipped with an action of $G$ by automorphisms, and let $\operatorname{\mathcal{C}}$ be the $2$-category obtained by applying the construction of Example 9.4.0.14 to some $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $. We can describe $\operatorname{\mathcal{C}}$ informally as follows: it is a $2$-category whose $1$-morphisms are the elements of the group $G$, whose $2$-morphisms are the elements of the abelian group $\Gamma $, and whose associativity constraint is the $3$-cocycle $\alpha $.