Proposition 5.6.6.21 (02J8)

Proposition 5.6.6.21. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small left fibration of $\infty $-categories and let $X \in \operatorname{\mathcal{C}}$ be an object. Then:

$(1)$

Let $\widetilde{X} \in \operatorname{\mathcal{D}}$ be an object satisfying $U( \widetilde{X} ) = X$. Then $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if, for every object $Y \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\theta } \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} ) \xrightarrow { \operatorname{ev}_{\widetilde{X}} } \operatorname{\mathcal{D}}_{Y} \]

is a homotopy equivalence, where $\theta $ is given by parametrized covariant transport (see Definition 5.2.8.1).

$(2)$

Let $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be the homotopy transport representation of $U$, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor (Variant 5.2.8.12). Then $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if it exhibits $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ as corepresented by $X$, in the sense of Definition 5.6.6.10.

$(3)$

The homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ is corepresentable by the object $X$ if and only if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X} ) = X$.

$(4)$

Let $\operatorname{Tr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Then $\operatorname{Tr}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}$ is corepresentable by the object $X$ if and only if there exists an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X} ) = X$.

Proof. Let $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ be the oriented fiber product of Definition 4.6.4.1, and let us regard $\operatorname{id}_{X}$ as an initial object of $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ (Proposition 4.6.7.22). Using Proposition 5.6.6.19, we can choose a functor of $\infty $-categories $F: \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F( \operatorname{id}_ X ) = \widetilde{X}$ which fits into a commutative diagram

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

Using Proposition 5.6.6.19, we see that $\widetilde{X}$ is an initial object of $\operatorname{\mathcal{D}}$ if and only if $F$ is an equivalence of left fibrations over $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.1.7.15, this is equivalent to the requirement that for each object $Y \in \operatorname{\mathcal{C}}$, the functor $F$ restricts to a homotopy equivalence of Kan complexes

\[ F_{Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{\mathcal{D}}_{Y} \]

Assertion $(1)$ follows from the observation that $F_{Y}$ is homotopic to the composition of the parametrized covariant transport morphism $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} )$ with the evaluation map $\operatorname{ev}_{ \widetilde{X} }: \operatorname{Fun}( \operatorname{\mathcal{D}}_{X}, \operatorname{\mathcal{D}}_{Y} ) \rightarrow \operatorname{\mathcal{D}}_{Y}$ (see Remark 5.2.8.5 and Proposition 5.2.8.7). The implication $(1) \Rightarrow (2)$ follows from Remark 5.6.5.8, the implication $(2) \Rightarrow (3)$ is immediate, and the implication $(3) \Rightarrow (4)$ follows from Remark 5.6.6.11. $\square$