Theorem 9.4.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories. Then $U$ is flat if and only if, for every morphism $f: X \rightarrow Z$ in $\operatorname{\mathcal{E}}$ satisfying $U(X) = 0$ and $U(Z) = 2$, the $\infty $-category $\{ 1\} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{ X/ \, /Z }$ is weakly contractible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.4.7 Flatness and Morphism Spaces
Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. For each morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{E}}$, we let $\operatorname{\mathcal{E}}_{ X/ \, /Z}$ denote the $\infty $-category whose objects are diagrams of the form
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr] & \\ X \ar [ur] \ar [rr]^-{f} & & Z } \]
(see Notation 4.6.6.1). Our goal in this section is to prove the following:
Stated more informally, Theorem 9.4.7.1 asserts that a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ is flat if every morphism from an object $X \in \operatorname{\mathcal{E}}_0$ to an object $Z \in \operatorname{\mathcal{E}}_2$ admits a factorization through an object $Y \in \operatorname{\mathcal{E}}_1$, which is unique up to (weakly) contractible choice. To carry out the proof, it will be useful to work with another formulation of this condition.
Lemma 9.4.7.2. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is essentially $\kappa $-small, let $X$ be an object of $\operatorname{\mathcal{E}}$, and let $h_{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor corepresented by $X$. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be an inner fibration of $\infty $-categories, let $\operatorname{\mathcal{E}}_{\geq 1}$ be the full subcategory of $\operatorname{\mathcal{E}}$ spanned by those objects $E$ satisfying $U(E) \geq 1$, and let $\operatorname{\mathcal{E}}_{1}$ denote the full subcategory spanned by those objects satisfying $U(E) = 1$. If $U$ is flat, then the functor $h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}_{1}$.
Proof. Choose a functor $\mathscr {F}_{\geq 1}: \operatorname{\mathcal{E}}_{\geq 1} \rightarrow \operatorname{\mathcal{S}}$, and set $\mathscr {F}_{1} = \mathscr {F}_{\geq 1}|_{ \operatorname{\mathcal{E}}_{1} }$. By virtue of Corollary 7.3.6.13, it will suffice to show that the restriction map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }, \mathscr {F}_{\geq 1} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{1} }, \mathscr {F}_{1} ) \]
is a homotopy equivalence of Kan complexes. Let $\operatorname{\mathcal{E}}_{\leq 1}$ be the full subcategory of $\operatorname{\mathcal{E}}$ spanned by those objects $E$ satisfying $U(E) \leq 1$, and choose a regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $. Then the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ is $\kappa $-complete (Example 7.6.6.4). Since $\operatorname{\mathcal{E}}_{\leq 1}$ is essentially $\kappa $-small, the functor $\mathscr {F}_{1}$ admits a right Kan extension $\mathscr {F}_{\leq 1}: \operatorname{\mathcal{E}}_{\leq 1} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$. Our assumption that $U$ is flat guarantees that the restriction functor
\[ T: \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{\leq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) \times _{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \lambda } ) } \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) \]
is a trivial Kan fibration, so that we can choose a functor $\mathscr {F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ satisfying $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{\leq 1} } = \mathscr {F}_{\leq 1}$ and $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{\geq 1} } = \mathscr {F}_{\geq 1}$. Since $T$ is a trivial Kan fibration, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}, \mathscr {F} ) \ar [r]^-{\theta '} \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\leq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\leq 1} }, \mathscr {F}_{\leq 1} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{\geq 1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }, \mathscr {F}_{\geq 1} ) \ar [r]^-{\theta } & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{E}}_{1}, \operatorname{\mathcal{S}}^{< \lambda } ) }( h_{X}|_{ \operatorname{\mathcal{E}}_{1} }, \mathscr {F}_{1} ). } \]
Our assumption that $\mathscr {F}_{\leq 1}$ is right Kan extended from $\mathscr {F}_{1}$ guarantees that the right vertical map is a homotopy equivalence (Corollary 7.3.6.13). Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that $\theta '$ is a homotopy equivalence. This follows from the $\infty $-categorical version of Yoneda's lemma (Proposition 8.3.1.3): the source and target of $\theta '$ can both be identified with the Kan complex $\mathscr {F}(X)$. $\square$
The converse of Lemma 9.4.7.2 is also true:
Lemma 9.4.7.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ and $\kappa $ be as in Lemma 9.4.7.2. Suppose that, for every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = 0$, the functor $h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}_{1}$. Then $U$ is flat.
Proof. Using Proposition 9.4.6.24 (or Exercise 9.4.6.25), we can choose a flat inner fibration $U': \operatorname{\mathcal{E}}' \rightarrow \Delta ^2$ and an isomorphism $F_0: \Lambda ^{2}_{1} \times _{\Delta ^2} \operatorname{\mathcal{E}}' \xrightarrow {\sim } \Lambda ^{2}_{1} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}$ of simplicial sets over $\Lambda ^{2}_{1}$. Since the inclusion map $\Lambda ^{2}_{1} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}' \hookrightarrow \operatorname{\mathcal{E}}'$ is a categorical equivalence of simplicial sets, we can extend $F_0$ to a functor of $\infty $-categories $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ (which automatically satisfies $U \circ F = U'$). To prove that $U$ is flat, it will suffice to show that $F$ is an equivalence of inner fibrations over $\Delta ^2$ (Remark 9.4.6.3).
By construction, $F$ is bijective on objects. By virtue of Corollary 5.1.7.10, we are reduced to showing that $F$ is fully faithful: that is, for every pair of objects $X',Y' \in \operatorname{\mathcal{E}}'$ having images $X = F(X')$ and $Y = F(Y')$, the morphism $\overline{F}$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{E}}'}( X', Y' ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{E}}}( X, Y)$. We may assume that $U(X) = 0$ (otherwise, the result follows immediately from the fact that $F$ is an isomorphism). Choose an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}'$ are essentially $\kappa $-small, let $h_{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be the functor represented by $X$, and define $h_{X'}: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ similarly. The functor $F$ then induces a natural transformation $\alpha : h_{X'} \rightarrow h_{X} \circ F$, and we wish to show that $\alpha $ is an isomorphism. Let $\operatorname{\mathcal{E}}'_{ \leq 1 }$ denote the full subcategory of $\operatorname{\mathcal{E}}'$ spanned by those objects $Y$ satisfying $U(Y) \leq 1$, and define $\operatorname{\mathcal{E}}'_{1}$ and $\operatorname{\mathcal{E}}'_{\geq 1}$ similarly. Since $F_0$ is an isomorphism, the natural transformation $\alpha $ is an isomorphism when restricted to $\operatorname{\mathcal{E}}'_{\leq 1}$. It will therefore suffice to show that $\alpha $ is also an isomorphism when restricted $\operatorname{\mathcal{E}}'_{\geq 1}$. Our assumption (and the fact that $F_0$ is an isomorphism) guarantees that $(h_{X} \circ F)|_{ \operatorname{\mathcal{E}}'_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}'_{1}$. Since $U'$ is flat, Lemma 9.4.7.2 guarantees that the functor $h_{X'}|_{ \operatorname{\mathcal{E}}'_{\geq 1} }$ is also left Kan extended from $\operatorname{\mathcal{E}}'_{1}$. The desired result now follows from the fact that $\alpha $ is an isomorphism when restricted to $\operatorname{\mathcal{E}}'_{1}$. $\square$
Proposition 9.4.7.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories. Then $U$ is flat if and only if, for every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = 0$, the inclusion functor is left cofinal.
\[ \iota _{X}: \{ 1\} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/} \hookrightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/ } \]
Proof. Choose an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Using Lemmas 9.4.7.2 and 9.4.7.3, we see that $U$ is flat if and only if, for every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = 0$, the corepresentable functor $h_{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ has the property that $h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}_{1}$. Since $h_{X}$ is a covariant transport representation for the left fibration $\operatorname{\mathcal{E}}_{X/} \rightarrow \operatorname{\mathcal{E}}$, this is equivalent to the requirement that $\iota _{X}$ is left cofinal (see Corollary 7.4.3.15). $\square$
Proof of Theorem 9.4.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories and let $X$ be an object of $\operatorname{\mathcal{E}}$ satisfying $U(X) = 0$. By virtue of Proposition 9.4.7.4, it will suffice to show that the following conditions are equivalent:
- $(1)$
The inclusion functor
\[ \iota _{X}: \{ 1\} \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/} \hookrightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times _{ \Delta ^2 } \operatorname{\mathcal{E}}_{X/ } \]is left cofinal.
- $(2)$
For every morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{E}}$ where $U(Z) = 2$, the $\infty $-category $\{ 1\} \times _{\Delta ^2} \operatorname{\mathcal{E}}_{ X/ \, /Z}$ is weakly contractible.
This follows from the cofinality criterion of Theorem 7.2.3.1. $\square$