# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

## 9.5 Tags Lost in Reorganization

These tags became irrelevant because they refer to an older organization of various materials.

*** work this in ***

Corollary 9.5.0.1. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{q} & \operatorname{\mathcal{C}}' \ar [d]^-{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}'. }$

Assume that:

$(1)$

The functors $q$ and $q'$ are isofibrations.

$(2)$

The isofibration $q$ is locally cartesian and the functor $F$ carries locally $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to locally $q'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.

$(3)$

The functor $\overline{F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is an equivalence of $\infty$-categories.

Then $F$ is an equivalence of $\infty$-categories if and only if, for every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D) \in \operatorname{\mathcal{D}}'$, the induced map of fibers $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is an equivalence of $\infty$-categories. Moreover, if this condition is satisfied, then $q'$ is also a locally cartesian fibration.

Proof. If $F$ is an equivalence of $\infty$-categories, then Corollary 4.5.2.27 guarantees that each $F_{D}$ is an equivalence of $\infty$-categories. The converse follows by combining Proposition 5.1.6.7. $\square$

*** snip

Corollary 9.5.0.2. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets, let $K$ be a simplicial set, and let $q': S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow S$ be the projection map onto the first factor. Then:

$(1)$

The morphism $q'$ is a locally cartesian fibration of simplicial sets.

$(2)$

Let $e$ be an edge of the simplicial set $S \times _{ \operatorname{Fun}(B,S)} \operatorname{Fun}(B,X)$. Then $e$ is locally $q'$-cartesian if and only if, for every vertex $b \in B$, the image of $e$ under the evaluation functor $S \times _{ \operatorname{Fun}(B,S) } \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}( \{ b\} , X) \simeq X$ is locally $q$-cartesian.

Proof. By virtue of Remark 5.1.5.6, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration. The desired result now follows by combining Theorem 5.2.1.1 with Remark 5.1.4.6. $\square$

Lemma 9.5.0.4. Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a fully faithful functor of $\infty$-categories. Suppose that every object $Y \in \widehat{\operatorname{\mathcal{C}}}$ is a retract of $H(X)$, for some object $X \in \operatorname{\mathcal{C}}$. Then, for any $\infty$-category $\operatorname{\mathcal{D}}$, precomposition with $H$ determines a fully faithful functor $\theta : \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Moreover, the essential image of $\theta$ consists of those functors $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfy the following condition:

$(\ast )$

For every functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$, if $H \circ F$ is a split idempotent in $\widehat{\operatorname{\mathcal{C}}}$, then $G \circ F$ is a split idempotent in $\operatorname{\mathcal{D}}$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is a full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ and that $H$ is the inclusion functor. Let $\operatorname{Fun}'( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admit an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$. It follows from Proposition 8.5.1.7 that, in this case, the functor $\widehat{G}$ is automatically left (and right) Kan extended from $\operatorname{\mathcal{C}}$. Applying Corollary 7.3.6.15, we deduce that the restriction functor $\theta : \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is a trivial Kan extension. Note that any functor $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ carries split idempotents in $\widehat{\operatorname{\mathcal{C}}}$ to split idempotents in $\operatorname{\mathcal{D}}$, so that $G = \widehat{G}|_{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast )$. To complete the proof, it will suffice to prove the reverse implication. Fix a functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfies condition $(\ast )$; we wish to show that $G$ admits an extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$.

Choose an uncountable regular cardinal $\kappa$ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa$-small. Using Proposition 8.5.4.7, we can choose a fully faithful functor $H': \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where the $\infty$-category $\widehat{\operatorname{\mathcal{D}}}$ admits $\kappa$-small colimits. Replacing $\operatorname{\mathcal{D}}$ by the essential image of $H'$, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{D}}}$. Invoking Proposition 7.6.7.13, we deduce that the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ admits a left Kan extension $\widehat{G}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$. We will complete the proof by showing that $\widehat{G}$ factors through $\operatorname{\mathcal{D}}$.

Fix an object $Y \in \widehat{\operatorname{\mathcal{C}}}$; we wish to show that $\widehat{G}(Y)$ belongs to $\operatorname{\mathcal{D}}$. By assumption, there exists a retraction diagram

9.5
\begin{equation} \begin{gathered}\label{equation:universal-mapping-property-of-idempotent-completion} \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{i} & \\ X \ar [ur]^{r} \ar [rr]^{ \operatorname{id}_{X} } & & X } \end{gathered} \end{equation}

in $\widehat{\operatorname{\mathcal{C}}}$, where the object $X$ belongs to $\operatorname{\mathcal{C}}$. Using Corollary 8.5.1.24, we can extend (9.5) to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \overline{\operatorname{\mathcal{C}}}$. Then $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) }$ is an idempotent in $\operatorname{\mathcal{C}}$ which splits in $\widehat{\operatorname{\mathcal{C}}}$. Invoking assumption $(\ast )$, we deduce that $G \circ F$ is a split idempotent in $\operatorname{\mathcal{D}}$. That is, there exists a functor $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{D}}$ satisfying $\overline{F}' |_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } = G \circ F$. Applying Corollary 8.5.3.10, we deduce that $\widehat{G} \circ \overline{F}$ is isomorphic to $\overline{F}'$ as an object of the $\infty$-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \widehat{\operatorname{\mathcal{D}}} )$. Evaluating on the final object of $\operatorname{Ret}$, we deduce that $\widehat{G}(Y)$ is isomorphic to an object of $\operatorname{\mathcal{D}}$ and therefore belongs to $\operatorname{\mathcal{D}}$ (since the full subcategory $\operatorname{\mathcal{D}}\subseteq \widehat{\operatorname{\mathcal{D}}}$ was assumed to be replete). $\square$

Remark 9.5.0.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\widehat{\operatorname{\mathcal{C}}}$ be an idempotent completion of $\operatorname{\mathcal{C}}$. Then any $\infty$-category $\widehat{\operatorname{\mathcal{C}}}'$ which is equivalent to $\widehat{\operatorname{\mathcal{C}}}$ is also an idempotent completion of $\operatorname{\mathcal{C}}$. More precisely, if $G: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ is an equivalence of $\infty$-categories, then a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$ if and only if the composite functor $(G \circ H): \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as an idempotent completion of $\operatorname{\mathcal{C}}$.

Remark 9.5.0.6 (Isomorphism Invariance). Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty$-categories, and let $H': \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be another functor which is isomorphic to $H$ (as an object of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \widehat{\operatorname{\mathcal{C}}} )$). Then $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$ if and only if $H'$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$.

Remark 9.5.0.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, and let $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ denote the relative join of Construction 5.2.3.1. Then $F$ is dense if and only if the projection map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}$. See Proposition 7.3.2.10.

Remark 9.5.0.8. Stated more informally, Theorem 8.4.0.3 asserts that the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ can be obtained from $\operatorname{\mathcal{C}}$ by “freely” adjoining small colimits.

Remark 9.5.0.9. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, where $\operatorname{\mathcal{C}}$ is essentially small and $\operatorname{\mathcal{D}}$ admits small colimits. Fix a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Theorem 8.4.0.3 implies that $f$ is isomorphic to the composition $F \circ h_{\bullet }$, for some functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which preserves small colimits. Moreover, the functor $F$ is uniquely determined up to isomorphism.

Remark 9.5.0.12. In the situation of Proposition 9.5.0.11, condition $(\star )$ is automatically satisfied if $V$ is a cartesian fibration of simplicial sets.

Example 9.5.0.13. Let $X$ be a Kan complex, which we regard as an object of the $\infty$-category $\operatorname{\mathcal{S}}$ of spaces (Construction 5.6.1.1). Then:

• The Kan complex $X$ is an initial object of the $\infty$-category $\operatorname{\mathcal{S}}$ if and only if it is empty.

• The Kan complex $X$ is a final object of the $\infty$-category $\operatorname{\mathcal{S}}$ if and only if it is contractible.

Corollary 9.5.0.14. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the pullback functor

$U^{\ast }: (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} \rightarrow (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} \quad \quad \operatorname{\mathcal{C}}' \mapsto \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$

has a right adjoint, given on objects by the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$.

Corollary 9.5.0.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors having restrictions $F_0 = F|_{\operatorname{\mathcal{C}}^{0}}$ and $G_0 = G|_{\operatorname{\mathcal{C}}^{0}}$, and suppose that $U \circ F = U \circ G$. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then the restriction map

$\operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0 )$

is a homotopy equivalence of Kan complexes.

Remark 9.5.0.16. Let $F: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty$-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, and let $\mathscr {K}'$ denote the composition

$\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}' \xrightarrow {\operatorname{id}\times F} \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow { \mathscr {K} } \operatorname{\mathcal{S}},$

which we regard as a profunctor from $\operatorname{\mathcal{D}}'$ to $\operatorname{\mathcal{C}}$. If $\beta : \underline{ \Delta ^0 } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}})}$ is a natural transformation which exhibits $\mathscr {K}$ as represented by $G$, then the restriction $\beta |_{ \operatorname{Tw}( \operatorname{\mathcal{D}}' )}$ exhibits $\mathscr {K}'$ as represented by $G \circ F$.

Corollary 9.5.0.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $(\mathscr {H}, \alpha )$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$

$(2)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$.

$(3)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$.

Remark 9.5.0.19. In the statement of Theorem 5.7.6.13, 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 $\mathscr {F}(Y)$.

Example 9.5.0.20. 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 8.1.8.5 to the case where $\operatorname{\mathcal{A}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \}$, we obtain a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{n-simplices of \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )} \} \ar [d]^{\sim } \\ \{ \textnormal{n-simplices of \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)} \} . }$

In other words, we can identify $n$-simplices of $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$ with commutative diagrams

$\xymatrix@C =20pt{ f_{0,0} \ar@ {=>}[dr] & & f_{1,1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n-1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{n,n} \ar@ {=>}[dl] \\ & \cdots \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dr] \ar@ {=>}[dl] & & \cdots \ar@ {=>}[dl] \ar@ {=>}[dr] & & \cdots \ar@ {=>}[dl] & \\ & & f_{0,n-2} \ar@ {=>}[dr] & & f_{1,n-1} \ar@ {=>}[dl] \ar@ {=>}[dr] & & f_{2,n} \ar@ {=>}[dl] & & \\ & & & f_{0,n-1} \ar@ {=>}[dr] & & f_{1,n} \ar@ {=>}[dl] & & & \\ & & & & f_{0,n} & & & & }$

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

Remark 9.5.0.21. 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 9.5.0.20, 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

$\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \quad \quad \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )^{\operatorname{op}}.$

Remark 9.5.0.23. 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

$\xymatrix@R =25pt@C=25pt{ & & & X_{0,n} \ar [dl] \ar [dr] & & & \\ & & \cdots \ar [dl] \ar [dr] & & \cdots \ar [dl] \ar [dr] & & \\ & X_{0,1} \ar [dl] \ar [dr] & & \cdots \ar [dl]\ar [dr] & & X_{n-1,n} \ar [dl] \ar [dr] & \\ X_{0,0} & & X_{1,1} & \cdots & X_{n-1,n-1} & & X_{n,n} }$

in the category $\operatorname{\mathcal{C}}$.

Question 9.5.0.24. 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 $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$?

Exercise 9.5.0.28. Suppose we are given a finite sequence of $\infty$-categories $\{ \operatorname{\mathcal{E}}(m) \} _{0 \leq im \leq n}$ and functors

$\operatorname{\mathcal{E}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{E}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {F(n)} \operatorname{\mathcal{E}}(n).$

Let $\operatorname{\mathcal{E}}$ denote the iterated relative join

$((( \operatorname{\mathcal{E}}(0) \star _{\operatorname{\mathcal{E}}(1)} \operatorname{\mathcal{E}}(1) ) \star _{\operatorname{\mathcal{E}}(2)} \operatorname{\mathcal{E}}(2)) \star \cdots ) \star _{\operatorname{\mathcal{E}}(n)} \operatorname{\mathcal{E}}(n).$

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

$\operatorname{\mathcal{E}}(0) \xrightarrow { [F(1)] } \operatorname{\mathcal{E}}(1) \xrightarrow { [F(2)] } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {[F(n)]} \operatorname{\mathcal{E}}(n).$

For a more general statement, see Proposition 9.6.0.6 and Remark 5.3.3.21.

Remark 9.5.0.29. In the special case $\operatorname{\mathcal{C}}= $, Proposition 9.6.0.6 reduces to Proposition 5.2.3.15. In the case $\operatorname{\mathcal{C}}= [n]$, it reduces to *** (see Variant 9.5.0.30).

Variant 9.5.0.30. 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

$C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_ n$

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

$\Delta ^ n \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq ((( \mathscr {F}(C_0) \star _{ \mathscr {F}(C_1)} \mathscr {F}(C_1)) \star _{ \mathscr {F}(C_2)} \mathscr {F}(C_2)) \star \cdots ) \star _{\mathscr {F}(C_ n)} \mathscr {F}(C_ n).$

Example 9.5.0.31. 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 5.7.2.1, so that the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map (see Example 5.7.2.8). 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 9.5.0.32. 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 5.7.5.1). Moreover, the refined analogues of Questions 9.5.0.24 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 5.7.2.1).

From this perspective, the negative answers to Questions 9.5.0.24 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.

Corollary 9.5.0.36. 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:

$(1)$

The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{C}}'$-coreflection of $Y$ (see Definition 6.2.2.1).

$(2)$

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 5.7.6.21. $\square$

Remark 9.5.0.37. 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 9.5.0.39. Suppose we are given a commutative diagram of simplicial sets

9.6
\begin{equation} \label{diagram:categorical-pushout-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

Using Exercise 3.1.7.10, 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 (9.6) 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 9.5.0.40. Suppose we are given a commutative diagram of simplicial sets

9.7
\begin{equation} \label{diagram:homotopy-pushout-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

Using Exercise 3.1.7.10, 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 (9.7) 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 $f = w \circ f'$.

Warning 9.5.0.43. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ C \ar [d] & A \ar [d] \ar [l] \ar [r] & B \ar [d] \\ C' & A' \ar [l] \ar [r] & B' }$

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

$\xymatrix@R =50pt@C=50pt{ \Delta ^1 \ar [d] & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] \\ \Delta ^0 & \operatorname{\partial \Delta }^{1} \ar [l] \ar [r] & \Delta ^0 }$

is not weakly contractible (it has nontrivial homology in degree $1$), but the pushout of the lower half is isomorphic to $\Delta ^0$.

Exercise 9.5.0.44 (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 9.5.0.45. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix { \operatorname{\mathcal{C}}' \ar [r]^-{F'} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [r]^-{G} & \operatorname{\mathcal{E}}}$

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

$\xymatrix { \operatorname{Fun}(X,\operatorname{\mathcal{C}}')^{\simeq } \ar [r]^-{F'_ X} \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{D}}')^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{E}}')^{\simeq } \ar [d] \\ \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } \ar [r]^-{F_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{D}})^{\simeq } \ar [r]^-{G_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{E}})^{\simeq }. }$

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. $\square$

Definition 9.5.0.47. 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 9.5.0.48. 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 9.5.0.49. 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

$U_{f/}: \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{D}}_{(U \circ f)/ } \quad \quad U_{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{D}}_{/ (U \circ f)}$

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 9.5.0.50. 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 9.5.0.51. 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 9.5.0.52. 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 9.5.0.53. The statement of Theorem 5.7.0.2 assumes that $\operatorname{\mathcal{Q}}$ is a full subcategory of $\operatorname{\mathcal{QC}}$. However, it will sometimes be convenient to apply Theorem 5.7.0.2 when $\operatorname{\mathcal{Q}}$ is an enlargement of $\operatorname{\mathcal{QC}}$, whose objects include $\infty$-categories which are not necessarily small.

Definition 9.5.0.54. Let $\operatorname{\mathcal{QC}}$ denote the $\infty$-category of small $\infty$-categories (Construction 5.6.4.1), 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 9.5.0.55. 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 5.7.2.18: 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 $Q$.

Remark 9.5.0.57. 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 9.5.0.58. In the statement of Theorem 5.7.0.2, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty$-category, or that it is small.

Remark 9.5.0.59 (Base Change). Suppose we are given a pullback diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^-{U} \ar [r] & \operatorname{\mathcal{E}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}', }$

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 9.5.0.60 (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 9.5.0.61 (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 9.5.0.64. 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 5.7.1.6).

Variant 9.5.0.72. 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 9.5.0.73. 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 9.5.0.74. 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 9.5.0.75. Suppose we are given a pullback diagram of categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{V} \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, }$

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 9.5.0.76. 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 9.5.0.77. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories.

Remark 9.5.0.78. 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 9.5.0.79. 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 9.5.0.73).

Remark 9.5.0.80. 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 9.5.0.81. 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 9.5.0.82. The converse of Remark 9.5.0.81 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 9.5.0.83. 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

$\underline{\operatorname{Hom}}_{ ( \operatorname{Set_{\Delta }})_{/S} }( X, Y) = \operatorname{Fun}_{S}(X,Y).$

Remark 9.5.0.86 (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 9.5.0.87 (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.