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11.10.1 Notational Junk

Notation 11.10.1.1. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. We let $\operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')$ denote the full subcategory of $\operatorname{Fun}_{/S}(X,X')$ spanned by those morphisms $f: X \rightarrow X'$ which carry $q$-cartesian edges of $X$ to $q'$-cartesian edges of $X'$. If $q: X \rightarrow S$ and $q': X' \rightarrow S$ are cocartesian fibrations, we let $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')$ denote the full subcategory of $\operatorname{Fun}_{/S}(X,X')$ spanned by those morphisms $f: X \rightarrow X'$ which carry $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$.

Definition 11.10.1.2. Let $S$ be a simplicial set. We define (non-full) simplicial subcategories

$\operatorname{CCart}(S) \subset (\operatorname{Set_{\Delta }})_{/S} \supset \operatorname{Cart}(S)$

as follows:

• Let $X$ be an object of the slice category $(\operatorname{Set_{\Delta }})_{/S}$: that is, a simplicial set equipped with a morphism $q: X \rightarrow S$. Then $X$ belongs to the subcategory $\operatorname{Cart}(S) \subset (\operatorname{Set_{\Delta }})_{/S}$ if and only if $q$ is a cartesian fibration, and $X$ belongs to $\operatorname{CCart}(S) \subset (\operatorname{Set_{\Delta }})_{/S}$ if and only if $q$ is a cocartesian fibration.

• Let $X$ and $X'$ be objects of $\operatorname{Cart}(S)$. Then $\operatorname{Hom}_{\operatorname{Cart}(S)}(X,X')_{\bullet } = \operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')^{\simeq }$ is the core of the $\infty$-category $\operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')$ of Notation 11.10.1.1. Similarly, if $X$ and $X'$ are objects of $\operatorname{CCart}(S)$, then $\operatorname{Hom}_{\operatorname{CCart}(S)}(X,X')_{\bullet } = \operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')^{\simeq }$ is the core of the $\infty$-category $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')$

We let $\mathrm{h} \mathit{\operatorname{Cart}(S)}$ and $\mathrm{h} \mathit{\operatorname{CCart}(S)}$ denote the homotopy categories of the simplicial categories $\operatorname{Cart}(S)$ and $\operatorname{CCart}(S)$, respectively (Construction 2.4.6.1).

Remark 11.10.1.3. Let $S$ be a simplicial set. Then the simplicial categories $\operatorname{Cart}(S)$ and $\operatorname{CCart}(S)$ are locally Kan.

Remark 11.10.1.4. Let $S$ be a simplicial set. Then the construction $X \mapsto X^{\operatorname{op}}$ determines an isomorphism of simplicial categories $\operatorname{CCart}(S) \simeq \operatorname{Cart}( S^{\operatorname{op}} )^{\operatorname{c}}$; here $\operatorname{Cart}(S^{\operatorname{op}})^{\operatorname{c}}$ denotes the conjugate of the simplicial category $\operatorname{Cart}(S^{\operatorname{op}})$ (see Example 2.4.2.12).

Example 11.10.1.5. Let $S$ be a Kan complex. Then a morphism of simplicial sets $q: X \rightarrow S$ is a (co)cartesian fibration if and only if it is an isofibration (Corollary 5.1.4.10). If this condition is satisfied, then $X$ is an $\infty$-category and a morphism of $X$ is $q$-(co)cartesian if and only if it is an isomorphism (Corollary 5.1.1.10). It follows that, if $q': X' \rightarrow S$ is another cartesian fibration, then every morphism $f: X \rightarrow X'$ in $(\operatorname{Set_{\Delta }})_{/S}$ carries $q$-(co)cartesian edges of $X$ to $q'$-(co)cartesian edges of $X'$ (Remark 1.5.1.6). In particular, $\operatorname{Cart}(S)$ coincides with $\operatorname{CCart}(S)$ as a simplicial subcategory of $(\operatorname{Set_{\Delta }})_{/S}$.

Example 11.10.1.6. Let $S = \Delta ^0$. Then the simplicial category $\operatorname{Cart}(S) = \operatorname{CCart}(S)$ can be described as follows:

• An objects of $\operatorname{Cart}(S)$ is a small $\infty$-category $\operatorname{\mathcal{C}}$.

• If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are small $\infty$-categories, then $\operatorname{Hom}_{\operatorname{Cart}(S)}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ is the core of the functor $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

In other words, $\operatorname{Cart}(S)$ can be identified with the simplicial category $\operatorname{QCat}$ of Construction 5.5.4.1.

Variant 11.10.1.7. Let $S$ be a simplicial set. We define full simplicial subcategories

$\operatorname{LFib}(S), \operatorname{KFib}(S), \operatorname{RFib}(S) \subset (\operatorname{Set_{\Delta }})_{/S}$

as follows:

• An object of $\operatorname{LFib}(S)$ is a simplicial set $X$ equipped with a left fibration $q: X \rightarrow S$.

• An object of $\operatorname{KFib}(S)$ is a simplicial set $X$ equipped with a Kan fibration $q: X \rightarrow S$.

• An object of $\operatorname{RFib}(S)$ is a simplicial set $X$ equipped with a right fibration $q: X \rightarrow S$.

Remark 11.10.1.8. Let $q: X \rightarrow S$ be a cocartesian fibration of simplicial sets and let $q': X' \rightarrow S$ be a left fibration. Then $q'$ is also a cocartesian fibration, and every edge of $X'$ is $q'$-cocartesian (Proposition 5.1.4.14). In particular:

• Every morphism $f: X \rightarrow X'$ in $(\operatorname{Set_{\Delta }})_{/S}$ carries $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$. That is, we have $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X') = \operatorname{Fun}_{/S}(X,X')$.

• The simplicial set $\operatorname{Fun}_{/S}(X,X')$ is a Kan complex (Corollary 4.4.2.5), and is therefore equal to its core $\operatorname{Fun}_{/S}(X,X')^{\simeq }$.

It follows that we can regard $\operatorname{LFib}(S)$ as a full subcategory of the simplicial category $\operatorname{CCart}(S)$. Similarly, we can regard $\operatorname{RFib}(S)$ as a full subcategory of the simplicial category $\operatorname{Cart}(S)$. We therefore obtain a diagram of fully faithful inclusions

$\operatorname{CCart}(S) \hookleftarrow \operatorname{LFib}(S) \hookleftarrow \operatorname{KFib}(S) \hookrightarrow \operatorname{RFib}(S) \hookrightarrow \operatorname{Cart}(S).$

Remark 11.10.1.9 (Functoriality). Let $f: S' \rightarrow S$ be a morphism of simplicial sets. Then $f$ determines a functor of simplicial categories $f^{\ast }: (\operatorname{Set_{\Delta }})_{/S} \rightarrow (\operatorname{Set_{\Delta }})_{/S'}$, given on objects by the formula $f^{\ast }(X) = S' \times _{S} X$. Since the collection of cartesian fibrations is stable under base change (Remark 5.1.4.6), $f^{\ast }$ restricts to a simplicial functor $\operatorname{Cart}(S) \rightarrow \operatorname{Cart}(S')$, which we will also denote by $f^{\ast }$. Similarly we obtain pullback functors

$f^{\ast }: \operatorname{CCart}(S) \rightarrow \operatorname{CCart}(S') \quad \quad f^{\ast }: \operatorname{KFib}(S) \rightarrow \operatorname{KFib}(S')$
$f^{\ast }: \operatorname{LFib}(S) \rightarrow \operatorname{LFib}(S') \quad \quad f^{\ast }: \operatorname{RFib}(S) \rightarrow \operatorname{RFib}(S').$

Definition 11.10.1.10. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of cocartesian fibrations over $S$ if it satisfies the following conditions:

• The morphism $f$ carries $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$ and satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{CCart}(S)$.

• The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{CCart}(S)}$. That is, there exists another morphism $g: X' \rightarrow X$ in $\operatorname{CCart}(S)$ for which the compositions $g \circ f$ and $f \circ g$ are homotopic to the identity maps $\operatorname{id}_{X}$ and $\operatorname{id}_{X'}$, respectively (as morphisms in the simplicial category $\operatorname{CCart}(S)$).

We say that the cocartesian fibrations $q: X \rightarrow S$ and $q': X' \rightarrow S$ are equivalent if there exists a morphism $f: X \rightarrow X'$ which is an equivalence of cocartesian fibrations over $S$: that is, if $X$ and $X'$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{CCart}(S)}$.

Variant 11.10.1.11. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of cartesian fibrations over $S$ if it satisfies the following conditions:

• The morphism $f$ carries $q$-cartesian edges of $X$ to $q'$-cartesian edges of $X'$ and satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{Cart}(S)$.

• The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Cart}(S)}$.

We say that the cartesian fibrations $q: X \rightarrow S$ and $q': X' \rightarrow S$ are equivalent if there exists a morphism $f: X \rightarrow X'$ which is an equivalence of cartesian fibrations over $S$: that is, if $X$ and $X'$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{Cart}(S)}$.

Remark 11.10.1.12. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. Then a morphism $f: X \rightarrow X'$ is an equivalence of cartesian fibrations over $S$ if and only if the opposite morphism $f^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow X'^{\operatorname{op}}$ is an equivalence of cocartesian fibrations over $S^{\operatorname{op}}$. In particular, the cartesian fibrations $q$ and $q'$ are equivalent if and only if the cocartesian fibrations $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ and $q'^{\operatorname{op}}: X'^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ are equivalent.

Example 11.10.1.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories, so that the projection maps $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ and $\operatorname{\mathcal{D}}\rightarrow \Delta ^0$ are both cartesian fibrations and cocartesian fibrations. For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the following conditions are equivalent:

• The functor $F$ is an equivalence of cartesian fibrations over $\Delta ^0$.

• The functor $F$ is an equivalence of cocartesian fibrations over $\Delta ^0$.

• The functor $F$ is an equivalence of $\infty$-categories.

Remark 11.10.1.14 (Pullback Stability). Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets, and let $f: X \rightarrow X'$ be an equivalence of cocartesian fibrations over $S$. For any morphism of simplicial sets $T \rightarrow S$, the induced map $f_{T}: T \times _{S} X \rightarrow T \times _{S} X'$ is an equivalence of cocartesian fibrations over $T$ (this follows immediately from Remark 11.10.1.9). In particular, for every vertex $s \in S$, the induced map of fibers $f_{s}: X_{s} \rightarrow X'_{s}$ is an equivalence of $\infty$-categories.

Corollary 11.10.1.15. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ X \ar [rr]^-{f} \ar [dr]_{q} & & X' \ar [dl]^{q'} \\ & S, & }$

where the morphisms $q$ and $q'$ are both cartesian fibrations and cocartesian fibrations. Then $f$ is an equivalence of cartesian fibrations over $S$ if and only if $f$ is an equivalence of cocartesian fibrations over $S$.

Corollary 11.10.1.16. Let $q: X \rightarrow S$ and $q: X' \rightarrow S$ be morphisms of simplicial sets which are both cartesian fibrations and cocartesian fibrations. Then $q$ and $q'$ are equivalent as cartesian fibrations over $S$ if and only if they are equivalent as cocartesian fibrations over $S$.

Corollary 11.10.1.17. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{U'} \\ & \operatorname{\mathcal{C}}. & }$

If $U$ and $U'$ are cartesian fibrations, then $F$ is an equivalence of cartesian fibrations over $\operatorname{\mathcal{C}}$ if and only if it is an equivalence of $\infty$-categories. If $U$ and $U'$ are cocartesian fibrations, then $F$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ if and only if it is an equivalence of $\infty$-categories.

Remark 11.10.1.18. We will see later that Corollary 11.10.1.17 remains valid without the assumption that $\operatorname{\mathcal{E}}$, $\operatorname{\mathcal{E}}'$, and $\operatorname{\mathcal{C}}$ are $\infty$-categories; see Corollary .

Corollary 11.10.1.19. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cocartesian fibrations of simplicial sets which are equivalent (in the sense of Definition 11.10.1.10). Then $q$ is a left fibration if and only if $q'$ is a left fibration.

Definition 11.10.1.20. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be left fibrations of simplicial sets. We will say that a morphism of simplicial sets $f: X \rightarrow X'$ is an equivalence of left fibrations over $S$ if it satisfies the following conditions:

• The morphism $f$ satisfies $q = q' \circ f$; that is, $f$ is a morphism in the category $\operatorname{LFib}(S)$.

• The homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{LFib}(S)}$.

We say that the left fibrations $q: X \rightarrow S$ and $q': X' \rightarrow S$ are equivalent if there exists a morphism $f: X \rightarrow X'$ which is an equivalence of left fibrations over $S$.

In the situation of Definition 11.10.1.20, a morphism $f: X \rightarrow X'$ is an equivalence of left fibrations over $S$ if and only if it is an equivalence of cocartesian fibrations over $S$ (in the sense of Definition 11.10.1.10). Applying Proposition 11.10.1.21, we obtain the following:

Corollary 11.10.1.21. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ X \ar [rr]^-{f} \ar [dr]_{q} & & X' \ar [dl]^{q'} \\ & S, & }$

where $q$ and $q'$ are left fibrations. The following conditions are equivalent:

• The morphism $f$ is an equivalence of left fibrations over $S$.

• For every vertex $s \in S$, the induced map of Kan complexes $f_{s}: X_{s} \rightarrow X'_{s}$ is a homotopy equivalence.