# Kerodon

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## 8.4 Tags Lost in Reorganization

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

Definition 8.4.0.1. 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 8.4.0.2. 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 8.4.0.3. 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 8.4.0.4. 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 8.4.0.5. 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 8.4.0.6. 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 8.4.0.7. The statement of Theorem 5.6.0.2 assumes that $\operatorname{\mathcal{Q}}$ is a full subcategory of $\operatorname{\mathcal{QC}}$. However, it will sometimes be convenient to apply Theorem 5.6.0.2 when $\operatorname{\mathcal{Q}}$ is an enlargement of $\operatorname{\mathcal{QC}}$, whose objects include $\infty$-categories which are not necessarily small.

Definition 8.4.0.8. Let $\operatorname{\mathcal{QC}}$ denote the $\infty$-category of small $\infty$-categories (Construction 5.4.4.1), 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 8.4.0.9. 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 5.5.4.18: 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 8.4.0.11. 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 8.4.0.12. In the statement of Theorem 5.6.0.2, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty$-category, or that it is small.

Remark 8.4.0.13 (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 8.4.0.14 (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 8.4.0.15 (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 8.4.0.18. 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 5.5.2.5).

Variant 8.4.0.27. 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 8.4.0.28. 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 8.4.0.29. 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 8.4.0.30. 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 8.4.0.31. 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 8.4.0.32. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories.

Remark 8.4.0.33. 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 8.4.0.34. 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 8.4.0.28).

Remark 8.4.0.35. 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 8.4.0.36. 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 8.4.0.37. The converse of Remark 8.4.0.36 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 8.4.0.38. 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 8.4.0.41 (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 8.4.0.42 (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.

Corollary 8.4.0.43. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of $\infty$-categories and let $C$ be an object of $\operatorname{\mathcal{C}}$ whose image $D = F(C)$ is an initial object of $\operatorname{\mathcal{D}}$. Then $C$ is an initial object of $\operatorname{\mathcal{C}}$ if and only if it is an initial object of the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.