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11.5 Tags Lost in Reorganization

Proposition 11.5.0.2. Let $f: K \rightarrow \operatorname{\mathcal{S}}$ be a diagram. Then:

  • An extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram if and only if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

  • An extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{S}}$ is a colimit diagram if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Corollary 11.5.0.3. See Remark 7.4.4.5.

Let $n$ be an integer, let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is locally $n$-truncated. Then the limit $\varprojlim ( \mathscr {F} )$ is a locally $n$-truncated $\infty $-category.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category and that $n \geq -2$. Let $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the $\infty $-category of elements of $\mathscr {F}$. It follows from Variant 5.1.5.17 that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an . Applying Corollary 4.8.6.22, we see that the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is locally $n$-truncated. Since $\varprojlim (\mathscr {F} )$ can be identified with a full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (Corollary 7.4.4.2), it is also locally $n$-truncated (Remark 4.8.2.3). $\square$

Remark 11.5.0.4. In the situation of Proposition 7.4.4.12, suppose that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration. Then the extension $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$ is also a left fibration. To prove this, it will suffice to show that the fiber $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ is a Kan complex (Proposition 5.1.4.15). This follows from the fact that the covariant diffraction functor

\[ \mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) = \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

is an equivalence of $\infty $-categories, since the simplicial set $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a Kan complex by (Corollary 4.4.2.5).

Remark 11.5.0.5. Let $\lambda $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\lambda $-small. Let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda $ and let $K$ be a $\kappa $-small simplicial set. Then, in the statements of Proposition 7.4.1.18 and Corollary 7.4.1.19, we can replace $\operatorname{\mathcal{S}}$ by the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ of $\lambda $-small spaces (see Variant 7.4.1.15).

Corollary 11.5.0.6. The $\infty $-category $\operatorname{\mathcal{S}}$ is complete and cocomplete.

Proof. See Corollaries None and None. $\square$

Variant 11.5.0.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, and suppose that the fibers of $U$ are essentially $\kappa $-small for some uncountable cardinal $\kappa $. Then, in the statements of Proposition 7.4.2.13 and Corollary 7.4.2.15, we can replace $\operatorname{\mathcal{S}}$ by the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ of $\kappa $-small spaces (see Variant 5.5.4.13).

Proposition 11.5.0.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. If $\operatorname{\mathcal{C}}$ has a final object $X$, then $\operatorname{\mathcal{C}}'$ contains an object which is isomorphic to $X$. In particular, if $\operatorname{\mathcal{C}}'$ is replete, then it contains every final object of $\operatorname{\mathcal{C}}$.

Remark 11.5.0.9. The conclusion of Corollary 11.5.0.10 does not require the assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category; see Corollary 8.6.3.14.

Corollary 11.5.0.10 (Uniqueness). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Then $U$ admits a cartesian conjugate, which is uniquely determined up to equivalence.

Variant 11.5.0.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and suppose we are given a pair of diagrams

\[ e_0, e_1: J \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} , \]

which we identify with morphisms of simplicial sets $F_0, F_1: J \diamond K \rightarrow \operatorname{\mathcal{C}}$ extending $F$ (Remark 4.6.4.9). The following conditions are equivalent:

$(1)$

The diagrams $e_0$ and $e_1$ are isomorphic when regarded as objects of the diagram $\infty $-category

\[ \operatorname{Fun}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} ). \]
$(2)$

The diagrams $F_0$ and $F_1$ are isomorphic when regarded as objects of the $\infty $-category

\[ \operatorname{Fun}_{K/}( J \diamond K, \operatorname{\mathcal{C}}). \]

Proof. We proceed as in Lemma 4.6.4.21. Choose a categorical mapping cylinder

\[ J \coprod J \xrightarrow {(s_0, s_1)} \overline{J} \xrightarrow {\pi } J \]

for the simplicial set $J$ (Definition 4.6.3.3). Using Remark 4.5.8.7, we see that the induced diagram

\[ (J \diamond K) \coprod _{K} (J \diamond K) \xrightarrow {(s'_0, s'_1)} \overline{J} \diamond K \xrightarrow {\pi '} J \diamond K \]

is a categorical mapping cylinder for the simplicial set $J \diamond K$ relative to $K$. Using the criterion of Corollary 4.6.3.11, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

There exists a diagram $\overline{e}: \overline{J} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $ satisfying $\overline{e} \circ s_0 = e_0$ and $\overline{e} \circ s_1 = e_1$.

$(2')$

There exists a diagram $\overline{F}: \overline{J} \diamond K \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F} \circ s'_0 = F_0$ and $\overline{F} \circ s'_1 = F_1$.

The equivalence of $(1')$ and $(2')$ follows from Remark 4.6.4.9. $\square$

Proposition 11.5.0.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the subterminal objects of $\operatorname{\mathcal{C}}$. Then the construction $X \mapsto [X]$ induces a trivial Kan fibration $U: \operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$.

Proof. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}'$. Since $Y$ is subterminal, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is either empty or contractible. It follows that the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )}( [X], [Y] )$ is a homotopy equivalence. Allowing $X$ and $Y$ to vary, we deduce that the functor $U$ is fully faithful. By construction, $U$ is surjective on objects, and therefore essentially surjective. Applying Theorem 4.6.2.21, we conclude that $U$ is an equivalence of $\infty $-categories. Since $\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ is the nerve of a category, $U$ is automatically an inner fibration (Proposition 4.1.1.10). Moreover, every isomorphism in $\operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ is an identity morphism, so $U$ is an isofibration. Applying Proposition 4.5.5.20, we conclude that $U$ is a trivial Kan fibration. $\square$

Corollary 11.5.0.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

There exists a partially ordered set $A$ and an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(A)$.

$(2)$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is either empty or contractible.

$(3)$

Every object of $\operatorname{\mathcal{C}}$ is subterminal.

Proof. The implications $(1) \Rightarrow (2)$ and $(2) \Leftrightarrow (3)$ follow immediately from the definitions. We conclude by observing that if condition $(3)$ is satisfied, then the construction $X \mapsto [X]$ induces a trivial Kan fibration $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ (Proposition 11.5.0.12). $\square$

Question 11.5.0.14. Given a topological space $X$, what can we say about the collection of sets $\{ \operatorname{Sing}_{n}(X) \} _{n \geq 0}$, together with the face and degeneracy operators

\[ d^{n}_{i}: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n-1}(X) \quad \quad s^{n}_ i: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n+1}(X)? \]

What sort of mathematical structure do they form?

Example 11.5.0.15. For $0 \leq i \leq n$, the horn $\Lambda ^{n}_{i}$ of Construction 1.2.4.1 is given by $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is the collection of all nonempty subsets of $[n]$ which are distinct from $[n]$ and $[n] \setminus \{ i \} $.

Example 11.5.0.16 (Singular Homology). For any topological space $X$, the singular homology groups $\operatorname{ \mathrm{H} }_{\ast }(X; \operatorname{\mathbf{Z}})$ are defined as the homology groups of a chain complex

\[ \cdots \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_{2}(X) ] \xrightarrow { \partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_1(X) ] \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_0(X) ], \]

where $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_ n(X) ]$ denotes the free abelian group generated by the set $\operatorname{Sing}_ n(X)$ and the differential is given on generators by the formula

\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i \sigma . \]

Example 11.5.0.17 (The Fundamental Group). Let $X$ be a topological space equipped with a base point $x \in X \simeq \operatorname{Sing}_0(X)$. Then continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x = p(1)$ can be identified with elements of the set $\{ \sigma \in \operatorname{Sing}_1(X): d^{1}_0(\sigma ) = x = d^{1}_1(\sigma ) \} $. The fundamental group $\pi _1(X,x)$ can then be described as the quotient

\[ \{ \sigma \in \operatorname{Sing}_1(X): d^{1}_0(\sigma ) = x = d^{1}_1(\sigma ) \} / \simeq , \]

where $\simeq $ is the equivalence relation on $\operatorname{Sing}_1(X)$ described by

\[ ( \sigma \simeq \sigma ' ) \Leftrightarrow ( \exists \tau \in \operatorname{Sing}_2(X) ) [ d^{2}_0(\tau ) = s^{0}_0(x) \text{ and } d^{2}_1(\tau ) = \sigma \text{ and } d^{2}_2(\tau ) = \sigma ' ]. \]

The datum of a $2$-simplex $\tau $ satisfying these conditions is equivalent to the datum of a continuous map $| \Delta ^2 | \rightarrow X$ with boundary behavior as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{ \underline{x} } \\ x \ar [ur]^{\sigma '} \ar [rr]^{\sigma } & & x; } \]

such a map can be identified with a homotopy between the paths determined by $\sigma $ and $\sigma '$.

Remark 11.5.0.18. Each of our proofs of *** gives additional information that the other does not. Our first proof shows that every simplicial set $S_{\bullet }$ can be built as a colimit of standard simplices in a very specific way: namely, by forming pushouts along boundary inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ (for a more precise assertion, see the proof of Proposition 1.5.5.14). This extra information was used in the proof of Proposition 1.2.3.4 to show that the geometric realization $| S_{\bullet } |$ is a CW complex (and not merely a topological space which is colimit of disks). On the other hand, our second proof shows that every simplicial set $S_{\bullet }$ can be built in a single step as the colimit of a diagram of standard simplices (which can be chosen in a specific, canonical way).

Proof. Using Proposition 3.1.7.1, we can choose a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d]^{f} & X' \ar [d]^{f'} \\ Y \ar [r] & Y' } \]

where the horizontal maps are inner anodyne, $Y'$ is a Kan complex, and $f'$ is a Kan fibration. Then $f$ is $n$-connective if and only if $f'$ is $n$-connective, and $f$ is a weak homotopy equivalence if and only if $f'$ is a homotopy equivalence (Proposition 3.1.6.13). The desired result now follows by applying Proposition 3.2.7.2 to the Kan fibration $f'$. $\square$

Proposition 11.5.0.30. Let $Y$ be a simplicial set. The following conditions are equivalent:

$(1)$

Every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic.

$(2)$

Every morphism of simplicial sets $g: Y \rightarrow Z$ is nullhomotopic.

$(3)$

The identity morphism $\operatorname{id}_{Y}: Y \rightarrow Y$ is nullhomotopic.

$(4)$

The simplicial set $Y$ is contractible.

Proof. The implications $(1) \Rightarrow (3)$ and $(2) \Rightarrow (3)$ are immediate, and the reverse implications follow from Remark 3.2.4.10. To see that $(3) \Leftrightarrow (4)$, it suffices to observe that a morphism $y: \Delta ^0 \rightarrow Y$ is homotopy inverse to the projection map $Y \rightarrow \Delta ^0$ if and only if the identity morphism $\operatorname{id}_{Y}$ is homotopic to the constant morphism $Y \twoheadrightarrow \{ y\} \hookrightarrow Y$. $\square$

Remark 11.5.0.31. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization functor. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to some localizing collection of morphisms $W$. The collection $W$ is then uniquely determined: it is the collection of all morphisms $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $F(u)$ is an isomorphism of $\operatorname{\mathcal{D}}$. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}[W^{-1}]$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects and that $F$ is a left adjoint to the inclusion functor $\operatorname{\mathcal{C}}[W^{-1}] \hookrightarrow \operatorname{\mathcal{C}}$, in which case it follows from Proposition 9.2.1.18.

Construction 11.5.0.33. Let $X$ be a braced simplicial set. Every nondegenerate simplex $\sigma : \Delta ^{n} \rightarrow X$ determines a functor

\[ \operatorname{Chain}[n] \simeq \operatorname{{\bf \Delta }}^{\mathrm{nd}}_{\Delta ^ n} \xrightarrow {\sigma } \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}, \]

which we can identify with an $n$-simplex $f_0(\sigma )$ of the simplicial set $\operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) )$ (Example 3.3.2.8). The construction $\sigma \mapsto f_0(\sigma )$ determines a morphism of semisimplicial sets $f_0: X^{\mathrm{nd}} \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) )$, which extends uniquely to a map of simplicial sets $f: X \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) )$ (Proposition 3.3.1.5).

Corollary 11.5.0.34. 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 4.5.2.32 guarantees that each $F_{D}$ is an equivalence of $\infty $-categories. The converse follows by combining Proposition 5.1.6.7. $\square$

*** snip

Corollary 11.5.0.35. 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 11.5.0.37. 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 8.5.1.8 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 7.6.6.13, 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

11.5
\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 8.5.1.28, we can extend (11.5) 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 11.5.0.38. 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 11.5.0.39 (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 11.5.0.40. 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 7.3.2.11.

Remark 11.5.0.41. 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 11.5.0.42. 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 11.5.0.45. In the situation of Proposition 11.5.0.44, condition $(\star )$ is automatically satisfied if $V$ is a cartesian fibration of simplicial sets.

Example 11.5.0.46. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (Construction 5.5.1.1). 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 11.5.0.47. 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 11.5.0.48. 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 11.5.0.49. 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 11.5.0.50. 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 11.5.0.52. In the statement of Theorem 5.6.6.13, 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 11.5.0.53. 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 11.5.0.54. 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 11.5.0.53, 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 11.5.0.56. 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 11.5.0.59. 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 11.5.0.63. 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 11.6.0.61 and Remark 5.3.3.22.

Remark 11.5.0.64. In the special case $\operatorname{\mathcal{C}}= [1]$, Proposition 11.6.0.61 reduces to Proposition 5.2.3.15. In the case $\operatorname{\mathcal{C}}= [n]$, it reduces to *** (see Variant 11.5.0.65).

Variant 11.5.0.65. 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 11.5.0.66. 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 5.6.2.1, so that the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map (see Example 5.6.2.8). 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 11.5.0.67. 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 5.6.5.1). Moreover, the refined analogues of Questions 11.5.0.59 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 5.6.2.1).

From this perspective, the negative answers to Questions 11.5.0.59 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 11.5.0.71. 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 5.6.6.21. $\square$

Remark 11.5.0.72. 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 4.6.9.13). 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 11.5.0.74. Suppose we are given a commutative diagram of simplicial sets

11.6
\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 3.1.7.11, 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 (11.6) 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 11.5.0.75. Suppose we are given a commutative diagram of simplicial sets

11.7
\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 3.1.7.11, 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 (11.7) 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 11.5.0.78. 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 11.5.0.79 (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 11.5.0.80. 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 4.4.3.19, 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 11.5.0.82. 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 11.5.0.83. 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 11.5.0.84. 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 11.5.0.85. 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 11.5.0.86. 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 11.5.0.87. 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 11.5.0.88. 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 11.5.0.89. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of small $\infty $-categories (Construction 5.5.4.1), 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 11.5.0.90. 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 5.6.2.19: 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 11.5.0.92. 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 11.5.0.93. 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 11.5.0.94 (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 11.5.0.95 (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 11.5.0.96 (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 11.5.0.98. The conclusion of Proposition 5.2.2.8 continues to hold under the more general assumption that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration (see Definition and Proposition ).

Remark 11.5.0.99. 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 5.6.1.6).

Variant 11.5.0.107. 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 11.5.0.108. 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 11.5.0.109. 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 11.5.0.110. 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 11.5.0.111. 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 11.5.0.112. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories.

Remark 11.5.0.113. 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 11.5.0.114. 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 11.5.0.108).

Remark 11.5.0.115. 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 11.5.0.116. 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 11.5.0.117. The converse of Remark 11.5.0.116 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 11.5.0.118. 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 11.5.0.121 (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 11.5.0.122 (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.