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9.1.5 Lifting Problems in $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be a category. Recall that a lifting problem in $\operatorname{\mathcal{C}}$ is a commutative diagram

9.6
\begin{equation} \begin{gathered}\label{equation:lifting-problem-ordinary} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{\overline{u}} & Y } \end{gathered} \end{equation}

In this case, a solution to the lifting problem (9.6) is a morphism $u: B \rightarrow X$ satisfying $u \circ f = u_0$ and $q \circ u = \overline{u}$ (see Definition 1.5.4.1). This definition has an obvious $\infty $-categorical counterpart:

Definition 9.1.5.1 (Lifting Problems in $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A lifting problem in $\operatorname{\mathcal{C}}$ is a diagram $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. In this case, a solution to the lifting problem $\sigma $ is a $3$-simplex $\overline{\sigma }: \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ for which the composition

\[ \Delta ^1 \times \Delta ^1 \xrightarrow {\alpha } \Delta ^3 \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}} \]

coincides with $\sigma $, where $\alpha $ denotes the map of simplicial sets given on vertices by $\alpha (i,j) = 2i+j$.

Remark 9.1.5.2. Let us informally display the standard simplex $\Delta ^3$ as a diagram

9.7
\begin{equation} \begin{gathered}\label{equation:square-in-simplex} \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar [dr] & \bullet \ar [d] \\ \bullet \ar [r] \ar@ {-->}[ur] & \bullet . } \end{gathered} \end{equation}

The morphism $\alpha : \Delta ^1 \times \Delta ^1 \rightarrow \Delta ^3$ appearing in Definition 9.1.5.1 is a monomorphism of simplicial sets, whose image is the simplicial subset $Q \subseteq \Delta ^3$ consisting of those simplices which do not contain the “inner” edge $\operatorname{N}_{\bullet }( \{ 1 < 2 \} )$ which is indicated by the dotted arrow in the diagram (9.7). Stated more informally, $Q$ is the subset of $\Delta ^3$ which is “visible from the top” in the diagram (9.7); in particular, $Q$ contains the inner faces $\operatorname{N}_{\bullet }( \{ 0 < 1 < 3 \} )$ and $\operatorname{N}_{\bullet }( \{ 0 < 2 < 3 \} )$, but not the outer faces $\operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )$ and $\operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )$.

Notation 9.1.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often denote a lifting problem $\sigma $ in $\operatorname{\mathcal{C}}$ by a diagram

9.8
\begin{equation} \begin{gathered}\label{equation:lifting-property-in-infinity2} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{ \overline{u} } \ar@ {-->}[ur] & Y. } \end{gathered} \end{equation}

Here the dotted arrow in the diagram does not indicate part of the data supplied by lifting problem $\sigma $; instead, it indicates part of the data of a hypothetical solution.

Stated more concretely, the lifting problem $\sigma $ is given by the following data:

  • Four objects of $\operatorname{\mathcal{C}}$, which are indicated by $A$, $B$, $X$, and $Y$ in the diagram (9.8).

  • Five morphisms of $\operatorname{\mathcal{C}}$, which we will denote by $f: A \rightarrow B$, $g: X \rightarrow Y$, $u_0: A \rightarrow X$, $\overline{u}: B \rightarrow Y$, and $\overline{u}_0: A \rightarrow Y$. Here the first four of these morphisms are indicated as outer edges of the diagram (9.8), while the fifth is left implicit.

  • A pair of $2$-simplices $\tau _{1}$ and $\tau _{2}$ of $\operatorname{\mathcal{C}}$, whose boundaries are indicated in the diagrams

    \[ \xymatrix@R =50pt@C=50pt{ A \ar [dr]_{\overline{u}_0} \ar [r]^-{u_0} & X \ar [d]^{g} & A \ar [d]^{f} \ar [dr]^{\overline{u}_0} & \\ & Y & B \ar [r]^-{\overline{u}} & Y. } \]

    In other words, $\tau _1$ and $\tau _2$ exhibit the morphism $\overline{u}_0$ as a composition $\overline{u} \circ f$ and a composition $g \circ u_0$, respectively.

A solution to the lifting problem $\sigma $ is given by the following additional data:

  • A morphism $u: B \rightarrow X$ (indicated by the dotted arrow in the diagram (9.8).

  • A pair of $2$-simplices $\tau _0$ and $\tau _3$ of $\operatorname{\mathcal{C}}$, whose boundaries are indicated in the diagrams

    \[ \xymatrix@R =50pt@C=50pt{ & X \ar [d]^{g} & A \ar [d]^{f} \ar [r]^-{u_0} & X \\ B \ar [ur]^{u} \ar [r]^-{\overline{u}} & Y & B. \ar [ur]_{u} & } \]

    In other words, $\tau _0$ exhibits $\overline{u}$ as a composition $g \circ u$, and $\tau _3$ exhibits $u_0$ as a composition $u \circ f$.

  • A $3$-simplex of $\operatorname{\mathcal{C}}$ having boundary $(\tau _0, \tau _1, \tau _2, \tau _3)$.

Example 9.1.5.4. Let $\operatorname{\mathcal{C}}_0$ be a category and let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ denote its nerve. Then lifting problems $\sigma $ in the $\infty $-category $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.1.5.1) can be identified with lifting problems $\sigma _0$ in the ordinary category $\operatorname{\mathcal{C}}_0$ (in the sense of Definition 1.5.4.1). In this case, we can also identify solutions to $\sigma $ with solutions to $\sigma _0$.

Warning 9.1.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every lifting problem

9.9
\begin{equation} \begin{gathered}\label{equation:lifting-problem-in-homotopy} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{ \overline{u} } \ar@ {-->}[ur] & Y } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$ determines a lifting problem in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, given by the diagram

9.10
\begin{equation} \begin{gathered}\label{equation:lifting-problem-in-homotopy2} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{[f]} \ar [r]^-{[u_0]} & X \ar [d]^{[g]} \\ B \ar [r]^-{ [\overline{u}] } \ar@ {-->}[ur] & Y. } \end{gathered} \end{equation}

Moreover, every solution to the lifting problem (9.9) determines a solution to the lifting problem (9.10). Beware that the converse is false: it is possible for the lifting problem (9.10) to admit a solution when the lifting problem (9.9) does not (Exercise 9.1.5.6).

Exercise 9.1.5.6. Let $g: X \rightarrow S$ be Kan fibration between Kan complexes, where $S$ is connected and $X$ is contractible. Choose a vertex $s \in S$ and let $X_{s}$ denote the fiber $\{ s\} \times _{S} X$, so that we have a commutative diagram of Kan complexes

9.11
\begin{equation} \begin{gathered}\label{equation:impossible-lifting-problem} \xymatrix@R =50pt@C=50pt{ X_{s} \ar [r] \ar [d] & X \ar [d]^{f} \\ \{ s\} \ar [r] & S } \end{gathered} \end{equation}

Show that:

  • In the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, the lifting problem determined by (9.11) admits a solution only if $S$ is contractible.

  • In the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}}$, the lifting problem determined by (9.11) always has a solution.

Remark 9.1.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Fix a morphism $\overline{u}_0: A \rightarrow Y$ of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_{ A/ \, /Y}$ denote the $\infty $-category of Remark 4.6.6.2. The datum of an extension of $\overline{u}_0$ to a lifting problem $\sigma :$

9.12
\begin{equation} \begin{gathered}\label{equation:lifting-problem-via-double-slice} \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & X \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & Y } \end{gathered} \end{equation}

can be identified with a pair of objects $\widetilde{B}$, $\widetilde{X} \in \operatorname{\mathcal{C}}_{A/ \, /Y}$. In this case, a solution to the lifting problem (9.12) is a morphism from $\widetilde{B}$ to $\widetilde{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{A/ \, /Y}$.

Definition 9.1.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f: A \rightarrow B$ and $g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$. We will say that $f$ is weakly left orthogonal to $g$ if every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$ admits a solution. In this case, we will also say that $g$ is weakly right orthogonal to $f$.

Example 9.1.5.9. In the situation of Definition 9.1.5.8, suppose that $\operatorname{\mathcal{C}}$ is the nerve of a category $\operatorname{\mathcal{C}}_0$. Then $f$ is weakly left orthogonal to $g$ (in the sense of Definition 9.1.5.8) if and only if it is weakly left orthogonal to $g$ when regarded as a morphism of the category $\operatorname{\mathcal{C}}_0$ (in the sense of Definition 1.5.4.3).

Warning 9.1.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. If $f$ is weakly left orthogonal to $g$ in the $\infty $-category $\operatorname{\mathcal{C}}$, then the homotopy class $[f]$ is weakly left orthogonal to $[g]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Exercise 1.5.2.10). Beware that the converse is false in general (see Warning 9.1.5.5 and Exercise 9.1.5.6).

Variant 9.1.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ and $T$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We say that $S$ is weakly left orthogonal to $T$ if every morphism $f \in S$ is weakly left orthogonal to every morphism $g \in T$. In this case, we also say that $T$ is weakly right orthogonal to $S$. In the special case where $S = \{ f\} $ is a singleton, we abbreviate this condition by saying that $f$ is weakly left orthogonal to $T$, or $T$ is weakly right orthogonal to $f$. In the special case $T = \{ g\} $ is a singleton, we abbreviate this condition by saying that $g$ is weakly right orthogonal to $S$, or $S$ is weakly left orthogonal to $g$.

Remark 9.1.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$, which we identify with an object $\widetilde{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. Let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\widetilde{S}$ denote its inverse image in $\operatorname{\mathcal{C}}_{/Y}$. The following conditions are equivalent:

  • The morphism $g$ is weakly right orthogonal to $S$ (in the sense of Variant 9.1.5.11).

  • The object $\widetilde{X}$ is weakly $\widetilde{S}$-local (in the sense of Definition 9.1.3.5).

Example 9.1.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$. Then every isomorphism $f$ of $\operatorname{\mathcal{C}}$ is weakly left orthogonal to $g$. This follows from the criterion of Remark 9.1.5.12, since every lift of $f$ to the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$ is also an isomorphism (Proposition 4.4.2.11).

We now record a few consequences of Remark 9.1.5.12.

Proposition 9.1.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. If another morphism $f: A \rightarrow B$ is weakly left orthogonal to $g$, then any retract of $f$ (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$) is also weakly left orthogonal to $g$.

Proof. Let $f': A' \rightarrow B'$ be a retract of $f$ (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$); we will show that $f'$ is weakly left orthogonal to $g$. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Remark 9.1.5.12, it will suffice to show that for any morphism $\widetilde{f}'$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}' ) = f'$, the object $\widetilde{X}$ is weakly $\widetilde{f}'$-local. It follows from Corollary 4.2.5.2 that $\pi $ induces a right fibration $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{/Y} ) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Applying Remark 8.5.1.23, we deduce that $\widetilde{f}'$ is a retract of a morphism $\widetilde{f}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $U( \widetilde{f} ) = f$. By virtue of Variant 9.1.3.13, it will suffice to show that the object $\widetilde{X}$ is weakly $\widetilde{f}$-local, which follows from our assumption that $f$ is weakly left orthogonal to $g$ (Remark 9.1.5.12). $\square$

Proposition 9.1.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pushout diagram

9.13
\begin{equation} \begin{gathered}\label{equation:pushout-of-weakly-left-orthogonal} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A' \ar [d]^{f'} \\ B \ar [r] & B'. } \end{gathered} \end{equation}

If $f$ is weakly left orthogonal to a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, then $f'$ is also weakly left orthogonal to $g$.

Proof. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Remark 9.1.5.12, it will suffice to show that for any morphism $\widetilde{f}'$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f}' ) = f'$, the object $\widetilde{X}$ is weakly $\widetilde{f}'$-local. Since $\pi $ is a right fibration, we can lift (9.13) to a diagram

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{A} \ar [r] \ar [d]^{ \widetilde{f} } & \widetilde{A}' \ar [d]^{ \widetilde{f}'} \\ \widetilde{B} \ar [r] & \widetilde{B}' } \]

in the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, which is also a pushout square (Proposition 7.1.3.19). By virtue of Proposition 9.1.3.14, it will suffice to show that $\widetilde{X}$ is weakly $\widetilde{f}$-local, which follows from our assumption that $f$ is weakly left orthogonal to $g$ (Remark 9.1.5.12). $\square$

Proposition 9.1.5.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$, and let $S$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are weakly left orthogonal to $g$. Then $S$ is closed under transfinite composition (see Definition 9.1.2.1).

Proof. Let $f: A \rightarrow B$ be a transfinite composition of morphisms of $S$; we wish to show that $f$ is weakly left orthogonal to $g$. Let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ be the projection map, and let us identify $g$ with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{X} ) = X$. By virtue of Remark 9.1.5.12, it will suffice to show that for every morphism $\widetilde{f}: \widetilde{A} \rightarrow \widetilde{B}$ of $\operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi ( \widetilde{f} ) = f$, the object $\widetilde{X}$ is weakly $\widetilde{f}$-local.

Choose an ordinal $\alpha $ and a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $S$ (see Definition 9.1.2.1). We will assume that $\alpha > 0$ (otherwise, $f$ is an identity morphism and the desired result follows from Example 9.1.5.13). In this case, Lemma 4.3.7.8 guarantees that the inclusion map $\operatorname{N}_{\bullet }( \{ 0 < \alpha \} ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )$ is right anodyne. Since $\pi $ is a right fibration (Proposition 4.3.6.1), we can lift $F$ to a diagram $\widetilde{F}: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ for which the associated morphism $\widetilde{F}(0) \rightarrow \widetilde{F}(\alpha )$ coincides with $\widetilde{f}$. For every nonzero limit ordinal $\lambda \leq \alpha $, Proposition 7.1.3.19 guarantees that the restriction $\widetilde{F}|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda } ) }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. Using Remark 9.1.5.12, we see that $\widetilde{F}$ exhibits $\widetilde{f}$ as a transfinite composition of morphisms of $\operatorname{\mathcal{C}}_{/Y}$ with respect to which $\widetilde{X}$ is weakly local. Applying Proposition 9.1.3.17, we conclude that $\widetilde{X}$ is weakly $\widetilde{f}$-local, as desired. $\square$

Corollary 9.1.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are weakly left orthogonal to $T$. Then $S$ is weakly saturated.