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5.4.2 Interior Fibrations

Recall that a morphism of simplicial sets $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration if it has the right lifting property with respect to the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ for every pair of integers $0 < i < n$. In the setting of $(\infty ,2)$-categories, it will be convenient to consider a variant of this condition.

Definition 5.4.2.1. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We will say that $q$ is an interior fibration if it satisfies the following conditions:

  • Every lifting problem

    \[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

    admits a solution, provided that $0 < i < n$ and the restriction $\overline{\sigma }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.

  • For every vertex $X \in \operatorname{\mathcal{C}}$, the degenerate edge $\operatorname{id}_{X}$ is $q$-cartesian and $q$-cocartesian.

Example 5.4.2.2. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is an interior fibration (in the sense of Definition 5.4.2.1.

$(2)$

The morphism $q$ is an inner fibration (in the sense of Definition 4.1.1.1).

The implication $(1) \Rightarrow (2)$ follows from the observation that every $2$-simplex of $\operatorname{\mathcal{D}}$ is thin, and the implication $(2) \Rightarrow (1)$ follows from Corollary 5.2.1.8. In particular, if either of these conditions is satisfied, then $\operatorname{\mathcal{C}}$ is an $\infty $-category.

Remark 5.4.2.3. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $q$ is an interior fibration if and only if the opposite morphism $q^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is an interior fibration.

Remark 5.4.2.4. Suppose we are given a pullback diagram of simplicial sets

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

Assume that $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{D}}'$ are $(\infty ,2)$-categories and that the morphism $F$ carries thin $2$-simplices of $\operatorname{\mathcal{D}}'$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$ (that is, that $F$ is a functor of $(\infty ,2)$-categories; see Definition 5.4.7.1). If $q$ is an interior fibration, then $q'$ is an interior fibration.

Remark 5.4.2.5. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration. Then, for every object $X \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{X} = \{ X\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an $\infty $-category (this follows by combining Example 5.4.2.2 with Remark 5.4.2.4).

Our goal in this section is to show that, if $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category and $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an interior fibration of simplicial sets, then $\operatorname{\mathcal{C}}$ is also an $(\infty ,2)$-category (Proposition 5.4.2.8). To prove this, we must exhibit a sufficiently large collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$.

Lemma 5.4.2.6. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category, let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of simplicial sets, and let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{C}}$. If $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, then $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. Suppose we are given a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$, where $n \geq 3$, $0 < i < n$, and $\tau _0$ carries $\operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )$ to the $2$-simplex $\sigma $. We wish to show that $\tau $ can be extended to an $n$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. Let $\overline{\tau }_0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{D}}$ be the composition $q \circ \tau _0$. Since $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, we can extend $\overline{\tau }_0$ to an $n$-simplex $\overline{\tau }: \Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$. To complete the proof, it suffices to find a solution to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{ \tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^{n} \ar@ {-->}[ur]^{\tau } \ar [r]^-{\overline{\tau }} & \operatorname{\mathcal{D}}, } \]

which exists by virtue of our assumption that $q$ is an interior fibration. $\square$

Remark 5.4.2.7. In the situation of Lemma 5.4.2.6, we will see later that the converse assertion is also true: if $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, then $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$ (see Proposition 5.4.7.9).

Proposition 5.4.2.8. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of simplicial sets. Then $\operatorname{\mathcal{C}}$ is also an $(\infty ,2)$-category.

Proof. We must verify that the simplicial set $\operatorname{\mathcal{C}}$ satisfies each of the axioms of Definition 5.4.1.3:

$(1)$

Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be edges of the simplicial set $\operatorname{\mathcal{C}}$; we wish to show that there exists a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ there exists a thin $2$-simplex $\Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ satisfying $d_2(\sigma ) = f$ and $d_0(\sigma ) = g$, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z. } \]

We first invoke our assumption that $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category to choose a thin $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ satisfying $d_2( \overline{\sigma } ) = q(f)$ and $d_0( \overline{\sigma } ) = q(g)$. Since $\overline{\sigma }$ is thin, our assumption that $q$ is an interior fibration guarantees that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r]^-{ (g, \bullet , f) } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^{2} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}} \]

admits a solution. It follows from Lemma 5.4.2.6 that $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(2)$

Let $\sigma $ be a degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Then $q(\sigma )$ is a degenerate $2$-simplex of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$. Applying Lemma 5.4.2.6, we conclude that $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(3)$

Let $n \geq 3$ and let $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1 < n\} ) }$ is left-degenerate; we wish to show that $\tau _0$ can be extended to an $n$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. Let $\overline{\tau }_0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ denote the composition $q \circ \tau _0$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, we can extend $\overline{\tau }_0$ to an $n$-simplex $\overline{\tau }: \Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$. To complete the proof, it will suffice to show that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\tau } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}} \]

admits a solution. We conclude by observing that the edge $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1\} ) }$ is degenerate and is therefore $q$-cocartesian by virtue of our assumption that $q$ is an interior fibration.

$(4)$

Let $n \geq 3$ and let $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< n-1 < n\} ) }$ is right-degenerate; we wish to show that $\tau _0$ can be extended to an $n$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. This follows by the argument given above, applied to the opposite interior fibration $q^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$.

$\square$

Proposition 5.4.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be interior fibrations of $(\infty ,2)$-categories. Then the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also an interior fibration.

Proof. Suppose we are given an integer $n \geq 2$ and a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{G \circ F} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{E}}. } \]

We wish to show that this lifting problem admits a solution if one of the following conditions is satisfied:

$(a)$

The integer $i$ is equal to $0$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}$.

$(b)$

The integer $i$ satisfies $0 < i < n$ and the restriction $\overline{\sigma }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{\mathcal{E}}$.

$(c)$

The integer $i$ is equal to $n$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ n-1 < n \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}$.

Since $G$ is an interior fibration, any of these hypotheses guarantee the existence of a solution to the associated lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{F \circ \sigma _0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{G} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{E}}. } \]

It will therefore suffice to construct a solution to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar [r]^-{\tau } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}. } \]

In cases $(a)$ and $(c)$, our assumption that $F$ is an interior fibration immediately guarantees the existence of $\sigma $. In case $(b)$, it suffices to verify that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )}$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, which follows from Lemma 5.4.2.6. $\square$

Proposition 5.4.2.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration between $(\infty ,2)$-categories, and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The induced map of left-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{L}}( F(X), F(Y) )$ is a right fibration of simplicial sets.

$(2)$

The induced map of right-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{R}}( F(X), F(Y) )$ is a left fibration of simplicial sets.

Proof. We will prove $(2)$; assertion $(1)$ follows from a similar argument. We wish to show that, for every pair of integers $0 \leq i < n$, every lifting problem

5.13
\begin{equation} \begin{gathered}\label{equation:interior-fibration-morphism-space} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \ar [d] \\ \Delta ^{n} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{R}}( F(X), F(Y) ) } \end{gathered} \end{equation}

admits a solution. Unwinding the definitions, we can rewrite (5.13) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i} \ar [r]^-{\tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n} \ar [r]^-{\overline{\tau }} \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}, } \]

where the restriction $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < \cdots < n \} )}$ is the constant map taking the value $X$. If $i=0$, then this lifting problem admits a solution because the edge $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is degenerate (and therefore $F$-cocartesian, by virtue of our assumption that $F$ is an interior fibration). If $0 < i < n$, the solution exists by virtue of the fact that $F$ is an interior fibration and $\overline{\tau }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a degenerate $2$-simplex of $\operatorname{\mathcal{D}}$ (and therefore thin). $\square$