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4.1.1 Inner Fibrations of Simplicial Sets

We now introduce the primary objects of interest in this section.

Definition 4.1.1.1. Let $q: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. We say that $q$ is an inner fibration if, for every pair of integers $0 < i < n$, every lifting problem

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

admits a solution (as indicated by the dotted arrow). That is, for every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X_{}$ and every $n$-simplex $\overline{\sigma }: \Delta ^ n \rightarrow S_{}$ extending $q \circ \sigma _0$, we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_{}$ satisfying $q \circ \sigma = \overline{\sigma }$.

Example 4.1.1.2. Let $X_{}$ be a simplicial set. Then the projection map $X_{} \rightarrow \Delta ^0$ is an inner fibration if and only if $X_{}$ is an $\infty $-category.

Remark 4.1.1.3. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if the opposite morphism $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ is an inner fibration.

Remark 4.1.1.4. The collection of inner fibrations is closed under retracts. That is, given a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_{} \ar [r] \ar [d]^{q} & X'_{} \ar [d]^{q'} \ar [r] & X_{} \ar [d]^{q} \\ S_{} \ar [r] & S'_{} \ar [r] & S_{} } \]

where both horizontal compositions are the identity, if $q'$ is an inner fibration, then so is $q$.

Remark 4.1.1.5. The collection inner fibrations is closed under pullback. That is, given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{q'} \ar [r] & X_{} \ar [d]^{q} \\ S'_{} \ar [r]^-{f} & S_{} } \]

where $q$ is an inner fibration, the morphism $q'$ is also an inner fibration. Conversely, if $q'$ is an inner fibration and $f$ is surjective, then $q$ is an inner fibration.

Remark 4.1.1.6. Let $q: X_{} \rightarrow S_{}$ be an inner fibration of simplicial sets. Then, for every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{ S_{} } X_{}$ is an $\infty $-category (this follows from Remark 4.1.1.5 and Example 4.1.1.2).

Remark 4.1.1.7. The collection of inner fibrations is closed under filtered colimits. That is, if $\{ q_{\alpha }: X_{\alpha } \rightarrow S_{\alpha } \} $ is a filtered diagram in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$ having colimit $q: X \rightarrow S$, and each $q_{\alpha }$ is an inner fibration of simplicial sets, then $q$ is also an inner fibration of simplicial sets.

Remark 4.1.1.8. Let $p: X_{} \rightarrow Y_{}$ and $q: Y_{} \rightarrow Z_{}$ be inner fibrations of simplicial sets. Then the composite map $(q \circ p): X_{} \rightarrow Z_{}$ is an inner fibration of simplicial sets.

Remark 4.1.1.9. Let $q: X \rightarrow Y$ be an inner fibration of simplicial sets. If $Y$ is an $\infty $-category, then $X$ is also an $\infty $-category (this follows by combining Remark 4.1.1.8 with Example 4.1.1.2).

Proposition 4.1.1.10. Let $\operatorname{\mathcal{C}}$ be a category, and let $q: X \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if $X$ is an $\infty $-category.

Proof. If $q$ is an inner fibration, then Remark 4.1.1.9 guarantees that $X$ is an $\infty $-category. Conversely, suppose that $X$ is an $\infty $-category and that we are given a lifting problem

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

for integers $0 < i < n$. Our assumption that $X$ is an $\infty $-category guarantees that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$. The equality $q \circ \sigma = \overline{\sigma }$ is automatic by virtue of Proposition 1.2.3.1). $\square$

Example 4.1.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Construction 1.3.5.1). Then the canonical map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an inner fibration.

Remark 4.1.1.12. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is an inner fibration.

$(2)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is an inner fibration.

$(3)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the fiber product $\Delta ^ n \times _{S} X$ is an $\infty $-category.

The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definition, and the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.1.1.10.