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4.2.1 Left and Right Fibrations of Simplicial Sets

We begin by introducing some terminology.

Definition 4.2.1.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We will say that $f$ is a left fibration if, for every pair of integers $0 \leq i < n$, every lifting problem

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

has 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 $f \circ \sigma _0$, we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_{}$ satisfying $f \circ \sigma = \overline{\sigma }$.

We say that $f$ is a right fibration if, for every pair of integers $0 < i \leq n$, every lifting problem

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

has a solution.

Example 4.2.1.2. Any isomorphism of simplicial sets is both a left fibration and a right fibration.

Remark 4.2.1.3. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a left fibration if and only if the opposite morphism $f^{\operatorname{op}}: X_{}^{\operatorname{op}} \rightarrow S_{}^{\operatorname{op}}$ is a right fibration.

Remark 4.2.1.4. 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 5.3.6.1 and Proposition 5.3.6.14). Similarly, $\operatorname{N}_{\bullet }(F)$ is a left fibration if and only if $F$ is an opfibration in groupoids.

Remark 4.2.1.5. Let $f: X \rightarrow S$ be a morphism of simplicial sets. If $f$ is either a left fibration or a right fibration, then it is an inner fibration. In this case, if $S$ is an $\infty$-category, then $X$ is also an $\infty$-category (Remark 4.1.1.9).

Example 4.2.1.6. A morphism of simplicial sets $f: X \rightarrow S$ is a Kan fibration if and only if it is both a left fibration and a right fibration.

Warning 4.2.1.7. In the statement of Example 4.2.1.6, both hypotheses are necessary: a left fibration of simplicial sets need not be a right fibration and vice versa. For example, the inclusion map $\{ 1 \} \hookrightarrow \Delta ^1$ is a left fibration, but not a right fibration (and therefore not a Kan fibration).

Remark 4.2.1.8. The collection of left and right fibrations is closed under retracts. That is, suppose we are given a diagram of simplicial sets

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

where both horizontal compositions are the identity. If $f'$ is a left fibration, then $f$ is a left fibration. If $f'$ is a right fibration, then $f$ is a right fibration.

Remark 4.2.1.9. The collections of left and right fibrations are closed under pullback. That is, suppose we are given a pullback diagram of simplicial sets

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

If $f$ is a left fibration, then $f'$ is also a left fibration. If $f$ is a right fibration, then $f'$ is a right fibration.

Remark 4.2.1.10. Let $f: X \rightarrow S$ be a map of simplicial sets. Suppose that, for every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is a left fibration (right fibration). Then $f$ is a left fibration (right fibration). Consequently, if we are given a pullback diagram of simplicial sets

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

where $g$ is surjective and $f'$ is a left fibration (right fibration), then $f$ is also a left fibration (right fibration).

Remark 4.2.1.11. The collections of left and right fibrations are closed under filtered colimits. That is, suppose we are given a filtered diagram $\{ f_{\alpha }: X_{\alpha } \rightarrow S_{\alpha } \}$ in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$, having colimit $f: X \rightarrow S$. If each $f_{\alpha }$ is a left fibration , then $f$ is also a left fibration. If each $f_{\alpha }$ is a right fibration, then $f$ is also a right fibration.

Remark 4.2.1.12. Let $f: X_{} \rightarrow Y_{}$ and $g: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If both $f$ and $g$ are left fibrations, then the composite map $(g \circ f): X_{} \rightarrow Z_{}$ is a left fibration. If both $f$ and $g$ are right fibrations, then $g \circ f$ is a right fibration.