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4.1 Left and Right Fibrations of Simplicial Sets

Let us begin by introducing the main objects of study in this chapter.

Definition 4.1.0.1. 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 $0 \leq i < n$, 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.

Example 4.1.0.2. Any isomorphism of simplicial sets is both a left fibration and a right fibration.

Remark 4.1.0.3. 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 4.1.0.4. Let $f: X \rightarrow S$ be a morphism of simplicial sets, where $S$ is an $\infty $-category. If $f$ is either a left or right fibration, then $X$ is also an $\infty $-category. To prove this, we must show that for $0 < i < n$, any map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow S$. Our assumption that $S$ is an $\infty $-category guarantees that the composite map $f \circ \sigma _0$ can be extended to an $n$-simplex $\overline{\sigma }: \Delta ^ n \rightarrow S$. The desired result now follows from the solvability of the lifting problem

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

Example 4.1.0.5. 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 4.1.0.6. In the statement of Example 4.1.0.5, 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 4.1.0.7. 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 4.1.0.8. The collections of left and right 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 4.1.0.9. 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).

Remark 4.1.0.10. The collections of left and right 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 4.1.0.11. 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.

Let $q: X \rightarrow S$ be a morphism of simplicial sets. Note that evaluation at the vertices of $\Delta ^1$ induces morphisms of simplicial sets

\[ \operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ) \]
\[ \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 1\} , X_{} ) \times _{ \operatorname{Fun}( \{ 1\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ). \]

In §4.1.3, we show that $f$ is a left fibration if and only if the evaluation map $\operatorname{ev}_0$ is a trivial Kan fibration, and that $f$ is a right fibration if and only if $\operatorname{ev}_{1}$ is a trivial Kan fibration (Proposition 4.1.3.1). The “only if” direction of this assertion is a special case of general stability properties of left and right fibrations under exponentiation, which we prove in §4.1.2 (Propositions 4.1.2.1 and 4.1.2.4). Our proofs will make use of some basic facts about left anodyne and right anodyne morphisms of simplicial sets, which we establish in §4.1.1.

The notions of left and right fibration introduced in this section have a counterpart in classical category theory. In §4.1.4, we recall the notion a fibration in groupoids (Definition 4.1.4.1), and show that a functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a fibration in groupoids if and only if the induced map of nerves $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a right fibration of simplicial sets (Proposition 4.1.4.12). Similarly, a functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an opfibration in groupoids if only if the map $\operatorname{N}_{\bullet }(F)$ is a left fibration of simplicial sets.

Structure

  • Subsection 4.1.1: Left Anodyne and Right Anodyne Morphisms
  • Subsection 4.1.2: Exponentiation for Left and Right Fibrations
  • Subsection 4.1.3: The Homotopy Extension Lifting Property
  • Subsection 4.1.4: Example: Fibrations in Groupoids