To study left and right fibrations between simplicial sets, it is useful to consider the following counterpart of Definitions 3.1.2.1 and 1.5.6.4:
Definition 4.2.4.1 (Left Anodyne Morphisms). Let $T_{L}$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:
For every pair of integers $0 \leq i < n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T_{L}$.
The collection $T_{L}$ is weakly saturated (Definition 1.5.4.12). That is, $T_{L}$ is closed under pushouts, retracts, and transfinite composition.
We say that a morphism of simplicial sets $f: A \rightarrow B$ is left anodyne if it belongs to $T_{L}$.
Variant 4.2.4.2 (Right Anodyne Morphisms). Let $T_{R}$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:
For every pair of integers $0 < i \leq n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T_{R}$.
The collection $T_{R}$ is weakly saturated (Definition 1.5.4.12). That is, $T_{R}$ is closed under pushouts, retracts, and transfinite composition.
We say that a morphism of simplicial sets $f: A \rightarrow B$ is right anodyne if it belongs to $T_{R}$.
Remark 4.2.4.3. Let $f: A \rightarrow B$ be a morphism of simplicial sets. Then $f$ is left anodyne if and only if the opposite morphism $f^{\operatorname{op}}: A^{\operatorname{op}} \rightarrow B^{\operatorname{op}}$ is right anodyne.
Remark 4.2.4.4. Let $f: A \rightarrow B$ be a morphism of simplicial sets. If $f$ is either left or right anodyne, then it is anodyne (Definition 3.1.2.1). In particular, any left or right anodyne morphism of simplicial sets is a monomorphism (Remark 3.1.2.3) and a weak homotopy equivalence (Proposition 3.1.6.14). Conversely, if $f$ is inner anodyne (Definition 1.5.6.4), then it is both left anodyne and right anodyne. That is, we have inclusions All of these inclusions are strict (see Example 4.2.4.7).
\[ \xymatrix@C =30pt@R=30pt{ \{ \textnormal{Inner anodyne morphisms} \} \ar@ {}[d]|{\cap } \ar@ {}[r]|{\subset } & \{ \textnormal{Left anodyne morphisms} \} \ar@ {}[d]|{\cap } \\ \{ \textnormal{Right anodyne morphisms} \} \ar@ {}[r]|{\subset } & \{ \textnormal{Anodyne morphisms} \} . } \]
Proposition 4.2.4.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then:
The morphism $q$ is a left fibration if and only if, for every square diagram of simplicial sets
where $i$ is left anodyne, there exists a dotted arrow rendering the diagram commutative.
The morphism $q$ is a right fibration if and only if, for every square diagram of simplicial sets
where $i$ is right anodyne, there exists a dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
Proof. The “only if” directions are immediate from the definitions, and the “if” directions follow from Proposition 1.5.4.13. $\square$
Corollary 4.2.4.6. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then:
The morphism $q$ is a left covering map if and only if, for every square diagram of simplicial sets
where $i$ is left anodyne, there exists a unique dotted arrow rendering the diagram commutative.
The morphism $q$ is a right covering map if and only if, for every square diagram of simplicial sets
where $i$ is right anodyne, there exists unique a dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
Proof. Combine Proposition 4.2.4.5 with Remark 4.2.3.11. $\square$
Example 4.2.4.7. The inclusion map $i_0: \{ 0\} \hookrightarrow \Delta ^1$ is left anodyne (and therefore anodyne). However, it is not right anodyne (and therefore not inner anodyne). This follows from Proposition 4.2.4.5, since the lifting problem does not admit a solution (note that the inclusion map $i_0: \{ 0\} \hookrightarrow \Delta ^1$ is a right fibration; see Warning 4.2.1.6).
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r]^-{\operatorname{id}} \ar [d]^{i_0} & \{ 0\} \ar [d]^{i_0} \\ \Delta ^1 \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & \Delta ^1 } \]
Proposition 4.2.4.8. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Then $f$ can be factored as a composition $X_{} \xrightarrow {f'} Q_{}(f) \xrightarrow {f''} Y_{}$, where $f''$ is a left fibration and $f'$ is left anodyne. Moreover, the simplicial set $Q_{}(f)$ (and the morphisms $f'$ and $f''$) can be chosen to depend functorially on $f$, in such a way that the functor commutes with filtered colimits.
\[ \operatorname{Fun}( [1], \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad (f: X_{} \rightarrow Y_{} ) \rightarrow Q_{}(f) \]
Proof. We proceed as in the proof of Proposition 3.1.7.1. We construct a sequence of simplicial sets $\{ X(m)_{} \} _{m \geq 0}$ and morphisms $f(m): X(m)_{} \rightarrow Y_{}$ by recursion. Set $X(0)_{} = X_{}$ and $f(0) = f$. Assuming that $f(m): X(m)_{} \rightarrow Y_{}$ has been defined, let $S(m)$ denote the set of all commutative diagrams $\sigma :$
where $0 \leq i < n$ and the left vertical map is the inclusion. For every such commutative diagram $\sigma $, let $C_{\sigma } = \Lambda ^ n_{i}$ denote the upper left hand corner of the diagram $\sigma $, and $D_{\sigma } = \Delta ^ n$ the lower left hand corner. Form a pushout diagram
and let $f(m+1): X(m+1)_{} \rightarrow Y_{}$ be the unique map whose restriction to $X(m)_{}$ is equal to $f(m)$ and whose restriction to each $D_{\sigma }$ is equal to $u_{\sigma }$. By construction, we have a direct system of left anodyne morphisms
Set $Q_{}(f) = \varinjlim _{m} X(m)_{}$. Then the natural map $f': X_{} \rightarrow Q_{}(f)$ is left anodyne (since the collection of left anodyne maps is closed under transfinite composition), and the system of morphisms $\{ f(m) \} _{m \geq 0}$ can be amalgamated to a single map $f'': Q_{}(f) \rightarrow Y_{}$ satisfying $f = f'' \circ f'$. It is clear from the definition that the construction $f \mapsto Q_{}(f)$ is functorial and commutes with filtered colimits. To complete the proof, it will suffice to show that $f''$ is a left fibration: that is, that every lifting problem $\sigma :$
admits a solution (provided that $0 \leq i < n$). Let us abuse notation by identifying each $X(m)_{}$ with its image in $Q_{}(f)$. Since $\Lambda ^{n}_{i}$ is a finite simplicial set, its image under $v$ is contained in $X(m)_{}$ for some $m \gg 0$. In this case, we can identify $\sigma $ with an element of the set $S(m)$, so that the lifting problem
admits a solution by construction. $\square$
Variant 4.2.4.9. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Then $f$ can be factored as a composition $X_{} \xrightarrow {f'} Q_{}(f) \xrightarrow {f''} Y_{}$, where $f''$ is a right fibration and $f'$ is right anodyne. Moreover, the simplicial set $Q_{}(f)$ (and the morphisms $f'$ and $f''$) can be chosen to depend functorially on $f$, in such a way that the functor commutes with filtered colimits.
\[ \operatorname{Fun}( [1], \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad (f: X_{} \rightarrow Y_{} ) \rightarrow Q_{}(f) \]
Using Proposition 4.2.4.8 (and Variant 4.2.4.9), we obtain the following converse of Proposition 4.2.4.5:
Corollary 4.2.4.10. Let $i: A \rightarrow B$ be a morphism of simplicial sets. Then:
The morphism $i$ is left anodyne if and only if, for every square diagram of simplicial sets
where $f$ is left fibration, there exists a dotted arrow rendering the diagram commutative.
The morphism $i$ is right anodyne if and only if, for every square diagram of simplicial sets
where $f$ is right fibration, there exists a dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
Proof. We will prove $(1)$; the proof of $(2)$ is similar. Using Proposition 4.2.4.8, we can factor $i$ as a composition $A \xrightarrow {i'} Q \xrightarrow {f} B$, where $i'$ is left anodyne and $f$ is a left fibration. If the lifting problem
admits a solution, then the map $r$ exhibits $i$ as a retract of $i'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of anodyne morphisms is closed under retracts, it follows that $i$ is anodyne. This proves the “if” direction of $(1)$; the reverse implication follows from Proposition 4.2.4.5. $\square$