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 4.2.4.6).
\[ \xymatrix@C =30pt@R=30pt{ \{ \textnormal{Inner anodyne morphisms} \} \ar@ {}[d]|{\cap } \ar@ {}[r]|{\subset } & \{ \textnormal{Left anodyne morphisms} \} \ar@ {}[d]|{\cap } \\ \{ \textnormal{Right anodyne morphisms} \} \ar@ {}[r]|{\subset } & \{ \textnormal{Anodyne morphisms} \} . } \]
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. $\square$
Example 4.2.4.6. 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 4.2.4.7. 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 4.2.4.8. 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 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 4.2.4.7 (and Variant 4.2.4.8), we obtain the following converse of Proposition 4.2.4.5:
Corollary 4.2.4.9. 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 4.2.4.7, 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. $\square$
For later use, we record the following:
Proposition 4.2.4.10. For every pair of integers $0 \leq j < n$, the horn $\Lambda ^{n}_{j}$ admits a finite filtration where each of the inclusion maps $X(a-1) \hookrightarrow X(a)$ can be realized as a pushout of a horn inclusion $\Lambda ^{m}_{i} \hookrightarrow \Delta ^ m$ for $0 \leq i < m < n$. In particular, the inclusion $\{ 0\} \hookrightarrow \Lambda ^{n}_{j}$ is left anodyne.
\[ \{ 0\} = X(0) \subset X(1) \subset \cdots \subset X(k) = \Lambda ^{n}_{j}, \]
Proof. Let us say that a monomorphism of simplicial sets $A \hookrightarrow B$ is good if it can be written as a composition of finitely many morphisms, each of which is a pushout of a horn inclusion $\Lambda ^{m}_{i} \hookrightarrow \Delta ^ m$ for $0 \leq i < m < n$. We wish to show that the inclusion map $\{ 0\} \hookrightarrow \Lambda ^{n}_{j}$ is good. Our proof proceeds by induction on $n$. It follows from our inductive hypothesis that the inclusion map $\{ 0\} \hookrightarrow \Lambda ^{j}_{0}$ is good; in particular, the inclusion map $\{ 0\} \hookrightarrow \Delta ^{j}$ is good. For every integer $d \geq 0$, let $Y(d) \subseteq \Delta ^ n$ be the simplicial subset whose nondegenerate simplices have vertex set $J \subseteq [n]$ satisfying one of the following conditions:
The set $J$ is contained in $[j] = \{ 0 < 1 < \cdots < j \} $.
The set $J \setminus \{ j\} $ has cardinality $< d$.
We have inclusion maps
It will therefore suffice to show that for $0 < d < n$, the inclusion map $Y(d-1) \hookrightarrow Y(d)$ is good. Let $S$ be the collection of all nondegenerate $d$-simplices of $Y(d)$. By construction, for each $\sigma \in S$, there is a unique integer $0 \leq i_{\sigma } < d$ satisfying $\sigma ( i_{\sigma } ) = j$. Unwinding the definitions, we see that there is a pushout square
so that the inclusion $Y(d-1) \hookrightarrow Y(d)$ can be written as a finite pushout of morphisms of the form $\Lambda ^{d}_{i} \hookrightarrow \Delta ^ d$ for $i < d$. $\square$
Corollary 4.2.4.11. For every integer $n \geq 0$, the inclusion map $\iota : \{ 0\} \hookrightarrow \Delta ^ n$ is left anodyne.
For a more general statement, see Example 4.3.7.11.
Proof of Corollary 4.2.4.11. We may assume $n > 0$ (otherwise the result is trivial). Choose an integer $0 \leq i < n$, so that the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ is left anodyne. It will therefore suffice to show that the inclusion $\{ 0\} \hookrightarrow \Lambda ^{n}_{i}$ is left anodyne, which follows from Proposition 4.2.4.10. $\square$
Corollary 4.2.4.12. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $q$ is a left covering map, in the sense of Definition 4.2.3.8.
Every lifting problem
admits a unique solution, provided that the morphism $i$ is inner anodyne.
For every integer $n \geq 0$, the diagram of sets
is a pullback square.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , X ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \{ 0\} , S) } \]
Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.2.4.5 and Remark 4.2.3.11, and the implication $(2) \Rightarrow (3)$ follows from Corollary 4.2.4.11. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied; we wish to show that, for $0 \leq j < n$, the left half of the diagram
is a pullback square. We proceed by induction on $n$. Assumption $(3)$ guarantees that the outer rectangle is a pullback, so we are reduced to showing that the square on the right is a pullback. This follows by combining our inductive hypothesis with Proposition 4.2.4.10. $\square$