Kerodon

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3.1.2 Anodyne Morphisms

By definition, a morphism of simplicial sets $f: X \rightarrow S$ is a Kan fibration if it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 \leq i \leq n$ and $n > 0$. If this condition is satisfied, then $f$ automatically has the right lifting property with respect to a much larger class of morphisms.

Definition 3.1.2.1 (Anodyne Morphisms). Let $T$ be the smallest collection of morphisms in the category $\operatorname{Set_{\Delta }}$ with the following properties:

  • For each $n > 0$ and each $0 \leq i \leq n$, the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T$.

  • The collection $T$ is weakly saturated (Definition 1.4.4.15). That is, $T$ is closed under pushouts, retracts, and transfinite composition.

We say that a morphism of simplicial sets $i: A_{} \rightarrow B_{}$ is anodyne if it belongs to the collection $T$.

Remark 3.1.2.3. Every anodyne morphism of simplicial sets $i: A_{} \rightarrow B_{}$ is a monomorphism. This follows from the observation that the collection of monomorphisms is weakly saturated (Proposition 1.4.5.12) and that every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a monomorphism.

Example 3.1.2.4. Let $i: A_{} \hookrightarrow B_{}$ be an inner anodyne morphism of simplicial sets (Definition 1.4.6.4). Then $i$ is anodyne. The converse is false in general. For example, the horn inclusions $\Lambda ^{n}_0 \hookrightarrow \Delta ^ n$ and $\Lambda ^{n}_{n} \hookrightarrow \Delta ^ n$ are anodyne (for $n > 0$), but are not inner anodyne.

Remark 3.1.2.5. By construction, the collection of anodyne morphisms is weakly saturated. In particular:

  • Every isomorphism of simplicial sets is anodyne.

  • If $i: A_{} \rightarrow B_{}$ and $j: B_{} \rightarrow C_{}$ are anodyne morphisms of simplicial sets, then the composition $g \circ f$ is anodyne.

  • For every pushout diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]_{i} \ar [r] & A'_{} \ar [d]^{i'} \\ B_{} \ar [r] & B'_{}, } \]

    if $i$ is anodyne, then $i'$ is also anodyne.

  • Suppose there exists a commutative diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & A'_{} \ar [d]^{i'} \ar [r] & A_{} \ar [d]^{i} \\ B_{} \ar [r] & B'_{} \ar [r] & B_{}, } \]

    where the horizontal compositions are the identity. If $i'$ is anodyne, then $i$ is anodyne.

Remark 3.1.2.6. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(a)$

The morphism $f$ is a Kan fibration (Definition 3.1.1.1).

$(b)$

For every square diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A_{} \ar [d]^{i} \ar [r] & X_{} \ar [d]^{f} \\ B_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

where $i$ is anodyne, there exists a dotted arrow rendering the diagram commutative.

The implication $(b) \Rightarrow (a)$ is immediate from the definitions (since the horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ are anodyne for $n > 0$). The reverse implication follows from the fact that the collection of those morphisms of simplicial sets $i: A_{} \rightarrow B_{}$ which have the left lifting property with respect to $f$ is weakly saturated (Proposition 1.4.4.16).

We will need the following stability properties for the class of anodyne morphisms:

Proposition 3.1.2.7. Let $f: A_{} \hookrightarrow B_{}$ and $f': A'_{} \hookrightarrow B'_{}$ be monomorphisms of simplicial sets. If either $f$ or $f'$ is anodyne, then the induced map

\[ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \hookrightarrow B_{} \times B'_{} \]

is anodyne.

The proof of Proposition 3.1.2.7 will require some preliminaries.

Lemma 3.1.2.8. For every pair of integers $0 < i \leq n$, the horn inclusion $f_0: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a retract of the inclusion map $f: (\Delta ^1 \times \Lambda ^ n_ i) \coprod _{ \{ 1\} \times \Lambda ^{n}_ i} ( \{ 1\} \times \Delta ^ n) \hookrightarrow \Delta ^1 \times \Delta ^ n$.

Proof. Let $A_{}$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\Delta ^1 \times \Lambda ^ n_ i$ with $\{ 1 \} \times \Delta ^ n$. To prove Lemma 3.1.2.8, it will suffice to show that there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \Lambda ^{n}_{i} \ar [r] \ar [d]^{f_0} & A_{} \ar [r] \ar [d]^{f} & \Lambda ^{n}_{i} \ar [d]^{f_0} \\ \{ 0\} \times \Delta ^ n \ar [r] & \Delta ^1 \times \Delta ^ n \ar [r]^-{r} & \Delta ^ n } \]

where the left horizontal maps are given by inclusion and the horizontal compositions are the identity maps. To achieve this, it suffices to choose $r$ to be given on vertices by the map of partially ordered sets

\[ r: [1] \times [n] \rightarrow [n] \quad \quad r(j,k) = \begin{cases} i & \text{ if $j=1$ and $k \leq i$ } \\ k & \text{ otherwise. } \end{cases} \]
$\square$

Lemma 3.1.2.9. Let $n$ be a nonnegative integer. Then there exists a chain of simplicial subsets

\[ X(0) \subset X(1) \subset \cdots \subset X(n) \subset X(n+1) = \Delta ^1 \times \Delta ^ n \]

with the following properties:

$(a)$

The simplicial $X(0)$ is given by the union of $\Delta ^1 \times \operatorname{\partial \Delta }^ n$ with $\{ 1\} \times \Delta ^ n$ (and can therefore be described abstractly as the pushout $(\Delta ^1 \times \operatorname{\partial \Delta }^ n) \coprod _{ \{ 1\} \times \operatorname{\partial \Delta }^ n} ( \{ 1 \} \times \Delta ^ n )$).

$(b)$

For $0 \leq i \leq n$, the inclusion map $X(i) \hookrightarrow X(i+1)$ fits into a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i+1} \ar [r] \ar [d] & X(i) \ar [d] \\ \Delta ^{n+1} \ar [r] & X(i+1). } \]

Proof. For $0 \leq i \leq n$, let $\sigma _ i: \Delta ^{n+1} \rightarrow \Delta ^1 \times \Delta ^ n$ denote the map of simplicial sets given on vertices by the formula $\sigma _ i(j) = \begin{cases} (0,j) & \text{ if } j \leq i \\ (1,j-1) & \text{ if } j > i. \end{cases}$ We define simplicial subsets $X(i) \subseteq \Delta ^1 \times \Delta ^ n$ inductively by the formulae

\[ X(0) = (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \cup (\{ 1\} \times \Delta ^ n) \quad \quad X(i+1) = X(i) \cup \operatorname{im}( \sigma _{i} ), \]

where $\operatorname{im}( \sigma _{i} )$ denotes the image of the morphism $\sigma _{i}$. Note that $\Delta ^1 \times \Delta ^ n$ is the union of the simplicial subsets $\{ \operatorname{im}(\sigma _ i) \} _{0 \leq i \leq n}$, and is therefore equal to $X(n+1)$. This definition satisfies condition $(a)$ by construction. To verify $(b)$, it will suffice to show that for $0 \leq i \leq n$, the inverse image $A_{} = \sigma _{i}^{-1} X(i)$ is equal to $\Lambda ^{n+1}_{i+1}$ (as a simplicial subset of $\Delta ^{n+1}$). Regarding $\sigma _{i}$ as an $(n+1)$-simplex of $\Delta ^1 \times \Delta ^ n$, we are reduced to showing that the faces $d_{j}( \sigma _{i} )$ belong to $X(i)$ if and only if $j \neq i+1$. One direction is clear: the face $d_{j}( \sigma _{i} )$ is contained in $\Delta ^1 \times \operatorname{\partial \Delta }^ n$ for $j \notin \{ i, i+1 \} $, the face $d_{i}(\sigma _ i) = d_{i}( \sigma _{i-1} )$ is contained in $\operatorname{im}( \sigma _{i-1} ) \subseteq X(i)$ for $i > 0$, and $d_0( \sigma _0)$ is contained in $\{ 1\} \times \Delta ^ n$. To complete the proof, it suffices to show that the face $d_{i+1}( \sigma _ i )$ is not contained in $X(i)$, which follows by inspection. $\square$

Proof of Proposition 3.1.2.7. Let us first regard the monomorphism $f': A'_{} \hookrightarrow B'_{}$ as fixed, and let $T$ be the collection of all maps $f: A_{} \rightarrow B_{}$ for which the induced map

\[ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \hookrightarrow B_{} \times B'_{} \]

is anodyne. We wish to show that every anodyne morphism belongs to $T$. Since $T$ is weakly saturated, it will suffice to show that every horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $T$ (for $n > 0$). Without loss of generality, we may assume that $0 < i$, so that $f$ is a retract of the map $g: (\Delta ^1 \times \Lambda ^ n_ i) \coprod _{ \{ 1\} \times \Lambda ^{n}_ i} ( \{ 1\} \times \Delta ^ n) \hookrightarrow \Delta ^1 \times \Delta ^ n$ (Lemma 3.1.2.8). It will therefore suffice to show that $g$ belongs to $T$. Replacing $f'$ by the monomorphism $(\Lambda ^{n}_{i} \times B'_{} ) \coprod _{ \Lambda ^{n}_{i} \times A'_{} } (\Delta ^ n \times A'_{} )$, we are reduced to showing that the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ belongs to $T$.

Let $T'$ denote the collection of all morphisms of simplicial sets $f'': A''_{} \rightarrow B''_{}$ for which the map $(\{ 1\} \times B''_{} ) \coprod _{ \{ 1\} \times A''_{} } ( \Delta ^1 \times A''_{} ) \rightarrow \Delta ^1 \times B''_{}$ is anodyne. We will complete the proof by showing that $T'$ contains all monomorphisms of simplicial sets. By virtue of Proposition 1.4.5.12, it will suffice to show that $T''$ contains the inclusion map $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^{m}$, for each $m > 0$. In other words, we are reduced to showing that the inclusion $(\{ 1\} \times \Delta ^ m ) \coprod _{ \{ 1\} \times \operatorname{\partial \Delta }^ m } ( \Delta ^1 \times \operatorname{\partial \Delta }^ m) \hookrightarrow \Delta ^1 \times \Delta ^ m$ is anodyne, which follows from Lemma 3.1.2.9. $\square$