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3.1.6 Anodyne Morphisms

We now introduce another large class of examples of weak homotopy equivalences between simplicial sets.

Definition 3.1.6.1 (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 3.1.6.3. 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 3.1.6.4. 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 3.1.6.5. 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 3.1.6.6. 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).

Our next goal is to prove the following:

Proposition 3.1.6.7. Let $f: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets. Then $f$ is a weak homotopy equivalence.

Remark 3.1.6.8. We will later prove a (partial) convere to Proposition 3.1.6.7: if a monomorphism of simplicial sets $f: A_{} \hookrightarrow B_{}$ is a weak homotopy equivalence, then $f$ is anodyne (see Corollary ).

The proof of Proposition 3.1.6.7 will require some preliminaries.

Lemma 3.1.6.9. 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.

Proof. 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. $\square$

Proposition 3.1.6.10. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets and let $f: X_{} \rightarrow S_{}$ be a Kan fibration. Then the induced map

\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

is a trivial Kan fibration.

Proof. Let $i': A'_{} \hookrightarrow B'_{}$ be a monomorphism of simplicial sets; we must show that every lifting problem

\[ \xymatrix@C =100pt{ A'_{} \ar [d]^{i'} \ar [r] & \operatorname{Fun}( B_{}, X_{} ) \ar [d] \\ B'_{} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) } \]

admits a solution. Equivalently, we must show that every lifting problem

\[ \xymatrix@C =100pt{ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ B_{} \times B'_{} \ar [r] \ar@ {-->}[ur] & S_{} } \]

admits a solution. This follows from Remark 3.1.6.6, since the left vertical map is anodyne (Lemma 3.1.6.9) and the right vertical map is a Kan fibration. $\square$

Remark 3.1.6.11. By swapping the roles of the monomorphisms $i: A_{} \hookrightarrow B_{}$ and $i': A'_{} \hookrightarrow B'_{}$ in the proof of Proposition 3.1.6.10, we obtain a proof of Theorem 3.1.3.1 (which is essentially the same as the proof given in ยง3.1.3).

Taking $S_{} = \Delta ^0$ in the statement of Proposition 3.1.6.10, we obtain the following:

Corollary 3.1.6.12. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets and let $X_{}$ be a Kan complex. Then the restriction map $\operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( A_{}, X_{} )$ is a trivial Kan fibration.

Proof of Proposition 3.1.6.7. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets; we wish to show that $i$ is a weak homotopy equivalence. Let $X_{}$ be any Kan complex. It follows from Corollary 3.1.6.12 that the restriction map $\theta : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}(A_{}, X_{} )$ is a trivial Kan fibration. In particular, $\theta $ is a homotopy equivalence (Proposition 3.1.5.9), and therefore induced a bijection on connected components $\pi _0( \operatorname{Fun}( B_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( A_{}, X_{} ) )$ (Remark 3.1.5.5). $\square$