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3.3.7 Application: Characterizations of Weak Homotopy Equivalences

Let $f: X \rightarrow S$ be a Kan fibration between Kan complexes. In §3.2.6, we proved that $f$ is a homotopy equivalence if and only if it is a trivial Kan fibration (Corollary 3.2.6.9). We now apply the machinery of §3.3.6 to extend this result to the case where $S$ is an arbitrary simplicial set. First, we need a slight generalization of Proposition 3.2.7.1.

Proposition 3.3.7.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { X \ar [r]^-{u} \ar [d] & X' \ar [d] \\ S \ar [r]^-{v} & S', } \]

where the vertical maps are Kan fibrations and $v$ is a weak homotopy equivalence. The following conditions are equivalent:

$(1)$

The morphism $u$ is a weak homotopy equivalence.

$(2)$

For every vertex $s \in S$, the induced map of fibers $u_{t}: X_{s} \rightarrow X'_{ v(s)}$ is a homotopy equivalence of Kan complexes.

Proof. Using Corollaries 3.3.6.8 and 3.3.6.5, we can replace $(1)$ and $(2)$ by the following assertions:

$(1')$

The morphism $\operatorname{Ex}^{\infty }(u): \operatorname{Ex}^{\infty }(X) \rightarrow \operatorname{Ex}^{\infty }(X')$ is a weak homotopy equivalence.

$(2')$

For every vertex $s \in S$, the induced map of fibers $u_{s}: \operatorname{Ex}^{\infty }(X)_{s} \rightarrow \operatorname{Ex}^{\infty }(X')_{v(s)}$ is a homotopy equivalence of Kan complexes.

The equivalence of $(1')$ and $(2')$ follows by applying Proposition 3.2.7.1 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Ex}^{\infty }(X) \ar [r]^-{\operatorname{Ex}^{\infty }(u) } \ar [d] & \operatorname{Ex}^{\infty }(X') \ar [d] \\ \operatorname{Ex}^{\infty }(S) \ar [r]^-{\operatorname{Ex}^{\infty }(v)} & \operatorname{Ex}^{\infty }(S'); } \]

note that every simplicial set appearing in this diagram is a Kan complex (Proposition 3.3.6.9), the vertical maps are Kan fibrations (Proposition 3.3.6.6) and $\operatorname{Ex}^{\infty }(v)$ is a homotopy equivalence by virtue of Corollary 3.3.6.8. $\square$

Corollary 3.3.7.2. Let $v: T \rightarrow S$ be a weak homotopy equivalence of simplicial sets. For every Kan fibration $f: X \rightarrow S$, the projection map $T \times _{S} X \rightarrow X$ is also a weak homotopy equivalence.

Corollary 3.3.7.3. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { Y \ar [rr]^{u} \ar [dr] & & X \ar [dl] \\ & S, & } \]

where the vertical maps are Kan fibrations. Then $u$ is a weak homotopy equivalence if and only if, for each vertex $s \in S$, the induced map $u_{s}: Y_{s} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes.

Proposition 3.3.7.4. Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. The following conditions are equivalent:

$(1)$

For every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a contractible Kan complex.

$(2)$

The morphism $f$ is a trivial Kan fibration.

$(3)$

The morphism $f$ is a homotopy equivalence.

$(4)$

The morphism $f$ is a weak homotopy equivalence.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 3.2.6.8 and the equivalence $(1) \Leftrightarrow (4)$ follows from Corollary 3.3.7.3. The implication $(2) \Rightarrow (3)$ is a special case of Corollary 3.3.7.3, and the implication $(3) \Rightarrow (4)$ follows from from Proposition 3.1.5.11, $\square$

Corollary 3.3.7.5. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is anodyne.

$(2)$

The morphism $f$ is both a monomorphism and a weak homotopy equivalence.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 3.1.5.12 and Remark 3.1.2.3. To prove the converse, assume that $f$ is a weak homotopy equivalence and apply Proposition 3.1.6.1 to write $f$ as a composition $X \xrightarrow {f'} Q \xrightarrow {f''} Y$, where $f'$ is anodyne and $f''$ is a Kan fibration. Then $f'$ is a weak homotopy equivalence (Proposition 3.1.5.12), so $f''$ is a weak homotopy equivalence (Remark 3.1.5.15). Invoking Proposition 3.3.7.4, we conclude that $f''$ is a trivial Kan fibration. If $f$ is a monomorphism, then the lifting problem

\[ \xymatrix { X \ar [d]^{f} \ar [r]^-{ f' } & Q \ar [d]^{f''} \\ Y \ar [r]^-{\operatorname{id}_ Y} \ar@ {-->}[ur] & Y } \]

admits a solution. It follows that $f$ is a retract of $f'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of anodyne morphisms is closed under retracts, we conclude that $f$ is anodyne. $\square$