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4.3.4 Relative Homotopy Equivalences

Recall that a morphism of simplicial sets $f: X \rightarrow Y$ is said to be a homotopy equivalence if there exists a morphism $g: Y \rightarrow X$ such that $f \circ g$ and $g \circ f$ are homotopic to the identity morphisms $\operatorname{id}_{Y}$ and $\operatorname{id}_{X}$, respectively (Definition 3.1.5.1). We now introduce a relative version of this notion.

Construction 4.3.4.1. Let $q_{X}: X \rightarrow S$ and $q_{Y}: Y \rightarrow S$ be morphisms of simplicial sets. We let $\operatorname{Fun}_{S}(X,Y)$ denote the fiber product $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}(X,S) } \{ q_ X \} $. More concretely, $\operatorname{Fun}_{S}(X,Y)$ is the simplicial set whose $n$-simplices can be identified with commutative diagrams

\[ \xymatrix { \Delta ^ n \times X \ar [rr] \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S. & } \]

Example 4.3.4.2. Let $q: Y \rightarrow S$ be a morphism of simplicial sets. Then, for each vertex $s \in S$, the simplicial set $\operatorname{Fun}_{S}( \{ s\} , Y)$ can be identified with the fiber $Y_{s} = \{ s\} \times _{S} Y$.

Remark 4.3.4.3. Let $S$ be a simplicial set, and let $(\operatorname{Set_{\Delta }})_{/S}$ denote the slice category of simplicial sets $X$ equipped with a morphism $q_{X}: X \rightarrow S$. Then we can regard $(\operatorname{Set_{\Delta }})_{/S}$ as a simplicially enriched category, with mapping simplicial sets given by

\[ \underline{\operatorname{Hom}}_{ ( \operatorname{Set_{\Delta }})_{/S} }( X, Y) = \operatorname{Fun}_{S}(X,Y). \]

Remark 4.3.4.4. Let $q: Y \rightarrow S$ be a morphism of simplicial sets. Then the construction $X \mapsto \operatorname{Fun}_{S}(X,Y)$ carries colimits in the slice category $( \operatorname{Set_{\Delta }})_{/S}$ to limits in the category of simplicial sets. In particular, $\operatorname{Fun}_{S}( \emptyset , Y)$ can be identified with the $0$-simplex $\Delta ^{0}$.

Definition 4.3.4.5. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S. & } \]

We say that $f$ is a homotopy equivalence relative to $S$ if induces an isomorphism in the homotopy category of the simplicial category $(\operatorname{Set_{\Delta }})_{/S}$. More concretely, $f$ is a homotopy equivalence relative to $S$ if there exists a morphism $g \in \operatorname{Fun}_{S}(Y,X)$ such that $g \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of $\operatorname{Fun}_{S}(X,X)$, and $f \circ g$ and $\operatorname{id}_{Y}$ belong to the same connected component of $\operatorname{Fun}_{S}(Y,Y)$. In this case, we will refer to $g$ as a homotopy inverse of $f$ relative to $S$.

Remark 4.3.4.6. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S. & } \]

The following conditions are equivalent:

  • The morphism $f$ is a homotopy equivalence relative to $S$ (Definition 4.3.4.5).

  • For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{S}(B,X) \rightarrow \operatorname{Fun}_{S}(B,Y)$.

  • For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, postcomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{S}(B,X) ) \rightarrow \pi _0( \operatorname{Fun}_{S}(B,Y) )$.

  • For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a homotopy equivalence of simplicial sets $\operatorname{Fun}_{S}(Y,Z) \rightarrow \operatorname{Fun}_{S}(X,Z)$.

  • For every object $Z$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}_{S}(Y,Z) )\rightarrow \pi _0( \operatorname{Fun}_{S}(X,Z) )$.

In the special case where $q_{X}$ and $q_{Y}$ are both left fibrations (or both right fibrations), then we can improve upon Remark 4.3.4.6:

Theorem 4.3.4.7. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S. & } \]

Assume either that $q_{X}$ and $q_{Y}$ are both left fibrations, or that they are both right fibrations. Then the following conditions are equivalent:

$(1)$

The morphism $f$ is a homotopy equivalence relative to $S$ (Definition 4.3.4.5).

$(2)$

For every object $B$ of the slice category $(\operatorname{Set_{\Delta }})_{/S}$, the induced map $f_{B}: \operatorname{Fun}_{S}(B,X) \rightarrow \operatorname{Fun}_{S}(B,Y)$ is a homotopy equivalence.

$(3)$

For each vertex $s \in S$, the induced map $f_{s}: X_{s} \rightarrow Y_{s}$ is a homotopy equivalence of Kan complexes.

The proof of Theorem 4.3.4.7 will require some preliminaries.

Proposition 4.3.4.8. Let $q_{X}: X \rightarrow S$ and $q_{Y}: Y \rightarrow S$ be morphisms of simplicial sets. If $q_{Y}$ is either a left fibration or a right fibration, then the simplicial set $\operatorname{Fun}_{S}(X,Y)$ is a Kan complex.

Proof. Without loss of generality, we may assume that $q_{Y}$ is a left fibration. Then postcomposition with $q_{Y}$ induces a left fibration $Q_{Y}: \operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}(X,S)$ (Corollary 4.1.2.2). By construction, $\operatorname{Fun}_{S}(X,Y)$ is a fiber of the morphism $Q_{Y}$, and is therefore a Kan complex by virtue of Corollary 4.3.1.2. $\square$

Proposition 4.3.4.9. Let $q_{X}: X \rightarrow S$ and $q_{Y}: Y \rightarrow S$ be morphisms of simplicial sets, and let $X' \subseteq X$ be a simplicial subset. If $q_{Y}$ is either a left fibration or a right fibration, then the restriction map $\theta : \operatorname{Fun}_{S}(X, Y) \rightarrow \operatorname{Fun}_{S}(X', Y)$ is a Kan fibration.

Proof. Without loss of generality, we may assume that $q_{Y}$ is a left fibration. By construction, we have a pullback diagram of simplicial sets

\[ \xymatrix { \operatorname{Fun}_{S}(X,Y) \ar [r] \ar [d]^{\theta } & \operatorname{Fun}(X,Y) \ar [d]^{\theta '} \\ \operatorname{Fun}_{S}(X', Y) \ar [r] & \operatorname{Fun}(X',Y) \times _{ \operatorname{Fun}(X',S)} \operatorname{Fun}(X,S). } \]

Proposition 4.1.2.1 guarantees $\theta '$ is a left fibration, so that $\theta $ is also a left fibration (Remark 4.1.0.8). Since $\operatorname{Fun}_{S}(X',Y)$ is a Kan complex (Proposition 4.3.4.8), it follows that $\theta $ is also a Kan fibration (Lemma 4.3.3.4). $\square$

Remark 4.3.4.10. In the situation of Proposition 4.3.4.9, suppose that $q_{Y}$ is a left fibration and the inclusion $X' \hookrightarrow X$ is left anodyne. Then the restriction map $\theta : \operatorname{Fun}_{S}(X, Y) \rightarrow \operatorname{Fun}_{S}(X', Y)$ is a trivial Kan fibration (this follows immediately from Proposition 4.1.2.4). Similarly, if $q_{Y}$ is a right fibration and the inclusion $X' \hookrightarrow X$ is right anodyne, then $\theta $ is a trivial Kan fibration.

Corollary 4.3.4.11. Let $q: Y \rightarrow S$ be a morphism of simplicial sets, and suppose we are given a pushout diagram

\[ \xymatrix { A \ar [r]^-{i} \ar [d] & A' \ar [d] \\ B \ar [r] & B' } \]

in the slice category $(\operatorname{Set_{\Delta }})_{/S}$. If $i$ is a monomorphism and $q$ is either a left or right fibration, then the induced diagram

4.17
\begin{equation} \begin{gathered}\label{equation:mapping-space-homotopy-pullback} \xymatrix { \operatorname{Fun}_{S}(A', Y) & \operatorname{Fun}_{S}(A, Y) \ar [l] \\ \operatorname{Fun}_{S}(B',Y) \ar [u] & \operatorname{Fun}_{S}(B, Y) \ar [u] \ar [l] } \end{gathered} \end{equation}

is a homotopy pullback square of Kan complexes.

Proof. Diagram (4.17) is automatically a pullback square (Remark 4.3.4.4). If $q$ is a right fibration and $i$ is a monomorphism, then the horizontal maps in (4.17) are Kan fibrations (Proposition 4.3.4.9). Invoking Example 3.4.1.5, we deduce that $(\ref{equation:mapping-space-homotopy-pullback})$ is also a homotopy pullback square. $\square$

Proof of Theorem 4.3.4.7. The equivalence $(1) \Leftrightarrow (2)$ follows from Remark 4.3.4.6, and the implication $(2) \Rightarrow (3)$ is immediate (Example 4.3.4.2). We will complete the proof by showing that $(3) \Rightarrow (2)$. Assume that we are given a commutative diagram of simplicial sets

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

where $q_{X}$ and $q_{Y}$ are left fibrations and, for every vertex $s \in S$, the induced map of fibers $f_{s}: X_{s} \rightarrow Y_{s}$ is a homotopy equivalence of Kan complexes. We wish to show that, for each object $B \in (\operatorname{Set_{\Delta }})_{/S}$, the induced map $f_{B}: \operatorname{Fun}_{S}(B,X) \rightarrow \operatorname{Fun}_{S}(B,Y)$ is a homotopy equivalence. For each $n \geq 0$, let $\operatorname{sk}_{n}(B)$ denote the $n$-skeleton of $B$ (Construction 1.1.3.5). Then $f_{A}$ can be realized as the inverse limit of a tower of morphisms $\{ f_{ \operatorname{sk}_ n(B) }: \operatorname{Fun}_{S}( \operatorname{sk}_ n(B), X) \rightarrow \operatorname{Fun}_{S}( \operatorname{sk}_ n(B), Y) \} _{n \geq 0}$. Using Proposition 4.3.4.9, we see that the transition morphisms $\operatorname{Fun}_{S}( \operatorname{sk}_{n+1}(B), X) \rightarrow \operatorname{Fun}_{S}( \operatorname{sk}_ n(B), X)$ and $\operatorname{Fun}_{S}( \operatorname{sk}_{n+1}(B), Y) \rightarrow \operatorname{Fun}_{S}( \operatorname{sk}_ n(B), Y)$ are Kan fibrations. Consequently, to show that $f_{B}$ is a homotopy equivalence, it will suffice to show that each of the morphisms $f_{ \operatorname{sk}_{n}(B) }$ is a homotopy equivalence (Proposition 3.3.9.1). We may therefore replace $B$ by $\operatorname{sk}_{n}(B)$ and thereby reduce to the case where the simplicial set $B$ has dimension $\leq n$, for some $n \geq -1$.

The proof now proceeds by induction on $n$. In the case $n=-1$, the simplicial set $B$ is empty, and the morphism $f_{B}$ is an isomorphism (see Remark 4.3.4.4). Assume that $n \geq 0$, let $B'$ denote the $(n-1)$-skeleton of $B$, and let $T$ denote the collection of all nondegenerate $n$-simplices of $B$. Then Proposition 1.1.3.13 supplies a pushout diagram of simplicial sets

\[ \xymatrix { A' \ar [r] \ar [d] & A \ar [d] \\ B' \ar [r] & B, } \]

where $A = \coprod _{ \tau \in T} \Delta ^{n}$ and $A' = \coprod _{\tau \in T} \operatorname{\partial \Delta }^ n$. This pushout square determines a commutative diagram of Kan complexes

\[ \xymatrix { \operatorname{Fun}_{S}(B, X) \ar [dd]^(.5){ f_{B} } \ar [rr] \ar [dr] & & \operatorname{Fun}_{S}( B', X) \ar [dd]^(.6){ f_{B'} } \ar [dr] & \\ & \operatorname{Fun}_{S}(A, X) \ar [rr] \ar [dd]^(.6){ f_{A} } & & \operatorname{Fun}_{S}(A', X) \ar [dd]^(.5){ f_{A'} } \\ \operatorname{Fun}_{S}(B, Y) \ar [rr] \ar [dr] & & \operatorname{Fun}_{S}(B', Y) \ar [dr] & \\ & \operatorname{Fun}_{S}(A, Y) \ar [rr] & & \operatorname{Fun}_{S}(A', Y), } \]

where the upper and lower squares are homotopy Cartesian by virtue of Corollary 4.3.4.11. Consequently, to show that $f_{B}$ is a homotopy equivalence of Kan complexes, it will suffice to show that $f_{B'}$, $f_{A'}$, and $f_{A}$ are homotopy equivalences (Corollary 3.4.1.10). In the first two cases, this follows from our inductive hypothesis. We may therefore replace $B$ by $A$, and thereby reduce to the case where is a coproduct of simplices. Using Remarks 4.3.4.4 and 3.1.5.8, we can further reduce to the case where $B = \Delta ^ n$ is a simplex of dimension $n$. In this case, the inclusion of the initial vertex $0 \in \Delta ^ n$ is left anodyne (Example 4.2.7.10). Let $s \in S$ denote the image of the vertex $0 \in \Delta ^ n$, so that we have a commutative diagram of Kan complexes

\[ \xymatrix { \operatorname{Fun}_{S}( B, X) \ar [r]^-{ f_{B} } \ar [d] & \operatorname{Fun}_{S}(B, Y) \ar [d] \\ \operatorname{Fun}_{S}( \{ s\} , X) \ar [r]^-{f_ s} & \operatorname{Fun}_{S}( \{ s\} , Y). } \]

Invoking Remark 4.3.4.10, we conclude that the vertical maps in this diagram are trivial Kan fibrations. Since $f_{s}$ is a homotopy equivalence by assumption, it follows that $f_{B}$ is also a homotopy equivalence. $\square$