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3.1.5 Homotopy Equivalences and Weak Homotopy Equivalences

Let $f: X_{} \rightarrow Y_{}$ be a morphism of Kan complexes. We will say that $f$ is a homotopy equivalence if the homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.4.7. This definition can be extended to more general simplicial sets in multiple ways.

Definition 3.1.5.1. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We will say that a morphism $g: Y_{} \rightarrow X_{}$ is a homotopy inverse to $f$ if the compositions $g \circ f$ and $f \circ g$ are homotopic to the identity morphisms $\operatorname{id}_{X_{}}$ and $\operatorname{id}_{ Y_{} }$, respectively (in the sense of Definition 3.1.4.2). We say that $f: X_{} \rightarrow Y_{}$ is a homotopy equivalence if it admits a homotopy inverse $g$.

Example 3.1.5.2. Let $f: X \rightarrow Y$ be a homotopy equivalence of topological spaces. Then the induced map of singular simplicial sets $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(Y)$ is a homotopy equivalence (see Example 3.1.4.5).

Remark 3.1.5.3. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. The condition that $f$ is a homotopy equivalence depends only on the homotopy class $[f] \in \pi _0( \operatorname{Fun}(X_{}, Y_{} ) )$. Moreover, if $f$ is a homotopy equivalence, then its homotopy inverse $g: Y_{} \rightarrow X_{}$ is determined uniquely up to homotopy.

Remark 3.1.5.4. Let $f: X_{} \rightarrow Y_{}$ be a morphism of Kan complexes. If $f$ is a homotopy equivalence, then the induced map of fundamental groupoids $\pi _{\leq 1}(f): \pi _{\leq 1}(X) \rightarrow \pi _{\leq 1}(Y)$ is an equivalence of categories. In particular, $f$ induces a bijection $\pi _0(f): \pi _0( X_{} ) \rightarrow \pi _0( Y_{} )$.

Remark 3.1.5.5. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

  • The morphism $f$ is a homotopy equivalence.

  • For every simplicial set $Z_{}$, composition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y_{}, Z_{})) \rightarrow \pi _0( \operatorname{Fun}( X_{}, Z_{}) ).$.

  • For every simplicial set $W_{}$, composition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(W_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( W_{}, Y_{} ))$.

In particular (taking $W_{} = \Delta ^{0}$), if $f$ is a homotopy equivalence, then the induced map $\pi _0(f): \pi _0( X_{} ) \rightarrow \pi _0( Y_{} )$ is a bijection.

Remark 3.1.5.6 (Two-out-of-Six). Let $f: W_{} \rightarrow X_{}$, $g: X_{} \rightarrow Y_{}$, and $h: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If $g \circ f$ and $h \circ g$ are homotopy equivalences, then $f$, $g$, and $h$ are all homotopy equivalences.

Remark 3.1.5.7 (Two-out-of-Three). Let $f: X_{} \rightarrow Y_{}$ and $g: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If any two of the morphisms $f$, $g$, and $g \circ f$ are homotopy equivalences, then so is the third.

We now give some more examples of homotopy equivalences.

Proposition 3.1.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories, and suppose that $F$ admits either a left or a right adjoint. Then the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a homotopy equivalence of simplicial sets.

Proof. Without loss of generality, we may assume that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then there exist natural transformations $u: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $v: F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ witnessing an adjunction between $F$ and $G$, which we can identify with functors $U: [1] \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ and $V: [1] \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$. Then the composite maps

\[ \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( [1] \times \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{N}_{\bullet }(U)} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]

\[ \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{N}_{\bullet }( [1] \times \operatorname{\mathcal{D}}) \xrightarrow { \operatorname{N}_{\bullet }(V)} \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

are homotopies from $\operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ to $\operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F)$ and from $\operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G)$ to $\operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})}$, respectively. It follows that $\operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a homotopy inverse to $\operatorname{N}_{\bullet }(F)$, so that $\operatorname{N}_{\bullet }(F)$ is a homotopy equivalence. $\square$

Proposition 3.1.5.9. Let $f: X_{} \rightarrow S_{}$ be a trivial Kan fibration of simplicial sets. Then $f$ is a homotopy equivalence.

Proof. Since $f$ is a trivial Kan fibration, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \emptyset \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ S_{} \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & S_{} } \]

admits a solution (Proposition 1.4.5.3). We can therefore choose a morphism of simplicial sets $g: S_{} \rightarrow X_{}$ which is a section of $f$: that is, $f \circ g$ is the identity morphism from $S_{}$ to itself. We will complete the proof by showing that $g$ is a homotopy inverse to $f$. In fact, we claim that there exists a homotopy $h$ from $\operatorname{id}_{X_{}}$ to the composition $g \circ f$. This follows from the solubility of the lifting problem

\[ \xymatrix@C =100pt{ \{ 0,1\} \times X_{} \ar [r]^-{(\operatorname{id}, g \circ f)} \ar [d] & X_{} \ar [d]^{f} \\ X_{} \ar [r]^-{f} \ar@ {-->}[ur]^-{h} & S_{}. } \]
$\square$

When working with simplicial sets which are not Kan complexes, it is usually better to work with a more liberal notion of homotopy equivalence.

Definition 3.1.5.10. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We will say that $f$ is a weak homotopy equivalence if, for every Kan complex $Z_{}$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y_{}, Z_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( X_{}, Z_{} ) )$.

Proposition 3.1.5.11. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. If $f$ is a homotopy equivalence, then it is a weak homotopy equivalence. The converse holds if $X_{}$ and $Y_{}$ are Kan complexes.

Proof. The first assertion follows from Remark 3.1.5.5. For the second, assume that $f$ is a weak homotopy equivalence. If $X_{}$ is a Kan complex, then precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( X_{}, X_{} ) )$. We can therefore choose a map of simplicial sets $g: Y_{} \rightarrow X_{}$ such that $g \circ f$ is homotopic to the identity on $X_{}$ It follows that $f \circ g \circ f$ is homotopic to $f = \operatorname{id}_{Y_{}} \circ f$. Invoking the injectivity of the map $\pi _0( \operatorname{Fun}(Y_{}, Y_{} ) ) \xrightarrow {\circ f} \pi _0( \operatorname{Fun}( X_{}, Y_{} ) )$, we conclude that $f \circ g$ is homotopic to $\operatorname{id}_{ Y_{} }$, so that $g$ is a homotopy inverse to $f$. $\square$

Remark 3.1.5.12 (Two-out-of-Six). Let $f: W_{} \rightarrow X_{}$, $g: X_{} \rightarrow Y_{}$, and $h: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If $g \circ f$ and $h \circ g$ are weak homotopy equivalences, then $f$, $g$, and $h$ are all weak homotopy equivalences.

Remark 3.1.5.13 (Two-out-of-Three). Let $f: X_{} \rightarrow Y_{}$ and $g: Y_{} \rightarrow Z_{}$ be morphisms of simplicial sets. If any two of the morphisms $f$, $g$, and $g \circ f$ are weak homotopy equivalences, then so is the third.

Remark 3.1.5.14 (Coproducts of Weak Homotopy Equivalences). Let $\{ f(i): X(i) \rightarrow Y(i) \} _{i \in I}$ be a collection of weak homotopy equivalences of simplicial sets indexed by a set $I$. For every Kan complex $Z$, we have a commutative diagram of Kan complexes

\[ \xymatrix { \operatorname{Fun}( \coprod _{i \in I} Y(i), Z) \ar [r] \ar [d]^{\sim } & \operatorname{Fun}( \coprod _{i \in I} X(i), Z) \ar [d]^{\sim } \\ \prod _{i \in I} \operatorname{Fun}( Y(i), Z) \ar [r] & \prod _{i \in I} \operatorname{Fun}(X(i), Z), } \]

where the vertical maps are isomorphisms. Passing to the connected components (and using the fact that the functor $Q \mapsto \pi _0(Q)$ preserves products when restricted to Kan complexes; see Corollary 1.1.9.11), we deduce that the map $\pi _0( \operatorname{Fun}( \coprod _{i \in I} Y(i), Z) ) \rightarrow \pi _0( \operatorname{Fun}(\coprod _{i \in I} X(i), Z) )$ is bijective. Allowing $Z$ to vary, we conclude that the induced map $\coprod _{i \in I} X(i) \rightarrow \coprod _{i \in I} Y(i)$ is also a weak homotopy equivalence.

Exercise 3.1.5.15. Let $G$ be the the directed graph depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & 3 \ar [r] & 4 \ar [r] & \cdots } \]

and let $G_{}$ denote the associated $1$-dimensional simplicial set (see Warning 1.1.6.27). Show that the projection map $G_{} \rightarrow \Delta ^{0}$ is a weak homotopy equivalence, but not a homotopy equivalence.

Warning 3.1.5.16. Let $X_{}$ and $Y_{}$ be simplicial sets. The existence of a weak homotopy equivalence $f: X_{} \rightarrow Y_{}$ does not guarantee the existence of a weak homotopy equivalence $g: Y_{} \rightarrow X_{}$.