Definition 3.1.6.1. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We will say that a morphism $g: Y_{} \rightarrow X_{}$ is a simplicial homotopy inverse of $f$ 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 3.1.5.1). In the case where $X$ and $Y$ are Kan complexes, we will say that $g$ is a homotopy inverse of $f$ if it is a simplicial homotopy inverse to $f$. We say that $f: X_{} \rightarrow Y_{}$ is a homotopy equivalence if it admits a simplicial homotopy inverse $g$.
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3.1.6 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 3.1.5.10. This definition can be extended to more general simplicial sets in multiple ways.
Warning 3.1.6.2. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Many authors refer to a morphism $g: Y \rightarrow X$ as 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. However, when $X$ and $Y$ are $\infty $-categories, it is natural to consider a different (and more restrictive) notion of homotopy inverse, which requires that $g \circ f$ and $f \circ g$ be isomorphic to $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ as objects of the $\infty $-categories $\operatorname{Fun}(X,X)$ and $\operatorname{Fun}(Y,Y)$, respectively (see Definition 4.5.1.10 and Warning 4.5.1.14). For this reason, we will use the term simplicial homotopy inverse in the setting of Definition 3.1.6.1 (unless $X$ and $Y$ are Kan complexes, in which case the distinction disappears).
Example 3.1.6.3. 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 3.1.5.6).
Remark 3.1.6.4. 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 simplicial homotopy inverse $g: Y_{} \rightarrow X_{}$ is determined uniquely up to homotopy.
Remark 3.1.6.5. 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 3.1.6.6. 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 3.1.6.7 (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.
Remark 3.1.6.8. Let $\{ f_ i: X_ i \rightarrow Y_ i \} _{i \in I}$ be a collection of homotopy equivalences of simplicial sets indexed by a set $I$, and let $f: \prod _{i \in I} X_ i \rightarrow \prod _{i \in I} Y_ i$ be their product. Then:
If $I$ is finite, then $f$ is a homotopy equivalence. This follows from Remark 3.1.6.6 and Corollary 1.2.1.27.
If each of the simplicial sets $X_{i}$ and $Y_{i}$ is a Kan complex, then $f$ is a homotopy equivalence. This follows from Remark 3.1.6.6 and Corollary 1.2.5.11.
The morphism $f$ need not be a homotopy equivalence in general (see Warning 1.2.1.28).
We now give some more examples of homotopy equivalences.
Proposition 3.1.6.9. 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 $G$, so that $\operatorname{N}_{\bullet }(F)$ is a simplicial homotopy inverse of $\operatorname{N}_{\bullet }(G)$ by virtue of Example 3.1.5.7. $\square$
Proposition 3.1.6.10. Let $f: X_{} \rightarrow S_{}$ be a trivial Kan fibration of simplicial sets. Then $f$ is a homotopy equivalence.
Proof. By virtue of Corollary 1.5.5.5, the morphism $f$ admits a section $s: S \rightarrow X$ such that $s \circ f$ is homotopic to the identity map $\operatorname{id}_{X}$. $\square$
Example 3.1.6.11. Let $S$ be a simplicial set and let $\mathrm{N}_{\ast }(S;\operatorname{\mathbf{Z}})$ for the normalized chain complex of $S$ (Construction 2.5.5.9). Let $M_{\ast }$ be a chain complex of abelian groups, let $\mathrm{K}( M_{\ast } )$ denote the associated (generalized) Eilenberg-MacLane space, and let denote the chain complex of maps from $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ to $M_{\ast }$. Then there is a map of Kan complexes which classifies the map of chain complexes where $\mathrm{AW}$ is the Alexander-Whitney map (see Construction 2.5.8.6). The morphism $\lambda $ is a homotopy equivalence of Kan complexes. To prove this, it will suffice to show that for every simplicial set $T$, composition with $\lambda $ induces a bijection Using Example 3.1.5.8 (and the definition of the chain complex $H_{\ast }$), we can identify the source of $\lambda _{T}$ with the set of chain homotopy classes of maps the tensor product $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T;\operatorname{\mathbf{Z}})$ into $M_{\ast }$, and the target of $\lambda _{T}$ with the set of chain homotopy classes of maps from $\mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}})$ into $M_{\ast }$. Under these identifications, we see that $\lambda _{T}$ is induced by precomposition with the Alexander-Whitney map This map is a quasi-isomorphism (Corollary 2.5.8.11), and therefore admit a chain homotopy inverse (since the source and target of $\mathrm{AW}$ are nonnegatively graded complexes of free abelian groups; see Remark
\[ H_{\ast } = \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( \mathrm{N}_{\ast }(S, \operatorname{\mathbf{Z}}), M_{\ast } )_{\ast } \]
\[ \lambda : \mathrm{K}( H_{\ast } ) \rightarrow \operatorname{Fun}(S, \mathrm{K}( M_{\ast } )), \]
\begin{eqnarray*} \mathrm{N}_{\ast }( S \times \mathrm{K}(H_{\ast }); \operatorname{\mathbf{Z}}) & \xrightarrow { \mathrm{AW} } & \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \mathrm{K}( H_{\ast } ); \operatorname{\mathbf{Z}}) \\ & \rightarrow & \mathrm{N}_{\ast }(S) \boxtimes H_{\ast } \\ & \xrightarrow {\operatorname{ev}} & M_{\ast } \end{eqnarray*}
\[ \lambda _{T}: \pi _0( \operatorname{Fun}(T, \mathrm{K}( H_{\ast } ) ) \rightarrow \pi _0( \operatorname{Fun}(S \times T, \mathrm{K}(M_{\ast } ) ) ). \]
\[ \mathrm{AW}: \mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(S ;\operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T;\operatorname{\mathbf{Z}}). \]
).
Definition 3.1.6.12. 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 3.1.6.13. 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 3.1.6.6. 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 $f$. $\square$
Proposition 3.1.6.14. Let $f: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets. Then $f$ is a weak homotopy equivalence.
Remark 3.1.6.15. We will later prove a (partial) converse to Proposition 3.1.6.14: if a monomorphism of simplicial sets $f: A_{} \hookrightarrow B_{}$ is a weak homotopy equivalence, then $f$ is anodyne (see Corollary 3.3.7.7).
Proof of Proposition 3.1.6.14. 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 3.1.3.6 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 3.1.6.10), and therefore induces a bijection on connected components $\pi _0( \operatorname{Fun}( B_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( A_{}, X_{} ) )$ (Remark 3.1.6.6). $\square$
Remark 3.1.6.16 (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.
Proposition 3.1.6.17. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $Z$ be a Kan complex. If $f$ is a weak homotopy equivalence, then composition with $f$ induces a homotopy equivalence $\operatorname{Fun}( Y, Z) \rightarrow \operatorname{Fun}(X,Z)$.
Proof. By virtue of Remark 3.1.6.6, it will suffice to show that for every simplicial set $A$, the induced map $\theta : \operatorname{Fun}( A, \operatorname{Fun}(Y, Z) ) \rightarrow \operatorname{Fun}( A, \operatorname{Fun}(X,Z) )$ induces a bijection on connected components. This follows by observing that $\theta $ can be identfied with the map $\operatorname{Fun}(Y, \operatorname{Fun}(A,Z) ) \rightarrow \operatorname{Fun}( X, \operatorname{Fun}(A,Z) )$ given be precomposition with $f$ (since Corollary 3.1.3.4 guarantees that the simplicial set $\operatorname{Fun}(A,Z)$ is a Kan complex). $\square$
Proposition 3.1.6.18. Let $f: X \rightarrow Y$ be a weak homotopy equivalence of simplicial sets. Then the induced map of normalized chain complexes $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(Y; \operatorname{\mathbf{Z}})$ is a chain homotopy equivalence. In particular, $f$ induces an isomorphism of homology groups $\mathrm{H}_{\ast }(X;\operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(Y; \operatorname{\mathbf{Z}})$.
Proof. Let $M_{\ast }$ be a chain complex of abelian groups. We wish to show that precomposition with $\mathrm{N}_{\ast }(f; \operatorname{\mathbf{Z}})$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Chain homotopy classes of maps $\mathrm{N}_{\ast }(Y; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$} \} \ar [d]^{\theta } \\ \{ \text{Chain homotopy classes of maps $\mathrm{N}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$} \} . } \]
Let $\mathrm{K}(M_{\ast })$ denote the Eilenberg-MacLane space associated to $M_{\ast }$ (Construction 2.5.6.3). Using Example 3.1.5.8, we can identify $\theta $ with the map
\[ \pi _{0}(\operatorname{Fun}(Y, \mathrm{K}(M_{\ast } ) )) \rightarrow \pi _0( \operatorname{Fun}(X, \mathrm{K}(M_{\ast } ) ) ) \]
given by precomposition with $f$. This map is bijective because $f$ is a weak homotopy equivalence (by assumption) and $\mathrm{K}(M_{\ast })$ is a Kan complex (Remark 2.5.6.4). $\square$
Remark 3.1.6.19. There is a partial converse to Proposition 3.1.6.18. If $f: X \rightarrow Y$ is a morphism between simply-connected simplicial sets and the induced map $\mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(Y; \operatorname{\mathbf{Z}})$ is an isomorphism, one can show that $f$ is a weak homotopy equivalence. Beware that this is not necessarily true if $X$ and $Y$ are not simply connected (see ยง for further discussion).
Remark 3.1.6.20 (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 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 1.2.5.11), 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.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \coprod _{i \in I} Y(i), Z) \ar [r] \ar [d]^{\sim } & \operatorname{Fun}( \coprod _{i \in I} X(i), Z) \ar [d]^{\sim } \\ \prod _{i \in I} \operatorname{Fun}( Y(i), Z) \ar [r] & \prod _{i \in I} \operatorname{Fun}(X(i), Z), } \]
Exercise 3.1.6.21. Let $G$ be the directed graph depicted in the diagram and let $G_{}$ denote the associated $1$-dimensional simplicial set (see Warning 1.2.1.28). Show that the projection map $G_{} \rightarrow \Delta ^{0}$ is a weak homotopy equivalence, but not a homotopy equivalence.
\[ \xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & 3 \ar [r] & 4 \ar [r] & \cdots } \]
Warning 3.1.6.22. 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 $g: Y_{} \rightarrow X_{}$.
Proposition 3.1.6.23. Let $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ be weak homotopy equivalences of simplicial sets. Then the induced map $(f \times f'): X \times X' \rightarrow Y \times Y'$ is also a weak homotopy equivalence.
Proof. By virtue of Remark 3.1.6.16, it will suffice to show that the morphisms $f \times \operatorname{id}_{X'}$ and $\operatorname{id}_{Y} \times f'$ are weak homotopy equivalences. We will give the proof for $f \times \operatorname{id}_{X'}$; the analogous statement for $\operatorname{id}_{Y} \times f'$ follows by a similar argument. Let $Z$ be a Kan complex; we wish to show that precomposition with $f$ induces a bijection
\begin{eqnarray*} \pi _0( \operatorname{Fun}(X \times X', Z) ) & \simeq & \pi _0( \operatorname{Fun}(X, \operatorname{Fun}(X',Z) ) ) \\ & \rightarrow & \pi _0( \operatorname{Fun}(Y, \operatorname{Fun}(X',Z) ) ) \\ & \simeq & \pi _0( \operatorname{Fun}(Y \times X', Z ) ). \end{eqnarray*}
This follows from our assumption that $f$ is a weak homotopy equivalence, since the simplicial set $\operatorname{Fun}(X',Z)$ is a Kan complex (Corollary 3.1.3.4). $\square$
Warning 3.1.6.24. The collection of weak homotopy equivalences is not closed under the formation of infinite products. For example, if $q: G \rightarrow \Delta ^0$ is the weak homotopy equivalence described in Exercise 3.1.6.21, then a product of infinitely many copies of $q$ with itself is not a weak homotopy equivalence (since a product of infinitely many copies of $G$ is not a connected simplicial set: see Warning 1.2.1.28).