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3.4.5 Digression: Weak Homotopy Equivalences of Semisimplicial Sets

Recall that a morphism of simplicial sets $f: X \rightarrow Y$ is a weak homotopy equivalence if, for every Kan complex $Z$, precomposition with $f$ induces a bijection $\pi _0( \operatorname{Fun}(Y,Z) ) \rightarrow \pi _0( \operatorname{Fun}(X,Z) )$ (Definition 3.1.5.10). Our goal in this section is to show that this condition depends only on the underlying morphism of semisimplicial sets. To see this, we begin by recalling that the forgetful functor

\[ \{ \text{Simplicial Sets} \} \rightarrow \{ \text{Semisimplicial Sets} \} \]

admits a left adjoint, which we denote by $X \mapsto X^{+}$ (Corollary 3.3.1.10).

Definition 3.4.5.1. Let $f: X \rightarrow Y$ be a morphism of semisimplicial sets. We will say that $f$ is a weak homotopy equivalence if the induced map of simplicial sets $f^{+}: X^{+} \rightarrow Y^{+}$ is a weak homotopy equivalence, in the sense of Definition 3.1.5.10.

Remark 3.4.5.2. The collection of weak homotopy equivalences of semisimplicial sets is closed under the formation of filtered colimits. This follows immediately from the corresponding assertion for simplicial sets (Proposition 3.2.7.3), since the construction $X \mapsto X^{+}$ commutes with filtered colimits.

Remark 3.4.5.3. Let $f: X_{} \rightarrow Y_{}$ and $g: Y_{} \rightarrow Z_{}$ be morphisms of semisimplicial sets. If any two of the morphisms $f$, $g$, and $g \circ f$ are weak homotopy equivalences, then so is the third (see Remark 3.1.5.13).

When $X$ is a simplicial set, we write $v_{X}: X^{+} \rightarrow X$ for the counit map (that is, the unique morphism of simplicial sets whose restriction to $(X^{+})^{\mathrm{nd}} \simeq X$ is the identity map). To compare Definition 3.4.5.1 with Definition 3.1.5.10, we need the following:

Proposition 3.4.5.4. For every simplicial set $X$, the counit map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence.

Corollary 3.4.5.5. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then $f$ is a weak homotopy equivalence (in the sense of Definition 3.1.5.10) if and only if the underlying morphism of semisimplicial sets is a weak homotopy equivalence (in the sense of Definition 3.4.5.1).

Proof. We have a commutative diagram of simplicial sets

\[ \xymatrix { X^{+} \ar [r]^-{f^{+}} \ar [d]^{ v_{X} } & Y^{+} \ar [d]^{ v_{Y} } \\ X \ar [r]^-{f} & Y, } \]

where the vertical maps are weak homotopy equivalences by virtue of Proposition 3.4.5.4. Invoking Remark 3.1.5.13, we deduce that $f$ is a weak homotopy equivalence if and only if $f^{+}$ is a weak homotopy equivalence. $\square$

Corollary 3.4.5.6. For every semisimplicial set $X$, the inclusion map $\iota : X \hookrightarrow X^{+}$ is a weak homotopy equivalence of semisimplicial sets.

Proof. We wish to show that the map $\iota ^{+}: X^{+} \rightarrow (X^{+})^{+}$ is a weak homotopy equivalence of simplicial sets. This is clear, since $\iota ^{+}$ is right inverse to the counit map $v_{ X^{+} }: (X^{+})^{+} \rightarrow X^{+}$, which is a weak homotopy equivalence of simplicial sets by virtue of Proposition 3.4.5.4. $\square$

Variant 3.4.5.7. Let $X$ be a simplicial set, and let $\iota : X \hookrightarrow X^{+}$ be the inclusion map. Then the map $\operatorname{Ex}(\iota ): \operatorname{Ex}(X) \hookrightarrow \operatorname{Ex}(X^{+} )$ is a weak homotopy equivalence of semisimplicial sets.

Proof. By virtue of Proposition 3.4.5.4, the counit map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence of simplicial sets. Applying Corollary 3.3.5.2, we deduce that the map $\operatorname{Ex}(v_{X}): \operatorname{Ex}( X^{+} ) \rightarrow \operatorname{Ex}(X)$ is a weak homotopy equivalence of simplicial sets, hence also a weak homotopy equivalence of the underlying semisimplicial sets (Corollary 3.4.5.5). Since the composite map

\[ \operatorname{Ex}(X) \xrightarrow { \operatorname{Ex}(\iota ) } \operatorname{Ex}(X^{+} ) \xrightarrow { \operatorname{Ex}(v_{X} )} \operatorname{Ex}(X) \]

is the identity, it follows that $\operatorname{Ex}(\iota )$ is also a weak homotopy equivalence of semisimplicial sets. $\square$

Corollary 3.4.5.8. Let $X$ and $Y$ be simplicial sets and let $f: X \rightarrow Y$ be a morphism of semisimplicial sets. Then $f$ is a weak homotopy equivalence of semisimplicial sets if and only if the induced map $\operatorname{Ex}(f): \operatorname{Ex}(X) \rightarrow \operatorname{Ex}(Y)$ is a weak homotopy equivalence of semisimplicial sets.

Proof. By definition, $f: X \rightarrow Y$ is a weak homotopy equivalence of semisimplicial sets if and only if the induced map $f^{+}: X^{+} \rightarrow Y^{+}$ is a weak homotopy equivalence of simplicial sets. By virtue of Corollary 3.3.5.2, this is equivalent to the assertion that $\operatorname{Ex}(f^{+}): \operatorname{Ex}(X^{+} ) \rightarrow \operatorname{Ex}(Y^{+} )$ is a weak homotopy equivalence when viewed as a morphism of simplicial sets, or equivalently when viewed as a morphism of semisimplicial sets (Corollary 3.4.5.5). The desired result now follows by inspecting the commutative diagram of semisimplicial sets

\[ \xymatrix { \operatorname{Ex}(X) \ar [r]^-{ \operatorname{Ex}(f) } \ar [d] & \operatorname{Ex}(Y) \ar [d] \\ \operatorname{Ex}(X^{+} ) \ar [r]^-{ \operatorname{Ex}(f^{+} )} & \operatorname{Ex}(Y^{+} ), } \]

since the vertical maps are weak homotopy equivalences by virtue of Variant 3.4.5.7. $\square$

We now turn to the proof of Proposition 3.4.5.4. The main ingredient we will need is the following is the following:

Lemma 3.4.5.9. Let $\operatorname{\mathcal{C}}$ be a category, and suppose that the collection of non-identity morphisms in $\operatorname{\mathcal{C}}$ is closed under composition. Then the counit map $v_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{+} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a homotopy equivalence of simplicial sets.

Proof. Let $\operatorname{\mathcal{C}}^{+}$ denote the category obtained from $\operatorname{\mathcal{C}}$ by formally adjoining a new identity morphism $\operatorname{id}_{X}^{+}$ for each object $X \in \operatorname{\mathcal{C}}$. More precisely, the category $\operatorname{\mathcal{C}}^{+}$ is defined as follows:

  • The objects of $\operatorname{\mathcal{C}}^{+}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}^{+}$, we have

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{+}}( X,Y ) = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \text{ if } X \neq Y \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \coprod \{ \operatorname{id}^{+}_{X} \} & \text{ if } X = Y. \end{cases} \]
  • If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are morphisms in $\operatorname{\mathcal{C}}^{+}$, then the composition $g \circ f$ is equal to $g$ if $f = \operatorname{id}_{Y}^{+}$, to the morphism $f$ if $g = \operatorname{id}_{Y}^{+}$, and is otherwise given by the composition law for morphisms in $\operatorname{\mathcal{C}}$.

Note that the collection of non-identity morphisms in $\operatorname{\mathcal{C}}^{+}$ is closed under composition, so that the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{+} )$ is a braced simplicial set (Exercise 3.3.1.2). Unwinding the definitions, we see that the semisimplicial subset $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{+})^{\mathrm{nd}} \subseteq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{+} )$ can be identified with the $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (as a semisimplicial set). Using Corollary 3.3.1.11, we obtain a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{+} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{+} )$. Under this isomorphism, the counit map $v_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ is induced by the functor $F: \operatorname{\mathcal{C}}^{+} \rightarrow \operatorname{\mathcal{C}}$ which is the identity on objects, and carries each morphism $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}^{+}}( X, Y)$ to itself.

Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{+}$ be the functor which is the identity on objects, and which carries a morphism $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to the morphism

\[ G(f) = \begin{cases} \operatorname{id}_{X}^{+} & \text{ if $X=Y$ and $f = \operatorname{id}_{X}$ } \\ f & \text{ otherwise. } \end{cases} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}^{+}}(X,Y); \]

this functor is well-defined by virtue of our assumption that the collection of non-identity morphisms of $\operatorname{\mathcal{C}}$ is closed under composition. We will complete the proof by showing that the induced map $\operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{+} )$ is a homotopy inverse of $\operatorname{N}_{\bullet }(F) = v_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$. One direction is clear: the composition $\operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{C}}^{+} \xrightarrow {F} \operatorname{\mathcal{C}}$ is the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$, so $\operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G)$ is equal to the identity. The composition $\operatorname{\mathcal{C}}^{+} \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{C}}^{+}$ is not the identity functor on $\operatorname{\mathcal{C}}^{+}$: for each object $X \in \operatorname{\mathcal{C}}$, it carries the morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \subset \operatorname{Hom}_{\operatorname{\mathcal{C}}^{+}}(X,X)$ to the “new” identity morphism $\operatorname{id}_{X}^{+}$. However, there is a natural transformation $\alpha : G \circ F \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}^{+}}$, given by the construction $(X \in \operatorname{\mathcal{C}}^{+}) \mapsto \operatorname{id}_{X}$. It follows that the map of simplicial sets $\operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F)$ is homotopic to the identity (Example 3.1.4.6). $\square$

Proof of Proposition 3.4.5.4. We proceed as in the proof of Proposition 3.3.4.8. For every simplicial set $X$, the counit map $v_{X}: X^{+} \rightarrow X$ can be realized as a filtered colimit of counit maps $\{ v_{ \operatorname{sk}_{n}(X)}: \operatorname{sk}_{n}(X)^{+} \rightarrow \operatorname{sk}_{n}(X) \} _{n \geq 0}$. Since the collection of weak homotopy equivalences is closed under the formation of filtered colimits (Proposition 3.2.7.3), it will suffice to show that each of the maps $v_{ \operatorname{sk}_{n}(X)}$ is a weak homotopy equivalence. We may therefore replace $X$ by $\operatorname{sk}_{n}(X)$, and thereby reduce to the case where $X$ is $n$-skeletal for some nonnegative integer $n \geq 0$. We now proceed by induction on $n$.

Let $Y = \operatorname{sk}_{n-1}(X)$ be the $(n-1)$-skeleton of $X$. Let $S$ denote the collection of nondegenerate $n$-simplices of $X$, let $X' = \coprod _{\sigma \in S} \Delta ^{n}$ denote their coproduct, and let $Y' = \coprod _{\sigma \in S} \operatorname{\partial \Delta }^{n}$ denote the boundary of $X'$. Proposition 1.1.3.11 then supplies a pushout diagram of simplicial sets

3.62
\begin{equation} \label{equation:pushout-aux-1} \begin{gathered} \xymatrix { Y' \ar [r] \ar [d] & X' \ar [d] \\ Y \ar [r] & X. } \end{gathered} \end{equation}

Note that both (3.62) and the induced diagram

\[ \xymatrix { Y'^{+} \ar [r] \ar [d] & X'^{+} \ar [d] \\ Y^{+} \ar [r] & X^{+} } \]

are homotopy coCartesian (this is a special case of Example 3.4.2.7, since the maps $Y' \hookrightarrow X'$ and $Y'^{+} \hookrightarrow X'^{+}$ are monomorphisms). Moreover, our inductive hypothesis guarantees that the maps $v_{Y}: Y^{+} \rightarrow Y$ and $v_{Y'}: Y'^{+} \rightarrow Y'$ are weak homotopy equivalences. Applying Proposition 3.4.2.4 to the commutative diagram

\[ \xymatrix { Y'^{+} \ar [rr] \ar [dd] \ar [dr]^{ v_{Y'} } & & Y^{+} \ar [dr]^{v_{Y}} \ar [dd] & \\ & Y' \ar [rr] \ar [dd] & & Y \ar [dd] \\ X'^{+} \ar [rr] \ar [dr]^{v_{X'} } & & X^{+} \ar [dr]^{v_{X}} & \\ & X' \ar [rr] & & X, } \]

we are reduced to proving that $v_{X'}$ is a weak homotopy equivalence is a homotopy equivalence (Proposition 3.2.7.1). Using Remark 3.1.5.16, we can reduce further to the problem of showing that the map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence in the special case $X = \Delta ^{n}$, which follows from Lemma 3.4.5.9. $\square$