Kerodon

Proof. We have a commutative diagram of simplicial sets

where the vertical maps are weak homotopy equivalences by virtue of Proposition 3.4.5.4. Invoking Remark 3.1.6.16, we deduce that $f$ is a weak homotopy equivalence if and only if $f^{+}$ is a weak homotopy equivalence. $\square$

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. $\square$

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

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

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

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

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

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 simplicial 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 3.1.5.7). $\square$

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 3.2.8.3), 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 1.1.4.12 then supplies a pushout diagram of simplicial sets

Note that both (3.61) and the induced diagram

are homotopy pushout squares (this is a special case of Example 3.4.2.12, 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 3.4.2.9 to the commutative diagram

we are reduced to proving that $v_{X'}$ is a weak homotopy equivalence. Using Remark 3.1.6.20, 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. $\square$