Kerodon

Proof. We define a category $\operatorname{{\bf \Delta }}_{\ast }$ as follows:

• The objects of $\operatorname{{\bf \Delta }}_{\ast }$ are pairs $([m], i)$, where $[m] = \{ 0 < 1 < \cdots < m \} $ is an object of $\operatorname{{\bf \Delta }}$ and $i$ is an integer satisfying $0 \leq i \leq m$.

• A morphism from $([m], i)$ to $([n], j)$ in $\operatorname{{\bf \Delta }}_{\ast }$ is a nondecreasing function $\alpha : [m] \rightarrow [n]$ satisfying $\alpha (i) \leq j$.

Alternatively, we describe $\operatorname{{\bf \Delta }}_{\ast }$ as the category of elements of the forgetful functor $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}$ (Definition 5.6.1.1). Then the forgetful functor $\operatorname{{\bf \Delta }}_{\ast } \rightarrow \operatorname{{\bf \Delta }}$ induces a cocartesian fibration $V: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})$ whose covariant transport representation is the standard simplex $\Delta ^{\bullet }$ (Proposition 5.6.3.4). Moreover, a morphism $\alpha : ([m], i) \rightarrow ([n], j)$ is $V$-cocartesian if and only if $\alpha (i) = j$. Let $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{\ast }$ spanned by objects $([m], i)$ where $i = m$. Note that the forgetful functor $\operatorname{{\bf \Delta }}_{\ast } \rightarrow \operatorname{{\bf \Delta }}$ restricts to an isomorphism from $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ to $\operatorname{{\bf \Delta }}$. Note that $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ is a coreflective subcategory of $\operatorname{{\bf \Delta }}_{\ast }$: the inclusion $\operatorname{{\bf \Delta }}_{\ast }^{\circ } \hookrightarrow \operatorname{{\bf \Delta }}_{\ast }$ admits a right adjoint, given on objects by the construction $([m], i) \mapsto ([i], i)$.

Let $\overline{\operatorname{\mathcal{E}}}$ denote the fiber product $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } ) \times _{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$ and let $\overline{W}$ denote the collection of morphisms $w$ of $\overline{\operatorname{\mathcal{E}}}$ such that the image of $w$ in $\operatorname{\mathcal{E}}$ is $U$-cartesian and the image of $w$ in $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$ is $V$-cocartesian. It follows from Theorem 8.6.9.9 that the coend $\operatorname{Coend}( \Delta ^{\bullet }, \mathscr {F} )$ can be identified with the localization $\overline{\operatorname{\mathcal{E}}}[ \overline{W}^{-1} ]$.

Let us abuse notation by identifying $\operatorname{\mathcal{E}}$ with the full subcategory of $\overline{\operatorname{\mathcal{E}}}$ given by the inverse image of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast }^{\circ } ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$. Since $U$ is a cartesian fibration, it follows that $\operatorname{\mathcal{E}}$ is a coreflective subcategory of $\overline{\operatorname{\mathcal{E}}}$ (Proposition 6.2.4.2): that is, the inclusion functor $\iota $ admits a right adjoint $L: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}$. Let $\overline{W}_0$ denote the collection of morphisms $\overline{w}$ in $\overline{\operatorname{\mathcal{E}}}$ such that $L( \overline{w} )$ is an isomorphism in $\operatorname{\mathcal{E}}$. Writing $\overline{w} = (w, \alpha )$, where $w$ is a morphism of $\operatorname{\mathcal{E}}$ and $\alpha : ([m], i) \rightarrow ([n], j)$ is a morphism in $\operatorname{{\bf \Delta }}_{\ast }$, we observe that $\overline{w}$ belongs to $\overline{W}_0$ if and only if $w$ is $U$-cartesian and $\alpha $ restricts to a bijection of $[i] \subseteq [m]$ with $[j] \subseteq [n]$. In particular, $\overline{W}_0$ is contained in $\overline{W}$.

Applying Proposition 6.3.3.7, we deduce that the functor $L$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\overline{\operatorname{\mathcal{E}}}$ with respect to $\overline{W}_0$. Note that for each morphism $\overline{w}$ of $\overline{W}$, the image $L( \overline{w} )$ belongs to $W$. Conversely, every morphism $w$ of $\operatorname{\mathcal{E}}$ which belongs to $W$ can be regarded as an element $\overline{W}$ (by means of the inclusion $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$), and is therefore isomorphic to an element of $L( \overline{W} )$. Applying Proposition 6.3.1.22, we conclude that $L$ induces an equivalence of localizations $\overline{\operatorname{\mathcal{E}}}[ \overline{W}^{-1} ] \rightarrow \operatorname{\mathcal{E}}[ W^{-1} ]$, so that the coend $\operatorname{Coend}( \Delta ^{\bullet }, \mathscr {F} )$ can also be identified with $\operatorname{\mathcal{E}}[W^{-1}]$. $\square$