Kerodon

Proof. We use the criterion of Corollary 10.1.1.9. Since the category $\operatorname{{\bf \Delta }}$ is nonempty, it will suffice to show that for every pair of nonnegative integers $m,n \geq 0$, the simplicial set

is weakly contractible. Unwinding the definitions, we can identify $\operatorname{{\bf \Delta }}_{/ [m]} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{ / [n] }$ with the category of simplices $\operatorname{{\bf \Delta }}_{S}$ of Construction 1.1.3.9, where $S$ is the product $\Delta ^{m} \times \Delta ^ n$. Note that $S$ can be identified with the nerve of a partially ordered set, and is therefore a braced simplicial set (Exercise 3.3.1.2). Let $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{S}$ spanned by the nondegenerate simplices of $S$ (Notation 3.3.3.12), so that the inclusion $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \hookrightarrow \operatorname{{\bf \Delta }}_{S}$ admits a left adjoint (Exercise 3.3.3.16). It follows that the inclusion map $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ is a homotopy equivalence of simplicial sets (Proposition 3.1.6.9). It will therefore suffice to show that the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}^{\mathrm{nd} })$ is weakly contractible. Using Proposition 3.3.3.17, we can identify $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} )$ with the subdivision $\operatorname{Sd}(S)$, so that Construction 3.3.4.3 supplies a weak homotopy equivalence $\lambda _{S}: \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}^{\mathrm{nd}}_{S} ) \rightarrow S$. We conclude by observing that the simplicial set $S = \Delta ^ m \times \Delta ^ n$ is weakly contractible (in fact, it is contractible, since it is the nerve of a partially ordered set having a smallest element). $\square$